Quantum Signatures of Chaos (Springer Series in Synergetics)

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Springer Complexity Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems – cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science. Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse “real-life” situations like the climate, the coherent emission of light from lasers, chemical reactiondiffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications. Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence. The two major book publication platforms of the Springer Complexity program are the monograph series “Understanding Complex Systems” focusing on the various applications of complexity, and the “Springer Series in Synergetics”, which is devoted to the quantitative theoretical and methodological foundations. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works.

Editorial and Programme Advisory Board Dan Braha, New England Complex Systems Institute and University of Massachusetts Dartmouth, USA ` , Center for Complex Systems Studies, Kalamazoo College, USA and Hungarian Academy of P´eter Erdi Sciences, Budapest, Hungary

Karl Friston, Institute of Cognitive Neuroscience, University College London, London, UK Hermann Haken, Center of Synergetics, University of Stuttgart, Stuttgart, Germany Janusz Kacprzyk, System Research, Polish Academy of Sciences, Warsaw, Poland Scott Kelso, Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, USA J¨urgen Kurths, Nonlinear Dynamics Group, University of Potsdam, Potsdam, Germany Linda Reichl, Center for Complex Quantum Systems, University of Texas, Austin, USA Peter Schuster, Theoretical Chemistry and Structural Biology, University of Vienna, Vienna, Austria Frank Schweitzer, System Design, ETH Zurich, Zurich, Switzerland Didier Sornette, Entrepreneurial Risk, ETH Zurich, Zurich, Switzerland

Springer Series in Synergetics Founding Editor: H. Haken

The Springer Series in Synergetics was founded by Herman Haken in 1977. Since then, the series has evolved into a substantial reference library for the quantitative, theoretical and methodological foundations of the science of complex systems. Through many enduring classic texts, such as Haken’s Synergetics and Information and Self-Organization, Gardiner’s Handbook of Stochastic Methods, Risken’s The Fokker Planck-Equation or Haake’s Quantum Signatures of Chaos, the series has made, and continues to make, important contributions to shaping the foundations of the field. The series publishes monographs and graduate-level textbooks of broad and general interest, with a pronounced emphasis on the physico-mathematical approach.

Fritz Haake

Quantum Signatures of Chaos Third Revised and Enlarged Edition

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Prof. Dr. Fritz Haake Universit¨at Duisburg-Essen FB 7 Physik Universit¨atsstr. 5 45117 Essen Germany [email protected] [email protected]

ISBN 978-3-642-05427-3 e-ISBN 978-3-642-05428-0 DOI 10.1007/978-3-642-05428-0 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010920833 c Springer-Verlag Berlin Heidelberg 2010, 2000, 1991  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Integra Software Services Pvt. Ltd., Pondicherry Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

F¨ur Gitta und Julia

Preface to the Third Edition

Nine years have passed since I dispatched the second edition, and the book still appears to be in demand. The time may be ripe for an update. As the perhaps most conspicable extension, I describe the understanding of universal spectral fluctuations recently reached on the basis of periodic-orbit theory. To make the presentation of those semiclassical developments selfcontained, I decided to to underpin them by a new short chapter on classical Hamiltonian mechanics. Inasmuch as the semiclassical theory not only draws inspiration from the nonlinear sigma model but actually aims at constructing that model in terms of periodic orbits, it appeared indicated to make small additions to the previous treatment within the chapter on superanalysis. Less voluminous but as close to my heart are additions to the chapter on level dynamics which close previous gaps in that approach to spectral universality. It was a pleasant duty to pay my respect to collegues in our TransregioSonderforschungsbereich, Martin Zirnbauer, Alex Altland, Alan Huckleberry, and Peter Heinzner, by including a short account of their beautiful work on nonstandard symmetry classes. The chapter on random matrices has not been expanded in proportion to the development of the field but now includes an up-to-date treatment of an old topic in algebra, Newton’s relations, to provide a background to the Riemann-Siegel lookalike of semiclassical periodic-orbit theory. The chapters on level clustering, localization, and dissipation are similarly preserved. I disciplined myself to just adding an occasional reference to recent work and to cutting some stuff of lesser relative importance. There was the temptation to rewrite the introduction, to no avail. Only a few additional words here and there annouce new topics taken up in the main text. So that chapter stands as a relic from the olden days when quantum chaos was just beginning to form as a field. Encouragement and help has come from Thomas Guhr, Dominique Spehner, Martin Zirnbauer, and, as always, from Hans–J¨urgen Sommers and Marek Ku´s.

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I owe special gratitude to Alex Altland, Peter Braun, Stefan Heusler, and Sebastian M¨uller. They have formed a dream team sharing search and finding, suffering and joy, row and laughter.

Essen August 2009

F. Haake

Preface to the Second Edition

The warm reception of the first edition, as well as the tumultuous development of the field of quantum chaos have tempted me to rewrite this book and include some of the important progress made during the past decade. Now we know that quantum signatures of chaos are paralleled by wave signatures. Whatever is undergoing wavy space-time variations, be it sound, electromagnetism, or quantum amplitudes, each shows exactly the same manifestations of chaos. The common origin is nonseparability of the pertinent wave equation; that latter “definition” of chaos, incidentally, also applies to classical mechanics if we see the Hamilton–Jacobi equation as the limiting case of a wave equation. At any rate, drums, concert halls, oscillating quartz blocks, microwave and optical oscillators, electrons moving ballistically or with impurity scattering through mesoscopic devices all provide evidence and data for wave or quantum chaos. All of these systems have deep analogies with billiards, much as the latter may have appeared of no more than academic interest only a decade ago. Of course, molecular, atomic, and nuclear spectroscopy also remain witnesses of chaos, while the chromodynamic innards of nucleons are beginning to attract interest as methods of treatment become available. Of the considerable theoretical progress lately achieved, the book focuses on the deeper statistical exploitation of level dynamics, improved control of semiclassical periodic-orbit expansions, and superanalytic techniques for dealing with various types of random matrices. These three fields are beginning, independently and in conjunction, to generate an understanding of why certain spectral fluctuations in classically nonintegrable systems are universal and why there are exceptions. Only the rudiments of periodic-orbit theory and superanalysis appeared in the first edition. More could not have been included here had I not enjoyed the privilege of individual instruction on periodic-orbit theory by Jon Keating and on superanalysis by Hans–J¨urgen Sommers and Yan Fyodorov. Hans–J¨urgen and Yan have even provided their lecture notes on the subject. While giving full credit and expressing my deep gratitude to these three colleagues, I must bear all blame for blunders. Reasonable limits of time and space had to be respected and have forced me to leave out much interesting material such as chaotic scattering and the semiclassical art of getting spectra for systems with mixed phase spaces. Equally regrettably, no justice could be done here to the wealth of experiments that have now been ix

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performed, but I am happy to see that gap filled by my much more competent colleague Hans–J¨urgen St¨ockmann. Incomplete as the book must be, it now contains more material than fits into a single course in quantum chaos theory. In some technical respects, it digs deeper than general introductory courses would go. I have held on to my original intention though, to provide a self-contained presentation that might help students and researchers to enter the field or parts thereof. The number of co-workers and colleagues from whose knowledge and work I could draw has increased considerably over the years. Having already mentioned Yan Fyodorov, Jon Keating, and Hans-J¨urgen Sommers, I must also express special gratitude to my partner and friend Marek Ku´s whose continuing help was equally crucial. My thanks for their invaluable influence go to Sergio Albeverio, Daniel Braun, Peter Braun, Eugene Bogomolny, Chang-qi Cao, Dominique Delande, Bruno Eckhardt, Pierre Gaspard, Sven Gnutzmann, Peter Goetsch, Siegfried Grossmann, Martin Gutzwiller, Gregor Hackenbroich, Alan Huckleberry, Micha Kolobov, Pavel Kurasov, Robert Littlejohn, Nils Lehmann, J¨org Main, Alexander Mirlin, Jan Mostowski, Alfredo Ozorio de Almeida, Pjotr Peplowski, Ravi Puri, Jonathan Robbins, Kazik Rza¸z˙ ewski, Henning Schomerus, Carsten Seeger, Thomas Seligmann, Frank Steiner, Hans-J¨urgen St¨ockmann, J¨urgen Vollmer, Joachim Weber, Harald Wiedemann, Christian Wiele, G¨unter Wunner, Dmitri Zaitsev, Kuba Zakrzewski, Martin Zirnbauer, Marek Zukowski, Wojtek Zurek, and, last but not ˙ at all least, Karol Zyczkowski. In part this book is an account of research done within the Sonderforschungsbereich “Unordnung und Große Fluktuationen” of the Deutsche Forschungsgemeinschaft. This fact needs to be gratefully acknowledged, since coherent long-term research of a large team of physicists and mathematicians could not be maintained without the generous funding we have enjoyed over the years through our Sonderforschungsbereich. Times do change. Like many present-day science authors I chose to pick up LATEX and key all changes and extensions into my little machine myself. As usually happens when learning a new language, the beginning is all effort, but one eventually begins to enjoy the new mode of expressing oneself. I must thank Peter Gerwinski, Heike Haschke, and R¨udiger Oberhage for their infinite patience in getting me going.

Essen July 2000

F. Haake

Preface to the First Edition

More than 60 years after its inception, quantum mechanics is still exerting fascination on every new generation of physicists. What began as the scandal of noncommuting observables and complex probability amplitudes has turned out to be the universal description of the micro-world. At no scale of energies accessible to observation have any findings emerged that suggest violation of quantum mechanics. Lingering doubts that some people have held about the universality of quantum mechanics have recently been resolved, at least in part. We have witnessed the serious blow dealt to competing hidden-variable theories by experiments on correlations of photon pairs. Such correlations were found to be in conflict with any local deterministic theory as expressed rigorously by Bell’s inequalities. – Doubts concerning the accommodation of dissipation in quantum mechanics have also been eased, in much the same way as in classical mechanics. Quantum observables can display effectively irreversible behavior when they are coupled to an appropriate environmental system containing many degrees of freedom. Even in closed quantum systems with relatively few degrees of freedom, behavior resembling damping is possible, provided the system displays chaotic motion in the classical limit. It has become clear that the relative phases of macroscopically distinguishable states tend, in the presence of damping, to become randomized in exceedingly short times; that remains true even when the damping is so weak that it is hardly noticeable for quantities with a well-defined classical limit. Consequently, a superposition (in the quantum sense) of different readings of a macroscopic measuring device would, even if one could be prepared momentarily, escape observation due to its practically instantaneous decay. While this behavior was conjectured early in the history of quantum mechanics it is only recently that we have been able to see it explicitly in rigorous solutions for specific model systems. There are many intricacies of the classical limit of quantum mechanics. They are by no means confined to abrupt decay processes or infinitely rapid oscillations of probability amplitudes. The classical distinction between regular and chaotic motion, for instance, makes itself felt in the semiclassical regime that is typically associated with high degrees of excitation. In that regime quantum effects like the discreteness of energy levels and interference phenomena are still discernible while the correspondence principle suggests the onset of validity of classical mechanics.

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The semiclassical world, which is intermediate between the microscopic and the macroscopic, is the topic of this book. It will deal with certain universal modes of behavior, both dynamical and spectral, which indicate whether their classical counterparts are regular or chaotic. Conservative as well as dissipative systems will be treated. The area under consideration often carries the label “quantum chaos”. It is a rapidly expanding one and therefore does not yet allow for a definite treatment. The material presented reflects subjective selections. Random-matrix theory will enjoy special emphasis. A possible alternative would have been to make current developments in periodic-orbit theory the backbone of the text. Much as I admire the latter theory for its beauty and its appeal to classical intuition, I do not understand it sufficiently well that I can trust myself to do it justice. With more learning, I might yet catch up and find out how to relate spectral fluctuations on an energy scale of a typical level spacing to classical properties. There are other regrettable omissions. Most notable among these may be the ionization of hydrogen atoms by microwaves, for which convergence of theory and experiment has been achieved recently. Also too late for inclusion is the quantum aspect of chaotic scattering, which has seen such fine progress in the months between the completion of the manuscript and the appearance of this book. This book grew out of lectures given at the universities of Essen and Bochum. Most of the problems listed at the end of each chapter have been solved by students attending those lectures. The level aimed at was typical of a course on advanced quantum mechanics. The book accordingly assumes the reader to have a good command of the elements of quantum mechanics and statistical mechanics, as well as some background knowledge of classical mechanics. A little acquaintance with classical nonlinear dynamics would not do any harm either. I could not have gone through with this project without the help of many colleagues and coworkers. They have posed many of the questions dealt with here and provided most of the answers. Perhaps more importantly, they have, within the theory group in Essen, sustained an atmosphere of dedication and curiosity, from which I keep drawing knowledge and stimulus. I can only hope that my young coworkers share my own experience of receiving more than one is able to give. I am especially indebted to Michael Berry, Oriol Bohigas, Giulio Casati, Boris Chirikov, Barbara Dietz, Thomas Dittrich, Mario Feingold, Shmuel Fishman, Dieter Forster, Robert Graham, Rainer Grobe, Italo Guarneri, Klaus-Dieter Harms, Michael H¨ohnerbach, Ralf H¨ubner, Felix Israilev, Marek Ku´s, Georg Lenz, Maciej Lewenstein, Madan ˇ Lal Mehta, Jan Mostowski, Akhilesh Pandey, Dirk Saher, Rainer Scharf, Petr Seba, Dima Shepelyansky, Uzy Smilansky, Hans-J¨urgen Sommers, Dan Walls, and Karol ˙ Zyczkowski. Angela Lahee has obliged me by smoothening out some clumsy Teutonisms and by her careful editing of the manuscript. My secretary, Barbara Sacha, deserves a big thank you for keying version upon version of the manuscript into her computer. My friend and untiring critic Roy Glauber has followed this work from a distance and provided invaluable advice. – I am grateful to Hermann Haken for his invitation to contribute this book to his series in synergetics, and I am all the more honored

Preface to the First Edition

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since it can fill but a tiny corner of Haken’s immense field. However, at least Chap. 8 does bear a strong relation to several other books in the series inasmuch as it touches upon adiabatic-elimination techniques and quantum stochastic processes. Moreover, that chapter represents variations on themes I learned as a young student in Stuttgart, as part of the set of ideas which has meanwhile grown to span the range of this series. The love of quantum mechanics was instilled in me by Hermann Haken and his younger colleagues, most notably Wolfgang Weidlich, as they were developing their quantum theory of the laser and thus making the first steps towards synergetics.

Essen January 1991

F. Haake

Foreword to the First Edition

The interdisciplinary field of synergetics grew out of the desire to find general principles that govern the spontaneous formation of ordered structures out of microscopic chaos. Indeed, large classes of classical and quantum systems have been found in which the emergence of ordered structures is governed by just a few degrees of freedom, the so-called order parameters. But then a surprise came with the observation that a few degrees of freedom may cause complicated behavior, nowadays generally subsumed under the title “deterministic chaos” (not to be confused with microscopic chaos, where many degrees of freedom are involved). One of the fundamental problems of chaos theory is the question of whether deterministic chaos can be exhibited by quantum systems, which, at first sight, seem to show no deterministic behavior at all because of the quantization rules. To be more precise, one can formulate the question as follows: How does the transition occur from quantum mechanical properties to classical properties showing deterministic chaos? Fritz Haake is one of the leading scientists investigating this field and he has contributed a number of important papers. I am therefore particularly happy that he agreed to write a book on this fascinating field of quantum chaos. I very much enjoyed reading the manuscript of this book, which is written in a highly lively style, and I am sure the book will appeal to many graduate students, teachers, and researchers in the field of physics. This book is an important addition to the Springer Series in Synergetics.

Stuttgart February 1991

H. Haken

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Time Reversal and Unitary Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Autonomous Classical Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Spinless Quanta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Spin-1/2 Quanta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Hamiltonians Without T Invariance . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 T Invariant Hamiltonians, T 2 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Kramers’ Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Kramers’ Degeneracy and Geometric Symmetries . . . . . . . . . . . . . 2.8 Kramers’ Degeneracy Without Geometric Symmetries . . . . . . . . . . 2.9 Nonconventional Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Stroboscopic Maps for Periodically Driven Systems . . . . . . . . . . . . 2.11 Time Reversal for Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Canonical Transformations for Floquet Operators . . . . . . . . . . . . . . 2.13 Beyond Dyson’s Threefold Way . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.1 Normal-Superconducting Hybrid Structures . . . . . . . . . . . 2.13.2 Systems with Chiral Symmetry . . . . . . . . . . . . . . . . . . . . . 2.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 16 17 19 21 22 22 25 27 29 31 33 36 38 43 44 45

3 Level Repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Symmetric Versus Nonsymmetric H or F . . . . . . . . . . . . . . . . . . . . 3.3 Kramers’ Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Universality Classes of Level Repulsion . . . . . . . . . . . . . . . . . . . . . . 3.5 Nonstandard Symmetry Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Experimental Observation of Level Repulsion . . . . . . . . . . . . . . . . . 3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Random-Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Gaussian Ensembles of Hermitian Matrices . . . . . . . . . . . . . . . . . . . 62 4.3 Eigenvalue Distributions for Dyson’s Ensembles . . . . . . . . . . . . . . 66 4.4 Eigenvalue Distributions for Nonstandard Symmetry Classes . . . . 68 4.5 Level Spacing Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.6 Invariance of the Integration Measure . . . . . . . . . . . . . . . . . . . . . . . . 71 4.7 Average Level Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.8 Unfolding Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.9 Eigenvector Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.9.1 Single-Vector Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.9.2 Joint Density of Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . 80 4.10 Ergodicity of the Level Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.11 Dyson’s Circular Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.12 Asymptotic Level Spacing Distributions . . . . . . . . . . . . . . . . . . . . . . 86 4.13 Determinants as Gaussian Grassmann Integrals . . . . . . . . . . . . . . . . 96 4.14 Two-Point Correlations of the Level Density . . . . . . . . . . . . . . . . . . 101 4.14.1 Two-Point Correlator and Form Factor . . . . . . . . . . . . . . . 101 4.14.2 Form Factor for the Poissonian Ensemble . . . . . . . . . . . . . 103 4.14.3 Form Factor for the CUE . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.14.4 Form Factor for the COE . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.14.5 Form Factor for the CSE . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.15 Newton’s Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.15.1 Traces Versus Secular Coefficients . . . . . . . . . . . . . . . . . . . 112 4.15.2 Solving Newton’s Relations . . . . . . . . . . . . . . . . . . . . . . . . 115 4.16 Selfinversiveness and Riemann–Siegel Lookalike . . . . . . . . . . . . . . 117 4.17 Higher Correlations of the Level Density . . . . . . . . . . . . . . . . . . . . . 118 4.17.1 Correlation and Cumulant Functions . . . . . . . . . . . . . . . . . 118 4.17.2 Ergodicity of the Two-Point Correlator . . . . . . . . . . . . . . . 121 4.17.3 Ergodicity of the Form Factor . . . . . . . . . . . . . . . . . . . . . . . 124 4.17.4 Joint Density of Traces of Large CUE Matrices . . . . . . . . 127 4.18 Correlations of Secular Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.19 Fidelity of Kicked Tops to Random-Matrix Theory . . . . . . . . . . . . 135 4.20 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5 Level Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.2 Invariant Tori of Classically Integrable Systems . . . . . . . . . . . . . . . 145 5.3 Einstein–Brillouin–Keller Approximation . . . . . . . . . . . . . . . . . . . . 147 5.4 Level Crossings for Integrable Systems . . . . . . . . . . . . . . . . . . . . . . 149 5.5 Poissonian Level Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.6 Superposition of Independent Spectra . . . . . . . . . . . . . . . . . . . . . . . . 151 5.7 Periodic Orbits and the Semiclassical Density of Levels . . . . . . . . 153

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5.8 Level Density Fluctuations for Integrable Systems . . . . . . . . . . . . . 158 5.9 Exponential Spacing Distribution for Integrable Systems . . . . . . . . 165 5.10 Equivalence of Different Unfoldings . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6 Level Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.2 Fictitious Particles (Pechukas-Yukawa Gas) . . . . . . . . . . . . . . . . . . . 170 6.3 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 6.4 Intermultiplet Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.5 Level Dynamics for Classically Integrable Dynamics . . . . . . . . . . . 181 6.6 Two-Body Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.7 Ergodicity of Level Dynamics and Universality of Spectral Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.7.1 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.7.2 Collision Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.7.3 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.8 Equilibrium Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.9 Random-Matrix Theory as Equilibrium Statistical Mechanics . . . . 195 6.9.1 General Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.9.2 A Typical Coordinate Integral . . . . . . . . . . . . . . . . . . . . . . . 199 6.9.3 Influence of a Typical Constant of the Motion . . . . . . . . . 205 6.9.4 The General Coordinate Integral . . . . . . . . . . . . . . . . . . . . 206 6.9.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 6.10 Dynamics of Rescaled Energy Levels . . . . . . . . . . . . . . . . . . . . . . . . 209 6.11 Level Curvature Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.12 Level Velocity Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.13 Dyson’s Brownian-Motion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 6.14 Local and Global Equilibrium in Spectra . . . . . . . . . . . . . . . . . . . . . 234 6.15 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 7 Quantum Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 7.2 Localization in Anderson’s Hopping Model . . . . . . . . . . . . . . . . . . . 248 7.3 The Kicked Rotator as a Variant of Anderson’s Model . . . . . . . . . . 251 7.4 Lloyd’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 7.5 The Classical Diffusion Constant as the Quantum Localization Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 7.6 Absence of Localization for the Kicked Top . . . . . . . . . . . . . . . . . . 264 7.7 The Rotator as a Limiting Case of the Top . . . . . . . . . . . . . . . . . . . . 274 7.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

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8 Dissipative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 8.2 Hamiltonian Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 8.3 Time-Scale Separation for Probabilities and Coherences . . . . . . . . 285 8.4 Dissipative Death of Quantum Recurrences . . . . . . . . . . . . . . . . . . . 288 8.5 Complex Energies and Quasi-Energies . . . . . . . . . . . . . . . . . . . . . . . 296 8.6 Different Degrees of Level Repulsion for Regular and Chaotic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 8.7 Poissonian Random Process in the Plane . . . . . . . . . . . . . . . . . . . . . 302 8.8 Ginibre’s Ensemble of Random Matrices . . . . . . . . . . . . . . . . . . . . . 303 8.8.1 Normalizing the Joint Density . . . . . . . . . . . . . . . . . . . . . . 304 8.8.2 The Density of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 306 8.8.3 The Reduced Joint Densities . . . . . . . . . . . . . . . . . . . . . . . . 308 8.8.4 The Spacing Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 8.9 General Properties of Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 8.10 Universality of Cubic Level Repulsion . . . . . . . . . . . . . . . . . . . . . . . 319 8.10.1 Antiunitary Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 8.10.2 Microreversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 8.11 Dissipation of Quantum Localization . . . . . . . . . . . . . . . . . . . . . . . . 326 8.11.1 Zaslavsky’s Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 8.11.2 Damped Rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 8.11.3 Destruction of Localization . . . . . . . . . . . . . . . . . . . . . . . . 332 8.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

9 Classical Hamiltonian Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 9.2 Phase Space, Hamilton’s Equations and All That . . . . . . . . . . . . . . 341 9.3 Action as a Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 9.4 Linearized Flow and Its Jacobian Matrix . . . . . . . . . . . . . . . . . . . . . 344 9.5 Liouville Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 9.6 Symplectic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 9.7 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 9.8 Stretching Factors and Local Stretching Rates . . . . . . . . . . . . . . . . . 350 9.9 Poincar´e Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 9.10 Stroboscopic Maps of Periodically Driven Systems . . . . . . . . . . . . 354 9.11 Varieties of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 9.12 The Sum Rule of Hannay and Ozorio de Almeida . . . . . . . . . . . . . . 355 9.12.1 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 9.12.2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 9.13 Propagator and Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 9.14 Exponential Stability of the Boundary Value Problem . . . . . . . . . . 362 9.15 Sieber–Richter Self-Encounter and Partner Orbit . . . . . . . . . . . . . . 363 9.15.1 Non-technical Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 363

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9.15.2 Quantitative Discussion of 2-Encounters . . . . . . . . . . . . . . 366 9.16 l-Encounters and Orbit Bunches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 9.17 Densities of Arbitrary Encounter Sets . . . . . . . . . . . . . . . . . . . . . . . . 379 9.18 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

10 Semiclassical Roles for Classical Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . 383 10.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 10.2 Van Vleck Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 10.2.1 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 10.2.2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 10.3 Gutzwiller’s Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 10.3.1 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 10.3.2 Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 10.3.3 Weyl’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 10.3.4 Limits of Validity and Outlook . . . . . . . . . . . . . . . . . . . . . . 405 10.4 Lagrangian Manifolds and Maslov Theory . . . . . . . . . . . . . . . . . . . . 407 10.4.1 Lagrangian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 10.4.2 Elements of Maslov Theory . . . . . . . . . . . . . . . . . . . . . . . . 414 10.4.3 Maslov Indices as Winding Numbers . . . . . . . . . . . . . . . . . 418 10.5 Riemann–Siegel Look-Alike . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 10.6 Spectral Two-Point Correlator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 10.6.1 Real and Complex Correlator . . . . . . . . . . . . . . . . . . . . . . . 428 10.6.2 Local Energy Average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 10.6.3 Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 10.6.4 Periodic-Orbit Representation . . . . . . . . . . . . . . . . . . . . . . . 433 10.7 Diagonal Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 10.7.1 Unitary Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 10.7.2 Orthogonal Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 10.8 Off-Diagonal Contributions, Unitary Symmetry Class . . . . . . . . . . 439 10.8.1 Structures of Pseudo-Orbit Quadruplets . . . . . . . . . . . . . . 441 10.8.2 Diagrammatic Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 10.8.3 Example of Structure Contributions: A Single 2-encounter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 10.8.4 Cancellation of all Encounter Contributions for the Unitary Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 10.9 Semiclassical Construction of a Sigma Model, Unitary Symmetry Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 10.9.1 Matrix Elements for Ports and Contraction Lines for Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 10.9.2 Wick’s Theorem and Link Summation . . . . . . . . . . . . . . . 452 10.9.3 Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 10.9.4 Proof of Contraction Rules, Unitary Case . . . . . . . . . . . . . 456 10.9.5 Emergence of a Sigma Model . . . . . . . . . . . . . . . . . . . . . . . 457

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10.10

Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 10.10.1 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 10.10.2 Leading-Order Contributions . . . . . . . . . . . . . . . . . . . . . . 462 10.10.3 Symbols for Ports and Contraction Lines for Links . . . . 464 10.10.4 Gauss and Wick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 10.10.5 Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 10.10.6 Proof of Contraction Rules, Orthogonal Case . . . . . . . . . 468 10.10.7 Sigma Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 10.11 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 10.12 Mixed Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 10.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

11 Superanalysis for Random-Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . 481 11.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 11.2 Semicircle Law for the Gaussian Unitary Ensemble . . . . . . . . . . . . 482 11.2.1 The Green Function and Its Average . . . . . . . . . . . . . . . . . 482 11.2.2 The GUE Average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 11.2.3 Doing the Superintegral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 11.2.4 Two Remaining Saddle-Point Integrals . . . . . . . . . . . . . . . 487 11.3 Superalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 11.3.1 Motivation and Generators of Grassmann Algebras . . . . . 490 11.3.2 Supervectors, Supermatrices . . . . . . . . . . . . . . . . . . . . . . . . 491 11.3.3 Superdeterminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 11.3.4 Complex Scalar Product, Hermitian and Unitary Supermatrices . . . . . . . . . . . . . . . . . . . . . . . . . 495 11.3.5 Diagonalizing Supermatrices . . . . . . . . . . . . . . . . . . . . . . . 497 11.4 Superintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 11.4.1 Some Bookkeeping for Ordinary Gaussian Integrals . . . . 498 11.4.2 Recalling Grassmann Integrals . . . . . . . . . . . . . . . . . . . . . . 499 11.4.3 Gaussian Superintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 11.4.4 Some Properties of General Superintegrals . . . . . . . . . . . . 502 11.4.5 Integrals over Supermatrices, Parisi–Sourlas–Efetov–Wegner Theorem . . . . . . . . . . . . . 504 11.5 The Semicircle Law Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 11.6 The Two-Point Function of the Gaussian Unitary Ensemble . . . . . 509 11.6.1 The Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 11.6.2 Unitary Versus Hyperbolic Symmetry . . . . . . . . . . . . . . . . 512 11.6.3 Efetov’s Nonlinear Sigma Model . . . . . . . . . . . . . . . . . . . . 515 11.6.4 Implementing the Zero-Dimensional Sigma Model . . . . . 522 11.6.5 Integration Measure of the Nonlinear Sigma Model . . . . 525 11.6.6 Back to the Generating Function . . . . . . . . . . . . . . . . . . . . 531 11.6.7 Rational Parametrization of the Sigma Model . . . . . . . . . 533

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11.6.8 High-Energy Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . 539 Universality of Spectral Fluctuations: Non-Gaussian Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 11.7.1 Delta Functions of Grassmann Variables . . . . . . . . . . . . . . 542 11.7.2 Generating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 11.8 Universal Spectral Fluctuations of Sparse Matrices . . . . . . . . . . . . . 546 11.9 Thick Wires, Banded Random Matrices, One-Dimensional Sigma Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 11.9.1 Banded Matrices Modelling Thick Wires . . . . . . . . . . . . . 547 11.9.2 Inverse Participation Ratio and Localization Length . . . . 549 11.9.3 One-Dimensional Nonlinear Sigma Model . . . . . . . . . . . . 550 11.9.4 Implementing the One-Dimensional Sigma Model . . . . . 555 11.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 11.7

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

Chapter 1

Introduction

Long as it may have taken to realize, we are now certain that there are two radically different types of motion in classical Hamiltonian mechanics: the regular motion of integrable systems and the chaotic motion of nonintegrable systems. The harmonic oscillator and the Kepler problem show regular motion, while systems as simple as a periodically driven pendulum or an autonomous conservative double pendulum can display chaotic dynamics. To identify the type of motion for a given system, one may look at a bundle of trajectories originating from a narrow cloud of points in phase space. The distance between any two such trajectories grows exponentially with time in the chaotic case; the growth rate is the so-called Lyapunov exponent. For regular motion, on the other hand, the distance in question may increase like a power of time but never exponentially; the corresponding Lyapunov exponent can thus be said to vanish. I should add that neither subexponential nor exponential separation can prevail indefinitely. Limits are set by Poincar´e recurrences or, in some cases, the accessible volume of phase space. However, such limits become effective only after regular or chaotic behavior is manifest. When quantum effects are important for a physical system under study, the notion of a phase-space trajectory loses its meaning and so does the notion of a Lyapunov exponent measuring the separation between trajectories. In cases with a discrete energy spectrum, exponential separation is strictly excluded even for expectation values of observables, on time scales on which level spacings are resolvable. The dynamics is then characterized by quasi-periodicity, i.e., recurrences rather than chaos in the classical sense. The quasi-period, i.e., a typical recurrence time, is inversely proportional to a typical level spacing and must of course tend to infinity in the classical limit (formally,  → 0). Having lost the classical distinction between regular motion and chaos when turning to quantum mechanics, one naturally wonders whether there are other, genuinely quantum mechanical criteria allowing one to distinguish two types of quantum dynamics. Such a distinction, if at all possible, should parallel the classical case: As  → 0, one group should become regular and the other chaotic. Intrinsically quantum mechanical distinction criteria do in fact exist. Some are based on the energy spectrum; others on the energy eigenvectors, or on the temporal evolution of suitable expectation values.

F. Haake, Quantum Signatures of Chaos, Springer Series in Synergetics, 3rd ed., C Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-05428-0 1, 

1

2

1 Introduction

A surprising lesson was taught to us by experiments with classical waves, electromagnetic and sound. Classical fields show much the same signatures of chaos as quantum probability amplitudes. Quantum or wave chaos arises whenever the pertinent wave equation is nonseparable for given boundary conditions. The character of the field is quite immaterial; it may be real, complex, or vector; the wave equation may be linear like Schr¨odinger’s or Maxwell’s or nonlinear like the Gross– Pitaevski equation for Bose–Einstein condensates. In all cases, classical trajectories or rays arise in the limit of short wavelengths, and these trajectories are chaotic in the sense of positive Lyapunov exponents whenever the underlying wave problem is nonseparable. Nonseparability is shared by the wave problem and the classical Hamilton–Jacobi equation ensuing in the short-wave limit and may indeed be seen as the deepest characterization of chaos. I shall discuss quantum (and wave) distinctions between “regular” and “irregular” motions at some length in the chapters to follow. It should be instructive to jump ahead a little and infer from Fig. 1.1 that the temporal quasi-periodicity in quantum systems with discrete spectra can manifest itself in a way that tells us whether the classical limit will reveal regular or chaotic behavior. The figure displays the time evolution of the expectation value of a typical observable of a periodically kicked top, i.e., a quantum spin J with fixed square, J 2 = j( j + 1) = const. In the classical limit,1 j → ∞, the classical angular momentum J/j is capable of regular or chaotic motion depending on the values of control parameters and on the initial orientation. Figure 1.1 pertains to values of the control parameters for which chaos and regular motion coexist in the classical phase space. Parts a and b of Fig. 1.1 refer to initial quantum states localized entirely within classically regular and classically chaotic parts of phase space, respectively. In both cases quasi-periodicity is manifest in the form of recurrences. Obviously, however, the quasi-periodicity in Fig. 1.1a is a nearly perfect periodicity (“collapse and revival”) while Fig. 1.1b shows an erratic sequence of recurrences. I shall show later how one can characterize this qualitative difference in quantitative terms. In any event, the difference in question is intrinsically quantum mechanical; the mean temporal separation of recurrences of

Fig. 1.1 Quasi-periodic behavior of a quantum mean value (angular momentum component for a periodically kicked top) under conditions of classically regular motion (a) and classical chaos (b). For details see Sect. 7.6

1

The classical limit of a periodically driven spin was experimentally realized by Waldner [1].

1 Introduction

3

either type is proportional to j and thus diverges in the classical limit. In contrast to classical chaos, quantum mechanically irregular motion cannot be characterized by extreme sensitivity to tiny changes of initial data: Due to the unitarity of quantum dynamics, the overlap of two wave functions remains time-independent, |φ(t)|ψ(t)|2 = |φ(0)|ψ(0)|2 , provided the time-dependences of φ(t) and ψ(t) are generated by the same Hamiltonian. However, an alternative characterization

Fig. 1.2 Time dependence of the overlap of two wave vectors of the kicked top, both originating from the same initial state but evolving with slightly different values of one control parameter. Upper and lower curves refer to conditions of classically regular and classically chaotic motion, respectively. For details see Sect. 7.6. Courtesy of A. Peres [10]

4

1 Introduction

of classical chaos, extreme sensitivity to slight changes of the dynamics, does carry over into quantum mechanics, as illustrated in Fig. 1.2. This figure refers to the same dynamical system as in Fig. 1.1 and shows the time-dependent overlap of two wave functions. For each curve, the two states involved originate from one and the same initial state but evolve with slightly different values (relative difference 10−4 ) of one control parameter; that (tiny!) difference apart, all control parameters are set as in Fig. 1.1, as are the two initial states used. The time-dependent overlap remains close to unity at all times, if the initial state is located in a classically regular part of the phase space. For the initial state residing in the classically chaotic region, however, the overlap falls exponentially, down to a level of order 1/j. Such sensitivity of the overlap to changes of a control parameter is quite striking and may indeed serve as a quantum criterion of irregular motion. In more recently estabished jargon the behavior in question is called fidelity decay. In a recent experimental realization of the kicked top [2] fidelity decay as well other distinctions of chaos and regular motion have been observed. Somewhat more widely known are the following two possibilities for the statistics of energy levels (or quasi-energy levels for periodically driven systems). Generic classically integrable systems with two or more degrees of freedom have quantum levels that tend to cluster and are not prohibited from crossing when a parameter in the Hamiltonian is varied [3]. The typical distribution of the spacings of neighboring levels is exponential, P(S) = exp (−S), just as if the levels arose as the uncorrelated events in a Poissonian random process. Classically nonintegrable systems with their phase spaces dominated by chaos, on the other hand, enjoy the privilege of levels that are correlated such that crossings are strongly resisted [4–9]. There are three universal degrees of level repulsion: linear, quadratic, and quartic [P(S) ∼ S β for S → 0 with β = 1, 2, or 4]. To which universality class a given nonintegrable system can belong is determined by the set of its symmetries. As will be explained in detail later, anti-unitary symmetries such as time-reversal invariance play an important role. Broadly speaking, in systems without any antiunitary symmetry generically, β = 2; in the presence of an antiunitary symmetry one typically finds linear level repulsion; the strongest resistance to level crossings, β = 4, is characteristic of time-reversal invariant systems possessing Kramers’ degeneracy but no geometric symmetry at all. Clearly, the alternative between level clustering and level repulsion belongs to the worlds of quanta and waves. It parallels, however, the classical distinction between predominantly regular and predominantly chaotic motion. Figure 1.3 illustrates the four generic possibilities mentioned, obtained from the numerically determined quasi-energies of various types of periodically kicked tops; the corresponding portraits of trajectories in the respective classical phase spaces (Fig. 7.3) show the correlation between the quantum and the classical distinction of the two types of dynamics. An experimentally determined spectrum of nuclear, atomic, or molecular levels will in general, if taken at face value, display statistics close to Poissonian, even when there is no reason to suppose that the corresponding classical many-body problem is integrable or at least nearly integrable. To uncover spectral correlations,

1 Introduction

5

Fig. 1.3 Level spacing distributions for kicked tops under conditions of classically regular motion β = 0 and classical chaos β = 1, 2, 4. The latter three curves pertain to tops from different universality classes and display linear β = 1, quadratic β = 2, and quartic level repulsion β = 4

the complete set of levels must be separated into subsets, each of which has fixed values of the quantum numbers related to the symmetries of the system. Subsets that are sufficiently large to allow for a statistical analysis generally reveal level repulsion of the degree expected on grounds of symmetry. Of the three universal degrees of level repulsion, only the linear and quadratic varieties have been observed experimentally to date. As for linear repulsion, the first data came from nuclear physics in the 1960s [7–9, 11–13]; mostly much later, confirmation came from microwaves [14–17], molecular [18] and atomic [19] spectroscopy, and sound waves [20, 21]. Figure 1.4 displays such results from various fields and provides evidence for the kinship of quantum and wave chaos. As regards quadratic level repulsion, it would be hopeless to look in nuclei; even though the weak interaction does break time-reversal invariance, that breaking is far too feeble to become visible in level-spacing distributions. Equally unwieldy are normal-size atoms with a magnetic field to break conventional time-reversal invariance. A homogeneous field would not change the linear degree of repulsion since it preserves antiunitary symmetry (generalized time-reversal); such a symmetry survives even in nonaligned electric and magnetic fields if both fields are homogeneous. Rydberg atoms exposed to strong and sufficiently inhomogeneous magnetic fields do possess quadratically repelling levels and will eventually reveal that property to spectroscopists. Promise also lies in systems with half-integer j that are invariant under conventional time reversal. One then confronts Kramers’ degeneracy. If all geometric symmetries could be broken, quartic repulsion would be expected, while sufficiently low geometric symmetries would still allow for the quadratic case. To reduce geometric symmetries, appropriately oriented magnetic and electric fields could be used. Needless to say, experimental data of such a quantum kind would be highly welcome. In view of the obstinacy of single atoms and molecules, the recent first observation of quadratic-level repulsion in microwave experiments [26, 27] was a most welcome achievement, all the more so since it once more underscored that wave chaos is not an exclusive privilege of quanta.

6

1 Introduction

Fig. 1.4 Level spacing distributions for (a) the Sinai billiard [9], (b) a hydrogen atom in a strong magnetic field [22], (c) an NO2 molecule [18], (d) a vibrating quartz block shaped like a three dimensional Sinai billiard [23], (e) the microwave spectrum of a three-dimensional chaotic cavity [24], (f) a vibrating elastic disc shaped like a quarter stadium [25]. Courtesy of St¨ockmann

As already emphasized, alternatives such as nearly periodic versus erratic quasiperiodicity and level clustering versus repulsion are quantum or wave concepts without meaning in the strict classical or ray limit. Conversely, the notion of the Lyapunov exponent as the logarithmic rate of separation between phase space trajectories is inapplicable when quantum effects are important. Such complementarity notwithstanding, the quantum distinctions of what becomes regular or chaotic motion classically are most pronounced in semiclassical situations, i.e., when the systems studied are highly excited and have action scales far exceeding Planck’s

1 Introduction

7

constant. Wave functions on smaller action scales are insensitive to the hierarchy of phase-space structures that distinguish regular and chaotic behavior. In fact, for few-body systems, the ground state and its first few neighbors in energy can often be calculated quite well by variational techniques; however, the classical trajectories of correspondingly low energy are accessible, if chaotic, only through considerable numerical effort. At high energies, Schr¨odinger’s equation for integrable systems yields to “torus quantization”. This beautiful semiclassical approximation scheme is based on the fact that the phase-space trajectory of an integrable system winds around a torus fixed by the set of constant actions. By expressing the Hamiltonian in terms of the actions and allowing each action to be an integer multiple of Planck’s constant, one usually obtains good approximations to the high-lying energy levels. As already mentioned, the spectrum so derived has an exponential spacing distribution (provided there are at least two degrees of freedom). A short exposition of the underlying phenomenon of level clustering and its explanation through torus quantization will be the object of Chap. 5. Similarly as unwieldy as Newton’s equations to nonnumerical techniques is Schr¨odinger’s equation with respect to highly excited states of nonintegrable systems, on the other hand. Torus quantization may still work reasonably in near integrable cases where slight perturbations of integrable limiting cases have only modified but not destroyed the vast majority of the original tori. When chaos is fully developed, however, Schr¨odinger’s equation can only be solved numerically. In numerical treatments of systems with global chaos, an interesting predictability paradox arises. As indicated by a positive Lyapunov exponent, small but inevitable numerical inaccuracies will amplify exponentially and thus render impossible longrange predictions of classical trajectories. Much more reliable by comparison can be the corresponding quantum predictions for mean values, at least in the case of discrete levels; quasi-periodic time evolution, even if very many sinusoidal terms contribute, is incapable of exponential runaway. A stupefying illustration of quantum predictions outdoing classical ones is given by the kinetic energy of the periodically kicked rotator in Fig. 1.5. The quantum mean (dotted) originates from the momentum eigenstate with a vanishing eigenvalue while the classical curve (full) represents an average over 250 000 trajectories starting from a cloud of points with zero momentum and equipartition for the conjugate angle. The classical average displays diffusive growth of the kinetic energy – which, as will be shown later, is tantamount to chaos – until time is reversed after 500 kicks. Instead of retracing its path back to the initial state, the classical mean soon turns again to diffusive growth. The quantum mean, on the other hand, is symmetric around t = 500 without noticeable error. Evidently, such a result suggests that a little caution is necessary with respect to our often too naive notions of classical determinism and quantum indeterminacy. Nuclear physicists have known for quite some time that mean-field theories not only give a rather satisfactory account of the low-energy excitations of nuclei, but also give reasonable approximations for the average level density and other average properties at high excitation energies. An average is meant as one over many excited

8

1 Introduction

Fig. 1.5 Classical (full curve) and quantum (dotted) mean kinetic energy of the periodically kicked rotator, determined by numerical iteration of the respective maps (see Chap. 7 for details). The quantum mean follows the classical diffusion for times up to some “break” time and then begins to display quasi-periodic fluctuations. After 500 kicks the direction of time was reserved; while the quantum mean accurately retraces its history, the classical mean reverts to diffusive growth, thus revealing the extreme sensitivity of chaotic systems to tiny perturbations (here round-off errors). For details see Chaps. 7 and 8. Courtesy of Graham and Dittrich [28]

states with neighboring energies. Such averages may be, and in general are, energydependent but they constitute sensible concepts only if they vary but little over energy scales of the order of a typical level spacing. In practice, an average level density can be defined for all but the lightest nuclei, at energies of about a few MeV and above. Relative to the average level density of a nucleus, the precise locations in energy of the individual levels appear, from a statistical point of view, to be determined by fluctuations; their calculation is beyond the scope of mean-field theories such as those based on the shell model. Theory would in fact be at a loss here, were it not for the fortunate fact that some statistical characteristics of the fluctuations turn out to be universal. The level-spacing distribution already discussed is among those characteristics that appear insensitive to details of the nuclear structure and are instead determined by no more than symmetries and some kind of maximumdisorder principle. The situation just described for nuclei is also found, either experimentally or from numerical work, in highly excited atoms and molecules and in fact, also in almost all systems displaying strong chaos in the classical limit. Following a conjecture of Bohigas, Giannoni, and Schmit [9], certain universal features of spectral fluctuations in classically chaotic systems have been found to be well described by random-matrix theory [29, 30]. Based on a suggestion by Wigner, this theory takes random Hermitian matrices as models of Hamiltonians H of autonomous systems. Unitary random matrices2 are employed similarly for 2 S-matrices of chaotic scatterers make for equally interesting applications of unitary random matrices; be warned, however, that scattering phenomena are not treated in this book.

1 Introduction

9

periodically driven systems, as models of the unitary “Floquet” operators F describing the change of the quantum state during one cycle of the driving; powers F n of the Floquet operator with n = 1, 2, 3, . . . yield a stroboscopic description of the driven dynamics under consideration. There are four important classes for both Hermitian and unitary random matrices, one related to classically integrable systems and three to nonintegrable ones. The four classes in question illustrate the quantum distinction between regular and irregular dynamics discussed above: their eigenvalues (energies in the Hermitian case and quasi-energies, i.e., eigenphases in the unitary case) cluster and repel according to one of three universal degrees, respectively. The matrix ensembles with level repulsion can be defined by the requirement of maximum statistical independence of all matrix elements within the constraints imposed by symmetries. I shall present a brief review of random-matrix theory in Chap. 4. Since several excellent texts on this subject are available [29–31], I shall attempt neither completeness nor rigor but keep to an introductory style and intuitive arguments. When slightly fancy machinery is employed for some more up-to-date issues, it is patiently developed rather than assumed as a prerequisite. Considerable emphasis will be given to an interesting reinterpretation of randommatrix theory as “level dynamics” in Chap. 6. Such a reinterpretation was already sought by Dyson but could be implemented only recently on the basis of a discovery of Pechukas [32]. The fate of the N eigenvalues and eigenvectors of an N × N Hermitian matrix, H = H0 + λV, is in one-to-one correspondence with the classical Hamiltonian dynamics of a particular one-dimensional N -particle system upon changing the weight λ of a perturbation V [32–34]. This fictitious system, now often called Pechukas–Yukawa gas, has λ as a time, the eigenvalues E n of H as coordinates, the diagonal elements Vnn of the perturbation V in the H representation as momenta, and the off-diagonal elements Vnm related to certain angular momenta. As I shall explain in Chap. 6, random-matrix theory now emerges as a result of standard equilibrium statistical mechanics for the fictitious N -particle system. Moreover, the universality of spectral fluctuations of chaotic dynamics will find a natural explanation on that basis. The fictitious N -particle model also sheds light on another important problem. A given Hamiltonian H0 +λV may have H0 integrable and V breaking integrability. The level dynamics will then display a transition from level clustering to level repulsion. We shall see that such transitions can be understood as equilibration processes for certain observables in the fictitious N -particle model. A related phenomenon is the transition from one universality class to another displayed by a Hamiltonian H (λ) = H0 + λV when H0 and V have different (antiunitary) symmetries. An ad hoc description of such transitions could previously be given in terms of Dyson’s Brownian-motion model, a certain “dynamic” generalization of random-matrix theory. As will become clear below, that venerable model is in fact rigorously implied by level dynamics, i.e., the λ-dependence of H0 + λV , provided the energy scale is reset in a suitable λ-dependent manner. A certain class of periodically driven systems of which the kicked rotator is a prototype displays an interesting quantum anomaly. Like all periodically driven systems, those in question are generically nonintegrable classically. Even under the

10

1 Introduction

conditions of fully developed classical chaos, however, the kicked rotator does not display level repulsion. The reason for this anomaly can be inferred from Fig. 1.5. The kinetic energy, and thus the quantum mechanical momentum uncertainty, does not follow the classical diffusive growth indefinitely. After a certain break time the quantum mean enters a regime of quasi-periodic behavior. Clearly, the eigenvectors of the corresponding Floquet operator then have an upper limit to their width in the momentum representation. They are, in fact, exponentially localized in that basis, and the width is interpretable as a “localization length”. Two eigenvectors much further apart in momentum than a localization length have no overlap and thus no matrix elements of noticeable magnitude with respect to any observable; they have no reason, therefore, to stay apart in quasi-energy and will display Poissonian level statistics. A theoretical understanding of the phenomenon has been generated by Grempel, Fishman, and Prange [35] who were able to map the Schr¨odinger equation for the kicked rotator onto that for Anderson’s one-dimensional tight-binding model of a particle in a random potential. Actually, the potential arrived at in the map is pseudorandom, but that restriction does not prevent exponential localization, which is rigorously established only for the strictly random case. It is amusing to find that number-theoretical considerations are relevant in a quantum context here. Quantum localization occurs only in the case where a certain dimensionless version of Planck’s constant is an irrational number. Otherwise, the equivalent tight-binding model has a periodic potential and thus extended eigenvectors of the Bloch type and eigenvalues forming continuous bands. I shall present a description of this fascinating situation in Chap. 7 on quantum localization. Next, I shall turn to dissipative systems. Instead of Schr¨odinger’s equation, we face master equations of Markovian processes. Real energies (or quasi-energies) are replaced with complex eigenvalues of generators of infinitesimal time translations (or nonunitary generalizations of the Floquet operator in the case of periodically driven systems) while the density operator or a suitable representative, like the Wigner function, takes over the role of the wave function. One would again like to identify genuinely quantum mechanical criteria to distinguish two types of motion, one becoming regular and the other chaotic in the classical limit. There is a general qualitative argument, however, suggesting that such quantum distinctions will be a lot harder to establish for dissipative than for Hamiltonian systems. The difference between, say, a complicated limit cycle and a strange attractor becomes apparent when phase-space structures are considered over several orders of magnitude of action scales. The density matrix or the Wigner function will of course reflect phase-space structures on action scales upward of Planck’s constant and will thus indicate, with reasonable certainty, the difference between a strange and a simple attractor. But for any representative of the density operator to reveal this difference in terms of genuinely quantum mechanical features without classical meaning, it would have to embody coherences with respect to states distinct on action scales that are large compared to Planck’s constant. Such coherences between or superpositions of “macroscopically distinct” states are often metaphorized as Schr¨odinger cat states.In the presence of even weak damping, such superpositions tend to decohere so rapidly in time to mixtures that observing them becomes difficult if not impossible.

1 Introduction

11

Fig. 1.6 Quantum recurrences for periodically kicked top without (left column) and with (right column) damping under conditions of classically regular motion. For details see Chap. 8

An illustration of the dissipative death of quantum coherences is presented in Fig. 1.6. As in Fig. 1.1a, column a in Fig. 1.6 shows quantum recurrences for an angular momentum component of a periodically kicked top without damping. The control parameters and the initial state are kept constant for all curves in the column and are chosen such that the classical limit would yield regular behavior. Proceeding down the column, the spin quantum number j is increased to demonstrate the

12

1 Introduction

rough proportionality of the quasi-period to j (which may be thought of as inversely proportional to Planck’s constant). The curves in column b of Fig. 1.6 correspond to their neighbors in column a in all respects except for the effect of weak dissipation. The damping mechanism is designed so as to leave j a good quantum number, and the damping constant is so small that classical trajectories would not be influenced noticeably during the times over which the plots extend. The sequences of quantum mechanical “collapses and revivals” is seen to be strongly altered by the dissipation, even though the quasi-period is in all cases smaller than the classical decay time. I shall argue below that the lifetime of the quantum coherences is shorter than the classical decay time and also shorter than the time constant for quantum observables not sensitive to coherences between “macroscopically” distinct states by a factor of the order of 1/j ∼ . Similarly dramatic is the effect of dissipation on the erratic recurrences found under conditions of classical chaos. Obviously, dissipation will then tend to wipe out the distinction between regular and erratic recurrences. The quantum localization in the periodically kicked rotator also involves coherences between states that are distinct on action scales far exceeding Planck’s constant. Indeed, if expressed in action units, the localization length is large compared to . One must therefore expect localization to be destroyed by dissipation. Figure 1.7 confirms that expectation. As a particular damping constant is increased, the time dependence of the kinetic energy of the rotator tends to resemble indefinite classical diffusion ever more closely. The influence of dissipation on level statistics is highly interesting. As long as the widths assigned to the individual levels by a damping mechanism are smaller than the spacings between neighboring levels, there is no noticeable change of the spacing distribution; in particular, the distinction between clustering and repulsion is still possible. The original concept of a spacing loses its meaning, however, when the typical level width exceeds a typical spacing in the absence of dissipation. A possible extension of this concept is the Euclidean distance in the complex plane between the eigenvalues of the relevant master equation or generalized Floquet operator. Numerical evidence for a damped version of the periodically kicked top suggests that these spacings display linear and cubic repulsion under conditions of classically regular and chaotic motion, respectively. Linear repulsion, incidentally, is a rigorous

Fig. 1.7 Classical (uppermost curve) and quantum mean kinetic energy of periodically kicked rotator with damping. The stronger the damping, the steeper are the curves. In all cases, the damping is so weak that the classical curve is still indistinguishable from that without dissipation. Courtesy of Graham and Dittrich [28]

References

13

property of Poissonian random processes in the plane. Cubic repulsion, on the other hand, will turn out to be characteristic of random matrices unrestricted by either Hermiticity or unitarity. A separate chapter will be devoted to the semiclassical approximation for chaotic dynamics, Gutzwiller’s periodic-orbit theory. The basic trace formulas for maps and autonomous flows will be derived and discussed. As a beautiful application I shall describe the recently established semiclassical explanation of universal spectral fluctuations [36]. Long periodic orbits, with periods of the order of the Heisenberg time TH , will be found “at work”. The latter time is related to the mean level spacing Δ, the energy scale on which universal spectral fluctuations arise, as TH = 2π /Δ; inasmuch as the mean spacing is small in the sense Δ ∝  f with f the number of degrees of freedom (note f ≥ 2 for chaos), the Heisenberg time diverges as TH ∝ − f +1 in the limit  → 0. The success of periodic-orbit theory on that time scale is quite remarkable in view of the infamous exponential proliferation of periodic orbits with increasing period. The key to success was provided by a new insight into classical chaotic dynamics: Long periodic orbits do not exist as mutually independent individuals but rather come in bunches with arbitrarily small orbit-to-orbit action differences. Under weak resolution the topological distinction between the orbits in a bunch is blurred such that a bunch looks like a single orbit. The construction principle of bunches is related to close self-encounters of an orbit where two or more orbit stretches run mutually close for a time span much longer than the inverse Lyapunov rate. The final chapter is devoted to superanalysis and its application to random matrices and disordered systems. This is in respectful reference to the effective merger of the fields of quantum chaos and disordered systems that we have witnessed during the past decade. The so-called supersymmetry technique has brought about many insights into various ensembles of random matrices, including banded and sparse matrices, and promises further progress. While several monographs on results and the method are available, a pedagogically oriented introduction seems missing and the present text aims to fill just that gap. Readers willing to carefully study Chap. 11 should end up motivated and equipped to carry on the game toward new applications.

References 1. 2. 3. 4. 5. 6.

F. Waldner, D.R. Barberis, H. Yamazaki: Phys. Rev. A31, 420 (1985) S. Chaudhury, A. Smith, B.E. Anderson, S. Ghose, P.S. Jessen: Nature 461, 768 (2009) M.V. Berry, M. Tabor: Proc. R. Soc. Lond. A356, 375 (1977) G.M. Zaslavskii, N.N. Filonenko: Sov. Phys. JETP, 8, 317 (1974) M.V. Berry, M. Tabor: Proc. R. Soc. Lond. A349, 101 (1976) M.V. Berry: In G. Iooss, R.H. Helleman, R. Stora (eds.) Les Houches Session XXXVI 1981, Chaotic Behavior of Deterministic Systems (North-Holland, Amsterdam, 1983) 7. O. Bohigas, R. Haq, A. Pandey: In K. B¨ockhoff (ed.) Nuclear Data for Science and Technology (Reidel, Dordrecht, 1983)

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8. G. Bohigas, M.-J. Giannoni: In Mathematical and Computational Methods in Nuclear Physics, Lecture Notes in Physics 209 (Springer, Berlin, Heidelberg, 1984) 9. O. Bohigas, M.J. Giannoni, C. Schmit: Phys. Rev. Lett. 52, 1 (1984) 10. A. Peres: Quantum Theory: Concepts and Methods (Kluwer Academic, New York, 1995) 11. T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, P.S.M. Wong: Rev. Mod. Phys. 53, 385 (1981) 12. N. Rosenzweig, C.E. Porter: Phys. Rev. 120, 1698 (1960) 13. H.S. Camarda, P.D. Georgopulos: Phys. Rev. Lett. 50, 492 (1983) 14. M.R. Schroeder: J. Audio Eng. Soc. 35, 307 (1987) 15. H.J. St¨ockmann, J. Stein: Phys. Rev. Lett. 64, 2215 (1990) 16. H. Alt, H.-D. Gr¨af, H.L. Harney, R. Hofferbert, H. Lengeler, A. Richter, P. Schart, H.A. Weidenm¨uller: Phys. Rev. Lett. 74, 62 (1995) 17. H. Alt, H.-D. Gr¨af, R. Hofferbert, C. Rangacharyulu, H. Rehfeld, A. Richter, P. Schart, A. Wirzba: Phys. Rev. E 54, 2303 (1996) 18. Th. Zimmermann, H. K¨oppel, L.S. Cederbaum, C. Persch, W. Demtr¨oder: Phys. Rev. Lett. 61, 3 (1988); G. Persch, E. Mehdizadeh, W. Demtr¨oder, T. Zimmermann, L.S. Cederbaum: Ber. Bunsenges. Phys. Chem. 92, 312 (1988) 19. H. Held, J. Schlichter, G. Raithel, H. Walther: Europhys. Lett. 43, 392 (1998) 20. C. Ellegaard, T. Guhr, K. Lindemann, H.Q. Lorensen, J. Nyg˚ard, M. Oxborrow: Phys. Rev. Lett. 75, 1546 (1995) 21. C. Ellegaard, T. Guhr, K. Lindemann, J. Nyg˚ard, M. Oxborrow: Phys. Rev. Lett. 77, 4918 (1996) 22. A. Hoenig, D. Wintgen: Phys. Rev. A39, 5642 (1989) 23. M. Oxborrow, C. Ellegaard: In Proceedings of the 3rd Experimental Chaos Conference (Edinburgh, 1995) 24. S. Deus, P.M. Koch, L. Sirko: Phys. Rev. E 52, 1146 (1995) 25. O. Legrand, C. Schmit, D. Sornette: Europhys. Lett. 18, 101 (1992) 26. U. Stoffregen, J. Stein, H.-J. St¨ockmann, M. Ku´s, F. Haake: Phys. Rev. Lett. 74, 2666 (1995) 27. P. So, S.M. Anlage, E. Ott, R.N. Oerter: Phys. Rev. Lett. 74, 2662 (1995) 28. T. Dittrich, R. Graham: Ann. Phys. 200, 363 (1990) 29. M.L. Mehta: Random Matrices (Academic, New York, 1967; 2nd edition 1991; 3rd edition Elsevier 2004) 30. C.E. Porter (ed.): Statistical Theory of Spectra (Academic, New York, 1965) 31. T. Guhr, A. M¨uller-Groeling, H.A. Weidenm¨uller: Phys. Rep. 299, 192 (1998) 32. P. Pechukas: Phys. Rev. Lett. 51, 943 (1983) 33. T. Yukawa: Phys. Rev. Lett. 54, 1883 (1985) 34. T. Yukawa: Phys. Lett. 116A, 227 (1986) 35. S. Fishman, D.R. Grempel, R.E. Prange: Phys. Rev. Lett. 49, 509 (1982); Phys. Rev. A29, 1639 (1984) 36. S. M¨uller, S. Heusler, A. Altland, P. Braun, F. Haake: New J. Phys. 11, 103025 (2009), arXiv:0906.1960v2

Chapter 2

Time Reversal and Unitary Symmetries

2.1 Autonomous Classical Flows A classical Hamiltonian system is called time-reversal invariant if from any given solution x(t), p(t) of Hamilton’s equations an independent solution x (t ), p (t ), is obtained with t = −t and some operation relating x and p to the original coordinates x and momenta p. The simplest such invariance, to be referred to as conventional, holds when the Hamiltonian is an even function of all momenta, t → −t, x → x, p → − p, H (x, p) = H (x, − p).

(2.1.1)

This is obviously not a canonical transformation since the Poisson brackets { pi , x j } = δi j are not left intact. The change of sign brought about for the Poisson brackets is often acknowledged by calling classical time reversal anticanonical. We should keep in mind that the angular momentum vector of a particle is bilinear in x and p and thus odd under conventional time reversal. The motion of a charged particle in an external magnetic field is not invariant under conventional time reversal since the minimal-coupling Hamiltonian ( p − (e/c) A)2 /2 m is not even in p. Such systems may nonetheless have some other, nonconventional time-reversal invariance, to be explained in Sect. 2.9. Hamiltonian systems with no time-reversal invariance must not be confused with dissipative systems. The differences between Hamiltonian and dissipative dynamics are drastic and well known. Most importantly from a theoretical point of view, all Hamiltonian motions conserve phase-space volumes according to Liouville’s theorem, while for dissipative processes such volumes contract in time. The difference between Hamiltonian systems with and without time-reversal invariance, on the other hand, is subtle and has never attracted much attention in the realm of classical physics. It will become clear below, however, that the latter difference plays an important role in the world of quanta [1–3].

F. Haake, Quantum Signatures of Chaos, Springer Series in Synergetics, 3rd ed., C Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-05428-0 2, 

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2 Time Reversal and Unitary Symmetries

2.2 Spinless Quanta The Schr¨odinger equation ˙ iψ(x, t) = H ψ(x, t)

(2.2.1)

is time-reversal invariant if, for any given solution ψ(x, t), there is another one, ψ (x, t ), with t = −t and ψ uniquely related to ψ. The simplest such invariance, again termed conventional, arises for a spinless particle with the real Hamiltonian H (x, p) =

p2 + V (x), V (x) = V ∗ (x), 2m

(2.2.2)

where the asterisk denotes complex conjugation. The conventional reversal is t → −t, x → x, p → − p, ψ(x) → ψ ∗ (x) = K ψ(x).

(2.2.3)

In other words, if ψ(x, t) solves (2.2.1) so does ψ (x, t) = K ψ(x, −t). The operator K of complex conjugation obviously fulfills K 2 = 1,

(2.2.4)

i.e., it equals its inverse, K = K −1 . Its definition also implies K (c1 ψ1 (x) + c2 ψ2 (x)) = c1∗ K ψ1 (x) + c2∗ K ψ2 (x),

(2.2.5)

a property commonly called antilinearity. The transformation ψ(x) → K ψ(x) does not change the modulus of the overlap of two wave functions, |K ψ|K φ|2 = |ψ|φ|2 ,

(2.2.6)

while the overlap itself is transformed into its complex conjugate, K ψ|K φ = ψ|φ∗ = φ|ψ.

(2.2.7)

The identity (2.2.7) defines the property of antiunitarity which implies antilinearity [1] (Problem 2.4). It is appropriate to emphasize that I have defined the operator K with respect to the position representation. Dirac’s notation makes this distinction of K especially obvious. If some state vector |ψ is expanded in terms of position eigenvectors |x,  |ψ = d xψ(x)|x, (2.2.8) the operator K acts as

2.3

Spin-1/2 Quanta

17

 K |ψ =

d xψ ∗ (x)|x,

(2.2.9)

i.e., as K |x = |x. A complex conjugation operator K can of course be defined with respect to any representation. It is illustrative to consider a discrete basis and introduce K |ψ = K



ψν |ν =

ν

 ν

ψν∗ |ν.

(2.2.10)

Conventional time reversal, i.e., complex conjugation in the position representation, can then be expressed as K = UK

(2.2.11)

with a certain symmetric unitary matrix U, the calculation of which is left to the reader as problem 2.5. Unless otherwise stated, the symbol K will be reserved for complex conjugation in the coordinate representation, as far as orbital wave functions are concerned. Moreover, antiunitary time-reversal operators will, for the most part, be denoted by T. Only the conventional time-reversal for spinless particles has the simple form T = K .

2.3 Spin-1/2 Quanta All time-reversal operators T must be antiunitary T ψ|T φ = φ|ψ,

(2.3.1)

because of (i) the explicit factor i in Schr¨odinger’s equation and (ii) since they should leave the modulus of the overlap of two wave vectors invariant. It follows from the definition (2.3.1) of antiunitarity that the product of two antiunitary operators is unitary. Consequently, any time-reversal operator T can be given the so-called standard form T = UK ,

(2.3.2)

where U is a suitable unitary operator and K the complex conjugation with respect to a standard representation (often chosen to be the position representation for the orbital part of wave functions). Another physically reasonable requirement for every time-reversal operator T is that any wave function should be reproduced, at least to within a phase factor, when acted upon twice by T, T 2 = α, |α| = 1.

(2.3.3)

18

2 Time Reversal and Unitary Symmetries

Inserting the standard form (2.3.2) in (2.3.3) yields1 UKUK = UU ∗ K 2 = UU ∗ = α, i.e., U ∗ = αU −1 = αU † = αU˜ ∗ . The latter identity once iterated gives U ∗ = α 2 U ∗ , i.e., α 2 = 1 or T 2 = ±1.

(2.3.4)

The positive sign holds for conventional time reversal with spinless particles. It will become clear presently that T 2 = −1 in the case of a spin-1/2 particle. See also Problem 2.9. A useful time-reversal operation for a spin-1/2 results from requiring that T J T −1 = − J

(2.3.5)

holds not only for the orbital angular momentum but likewise for the spin. With respect to the spin, however, T cannot simply be the complex conjugation operation since all purely imaginary Hermitian 2 × 2 matrices commute with one another. The more general structure (2.3.2) must therefore be considered. Just as a matter of convenience, I shall choose K as the complex conjugation in the standard representation where the spin operator S takes the form S = 2 σ ,       01 0 −i 1 0 , σy = , σz = . σx = 10 i 0 0 −1

(2.3.6)

The matrix U is then constrained by (2.3.5) to obey T σx T −1 = U K σx K U −1 = U σx U −1 = −σx T σ y T −1 = U K σ y K U −1 = −U σ y U −1 = −σ y

(2.3.7)

T σz T −1 = U K σz K U −1 = U σz U −1 = −σz , i.e., U must commute with σ y and anticommute with σx and σz . Because any 2 × 2 matrix U can be represented as a sum of Pauli matrices, we can write U = ασx + βσ y + γ σz + δ.

(2.3.8)

The first of the Eq. (2.3.7) immediately gives α = δ = 0; the second yields γ = 0, whereas β remains unrestricted by (2.3.7). However, since U is unitary, β must have unit modulus. It is thus possible to choose β = i whereupon the time-reversal operation reads

Matrix transposition will always be represented by a tilde, while the dagger † will denote Hermitian conjugation.

1

2.4

Hamiltonians Without T Invariance

19

T = iσ y K = eiπσ y /2 K .

(2.3.9)

This may be taken to include, if necessary, the time reversal for the orbital part of wave vectors by interpreting K as complex conjugation both in the position representation and in the standard spin representation. In this sense I shall refer to (2.3.9) as conventional time reversal for spin-1/2 quanta. The operation (2.3.9) squares to minus unity, in contrast to conventional time reversal for spinless particles. Indeed, T 2 = iσ y K iσ y K = (iσ y )2 = −1. If one is dealing with N particles with spin 1/2, the matrix U must obviously be taken as the direct product of N single-particle matrices, T = iσ1y iσ2y . . . iσ N y K   π  Sy σ1y + σ2y + . . . + σ N y K = exp iπ K , (2.3.10) = exp i 2  where S y now is the y-component of the total spin S = (σ 1 + σ2 + . . . + σ N )/2. The square of T depends on the number of particles according to

T = 2

+1 N even −1 N odd.

(2.3.11)

I shall ocasionally refer to “kicked tops”, dynamical systems involving only components of an angular momentum J = (Jx , Jy , Jz ) as dynamical variables. The square J 2 is then conserved and the Hilbert space can be chosen as the 2 j + 1 dimensional space with J 2 = j( j + 1) spanned the eigenvectors | j, m of, say, Jz with m = − j, − j + 1, . . . , j. The standard time reversal operator is the given by (2.3.10) as T = eiπ Jz / K with K | j, m = | j, m and squares to +1 or −1 when the qantum number j is integer and half-integer, respectively.

2.4 Hamiltonians Without T Invariance All Hamiltonians can be represented by Hermitian matrices. Before proceeding to identify the subclasses of Hermitian matrices to which time-reversal invariant Hamiltonians belong, it is appropriate to pause and make a few remarks about Hamiltonians unrestricted by antiunitary symmetries. Any Hamiltonian becomes real in its eigenrepresentation, diag (E 1 , E 2 , . . . ). Under a unitary transformation U, †

∗ , Hμν = Uμλ E λ Uλν = Uμλ E λ Uνλ

(2.4.1)

˜ μν = Hνμ , but ceases, in general, to be real. H preserves Hermiticity, (Hμν )∗ = H Now, I propose to construct the class of “canonical transformations” that change a Hamiltonian matrix without destroying its Hermiticity and without altering its

20

2 Time Reversal and Unitary Symmetries

eigenvalues. To this end it is important to look at each irreducible part of the matrix H separately, i.e., to think of good quantum numbers related to a complete set of mutually commuting conserved observables (other than H itself) as fixed. Eigenvalues are preserved under a similarity transformation with an arbitrary nonsingular matrix A. To show that H = A H A−1 has the same eigenvalues as H , it suffices to write out H in the H representation,

= Hμν

 λ

  Aμλ E λ A−1 λν ,

(2.4.2)

and to multiply from the right by A. The columns of A are then recognized as eigenvectors of H , and the eigenvalues of H turn out to be those of H as well. For A to qualify as a canonical transformation, H must also be Hermitian, 

†  A H A−1 = A H A−1 ⇔ H, A† A = 0.

(2.4.3)

Excluding the trivial solution where A† A is a function of H and recalling that all other mutually commuting conserved observables are multiples of the unit matrix in the space considered, one concludes that A† A must be the unit matrix, at least to within a positive factor. That factor must itself be unity if A is subjected to the additional constraint that it should preserve the normalization of vectors, A† A = 11.

(2.4.4)

The class of canonical transformations of Hamiltonians unrestricted by antiunitary symmetries is thus constituted by unitary matrices. Obviously, for an N -dimensional Hilbert space that class is the group U (N ). It is noteworthy that Hamiltonian matrices unrestricted by antiunitary symmetries are in general complex. They can, of course, be given real representations, but any such representation will become complex under a general canonical (i.e., unitary) transformation. A few more formal remarks may be permissible. The time evolution operators U = e−iH t/ generated by complex Hermitian Hamiltonians form the Lie group U (N ). The Hamiltionians themselves can be associated with the generators X = iH of the Lie algebra u(N ), the tangent space to the group U (N ). — Moreover, Eq. (2.4.1) reveals that a general complex Hermitian N×N Hamiltonian can be diagonalized by a unitary transformation. Once such a diagonalizing transformation U is found, others can be obtained by splitting off an arbitrary diagonal unitary matrix as U diag (e−iφ1 , . . . e−iφ N ). By identifying all such matrices one arrives at the coset space U (N )/U (1) N . — For all of these reasons, complex Hermitian Hamiltonians are said to form the “unitary symmetry class”.

2.5

T Invariant Hamiltonians, T 2 = 1

21

2.5 T Invariant Hamiltonians, T 2 = 1 When we have an antiunitary operator T with [H, T ] = 0, T 2 = 1

(2.5.1)

the Hamiltonian H can always be given a real matrix representation and such a representation can be found without diagonalizing H. As a first step toward proving the above statement, I demonstrate that, with the help of an antiunitary T squaring to plus unity, T invariant basis vectors ψν can be constructed. Take any vector φ1 and a complex number a1 . The vector ψ1 = a1 φ1 + T a1 φ1

(2.5.2)

is then T invariant, T ψ1 = ψ1 . Next, take any vector φ2 orthogonal to ψ1 and a complex number a2 . The combination ψ2 = a 2 φ2 + T a 2 φ2

(2.5.3)

is again T invariant. Moreover, ψ2 is orthogonal to ψ1 since ψ2 |ψ1  = a2∗ φ2 |ψ1  + a2 T φ2 |ψ1 

∗ = a2 T 2 φ2 |ψ1 = a2 φ2 |ψ1 ∗ = 0.

(2.5.4)

By so proceeding we eventually arrive at a complete set of mutually orthogonal vectors. If desired, the numbers aν can be chosen to normalize as ψμ |ψν  = δμν . With respect to a T invariant basis, the Hamiltonian H = T H T is real, Hμν = ψμ |H ψν  = T ψμ |T H ψν ∗ ∗ = ψμ |T H T 2 ψν ∗ = ψμ |T H T ψν ∗ = Hμν .

(2.5.5)

Note that the Hamiltonians in question can be made real without being diagonalized first. It is therefore quite legitimate to say that they are generically real matrices. The canonical transformations that are admissible now form the Lie group O(N ) of ˜ = 1. Beyond preserving eigenvalues and Hermiticreal orthogonal matrices O, O O ity, an orthogonal transformation also transforms a real matrix H into another real ˜ The orthogonal group is obviously a subgroup of the unitary matrix H = O H O. group considered in the last section. A T invariant N × N Hamiltonian can be diagonalized by a matrix from the yet smaller group S O(N ) of unit-determinant orthogonal matrices if T 2 = 1. It is therefore customary to say that the Hamiltonians under scrutiny form the “orthogonal symmetry class”.

22

2 Time Reversal and Unitary Symmetries

It may be worthwhile to look back at Sect. 2.2 where it was shown that the Schr¨odinger equation of a spinless particle is time-reversal invariant provided the Hamiltonian is a real operator in the position representation. The present section generalizes that previous statement. The time evolution operators for the orthogonal class can be characterized from the point of view of Lie groups, in analogy to e−iH t/ ∈ U (N ) for the unitary class. ˜ entails e−iH t/ to be symmetric as well. The time To that end I argue that H = H evolution operators can thus be written as U U˜ with U ∈ U (N ). Now the product U U˜ remains unchanged when U is replaced by U O with O any orthogonal matrix, and therefore the coset space U (N )/O(N ) houses the time evolution operators from the orthogonal class.

2.6 Kramers’ Degeneracy For any Hamiltonian invariant under a time reversal T, [H, T ] = 0,

(2.6.1)

i.e., if ψ is an eigenfunction with eigenvalue E, so is T ψ. As shown above, we may choose the equality T ψ = ψ without loss of generality if T 2 = +1. Here, I propose to consider time-reversal operators squaring to minus unity, T 2 = −1.

(2.6.2)

In this case, ψ and T ψ are orthogonal, ∗

ψ|T ψ = T ψ|T 2 ψ = −T ψ|ψ∗ = −ψ|T ψ = 0,

(2.6.3)

and therefore all eigenvalues of H are doubly degenerate. This is Kramers’ degeneracy. It follows that the dimension of the Hilbert space must, if finite, be even. This fits with the result of Sect. 2.3 that T 2 = −1 is possible only if the number of spin-1/2 particles in the system is odd; the total-spin quantum number s is then a half-integer and 2s + 1 is even. In the next two sections, I shall discuss the structure of Hamiltonian matrices with Kramers’ degeneracy, first for the case with additional geometric symmetries and then for the case in which T is the only invariance.

2.7 Kramers’ Degeneracy and Geometric Symmetries As an example of geometric symmetry, let us consider a parity such that [Rx , H ] = 0, [Rx , T ] = 0, Rx2 = −1.

(2.7.1)

2.7

Kramers’ Degeneracy and Geometric Symmetries

23

This could be realized, for example, by a rotation through π about, say, the xaxis, Rx = exp (iπ Jx /); note that since T 2 = −1 only half-integer values of the total angular momentum quantum number are admitted. To reveal the structure of the matrix H , it is convenient to employ a basis ordered by parity, Rx |n± = ±i|n±.

(2.7.2)

Moreover, since T changes the parity, Rx T |n± = T Rx |n± = ∓iT |n±,

(2.7.3)

the basis can be organized such that T |n± = ±|n∓.

(2.7.4)

For the sake of simplicity, let us assume a finite dimension 2N . The matrix H then falls into four N × N blocks   + 0 H (2.7.5) H= 0 H− two of which are zero since H has vanishing matrix elements between states of different parity. Indeed, m + |Rx H Rx−1 |n− is equal to +m + |H |n− due to the invariance of H under Rx and equal to −m + |H |n− because of (2.7.2). The T invariance relates the two blocks H ± :   

m + |H |n+ = m + T H T −1  n+ = −m + |T H |n− = −T (m+)|T 2 H |n−∗ = +m − |H |n−∗ = n − |H |m−.

(2.7.6)

At this point Kramers’ degeneracy emerges: Since they are the transposes of one another, H + and H − have the same eigenvalues. Moreover, they are in general complex and thus have U (N ) as their group of canonical transformations. Further restrictions on the matrices H ± arise from additional symmetries. It is illustrative to admit one further parity R y with 

 R y , H = 0, R y , T = 0, Rx R y + R y Rx = 0, R 2y = −1

(2.7.7)

which might be realized as R y = exp (iπ Jy /). The anticommutativity of Rx and R y immediately tells us that R y changes the Rx parity, just as T does, Rx R y |m± = ∓iR y |m±. The basis may thus be chosen according to R y |n± = ±|n∓

(2.7.8)

24

2 Time Reversal and Unitary Symmetries

which is indeed the same as (2.7.4) but with R y instead of T. Despite this similarity, the R y invariance imposes a restriction on H that goes beyond those achieved by T precisely because R y is unitary while T is antiunitary. The R y invariance implies, together with (2.7.8), that m + |H |n+ = m + |R y H R −1 y |n+ = m − |H |n−,

(2.7.9)

i.e., H + = H − while the T invariance had given H + = (H − )∗ [see (2.7.6)]. Thus we have the result that H + and H − are identical real matrices. Their group of canonical transformation is reduced by the new parity from U (N ) to O(N ). As a final illustration of the cooperation of time-reversal invariance with geometrical symmetries, the case of full isotropy, [H, J] = 0, deserves mention. The appropriate basis here is |α jm with m and 2 j( j + 1) the eigenvalues of Jz and J 2 , respectively. The Hamiltonian matrix then falls into blocks given by   

 α jm|H |β j m = δ j j δmm α  H ( j,m)  β ,

j=

1 2

,

3 , 2

5 2

, . . . , m = ± 12 , ± 32 , . . . , ± j.

(2.7.10)

It is left to the reader as Problem 2.10 to show that (i) due to T invariance, for any fixed value of j, the two blocks with differing signs of m are transposes of one another, ˜ ( j,−m) , H ( j,m) = H

(2.7.11)

and thus have identical eigenvalues (Kramers’ degeneracy!), and (ii) invariance of H under rotations about the y-axis makes the two blocks equal. The two statements above imply that the blocks H ( j,m) are all real and thus have the orthogonal transformations as their canonical transformations. To summarize, unitary transformations are canonical both when there is no timereversal invariance and when a time-reversal invariance with Kramers’ degeneracy (T 2 = −1) is combined with one parity. Orthogonal transformations constitute the canonical group when time-reversal invariance holds, either with or without Kramers’ degeneracy, in the first case, however, only in the presence of certain geometric symmetries. An altogether different group of canonical transformations will be encountered in Sect. 2.8.

2.8

Kramers’ Degeneracy Without Geometric Symmetries

25

2.8 Kramers’ Degeneracy Without Geometric Symmetries When a time reversal with T 2 = −1 is the only symmetry of H, it is convenient to adopt a basis of the form |1 , T |1 , |2 , T |2 , . . . |N  , T |N .

(2.8.1)

[Note that in (2.7.5) another ordering of states |n+ and T |n+ = |n− was chosen.] Sometimes I shall write |T n for T |n and T n| for the corresponding Dirac bra. For the sake of simplicity, the Hilbert space is again assumed to have the finite dimension 2N . If the complex conjugation operation K is defined relative to the basis (2.8.1), the unitary matrix U in T =U K takes a simple form which is easily found by letting T act on an arbitrary state vector |ψ =



(ψm+ |m + ψm− |T m) ,

m

T |ψ =

 m

 ∗ ∗ ψm+ |T m − ψm− |m .

(2.8.2)

Clearly, in each of the two-dimensional subspaces spanned by |m and |T m, the matrix U, to be called Z from now on, takes the form 

Z mm

 0 −1 = ≡ τ2 1 0

(2.8.3)

while different such subspaces are unconnected, Z mn = 0 for m = n.

(2.8.4)

The 2N × 2N matrix Z is obviously block diagonal with the 2 × 2 blocks (2.8.3) along the diagonal. In fact it will be convenient to consider Z as a diagonal N × N matrix, whose nonzero elements are themselves 2 × 2 matrices given by (2.8.3). Similarly, the two pairs of states |m, T |m, and |n, T |n give a 2 × 2 submatrix of the Hamiltonian   m|H |n m|H |T n (2.8.5) ≡ h mn . T m|H |n T m|H |T n The full 2N × 2N matrix H may be considered as an N × N matrix each element of which is itself a 2 × 2 block h mn . The reason for the pairwise ordering of the basis (2.8.1) is, as will become clear presently, that the restriction imposed on H by time-reversal invariance can be expressed as a simple property of h mn . As is the case for any 2 × 2 matrix, the block h mn can be represented as a linear combination of four independent matrices. Unity and the three Pauli matrices σ may

26

2 Time Reversal and Unitary Symmetries

come to mind first, but the condition of time-reversal invariance will take a nicer form if the anti-Hermitian matrices τ = −iσ are employed,  τ1 =

     0 −i 0 −1 −i 0 , τ2 = , τ3 = −i 0 1 0 0 +i

(2.8.6)

τi τ j = εi jk τk , τi τ j + τ j τi = −2δi j . Four coefficients h (μ) mn , μ = 0, 1, 2, 3, characterize the block h mn , h mn = h (0) mn 1 + hmn · τ .

(2.8.7)

Now, time-reversal invariance gives   h mn = T H T −1 mn   = Z K H K Z −1 mn   = Z H ∗ Z −1 mn

= −τ2 h ∗mn τ2  ∗  ∗ ∗ τ2 = −τ2 h (0) mn 1 + hmn · τ ∗

∗ = h (0) mn 1 + hmn · τ ,

(2.8.8)

(μ)

which simply means that the four amplitudes h mn are all real: ∗

(μ) h (μ) mn = h mn .

(2.8.9)

For historical reasons, matrices with the property (2.8.9) are called “quaternion real”. Note that this property does look nicer than the one that would have been obtained if we had used Pauli’s triple σ instead of the anti-Hermitian τ . The Hermiticity of H implies the relation h mn = h †nm ,

(2.8.10) (μ)

which in turn means that the four real amplitudes h mn obey (0) h (0) mn = h nm (k) h (k) mn = −h nm , k = 1, 2, 3.

(2.8.11)

It follows that the 2N × 2N matrix H is determined by N (2N − 1) independent real parameters. With the structure of the Hamiltonian now clarified, it remains to identify the canonical transformations that leave this structure intact. To that end we must find the subgroup of unitary matrices that preserve the form T = Z K of the time-reversal operator. In other words, the question is to what extent there is freedom in choosing

2.9

Nonconventional Time Reversal

27

a basis with the properties (2.8.1). The allowable unitary basis transformations S have to obey T = ST S −1 = S Z K S −1 = S Z S˜ K

=⇒

S Z S˜ = Z .

(2.8.12)

The requirement S Z S˜ = Z defines the Lie group Sp(2N ), see Problem 2.12. The symplectic transformations just found are in fact the relevant canonical transformations since they leave a quaternion real Hamiltonian quaternion real. To prove that statement, I shall show that if H is T invariant, then so is S H S −1 : With the help of the identities Z S ∗ = S Z and S˜ Z = Z S −1 , both of which reformulations of (2.8.12), I have ∗  T S H S −1 T −1 = Z K S H S −1 K Z −1 = Z S ∗ K H K S −1 Z −1 ˜ = S Z K H K S(−Z ) = S Z K H K (−Z )S −1 = ST H T −1 S −1 = S H S −1 .

(2.8.13)

Of course, a T invariant 2N × 2N Hamiltonian is diagonalizable by a symplectic transformation from Sp(2N ) if T 2 = −1. Time reversal invariant Hamiltonians with T 2 = −1 are said to form the “symplectic symmetry class”. Some readers may want to check that the pertinent time evolution operators live in the coset space U (2N )/Sp(2N ).

2.9 Nonconventional Time Reversal We have defined conventional time reversal by T xT −1 = x T pT −1 = − p T J T −1 = − J

(2.9.1)

and, for any pair of states, T φ|T ψ = ψ|φ T 2 = ±1.

(2.9.2)

The motivation for this definition is that many Hamiltonians of practical importance are invariant under conventional time reversal, [H, T ] = 0. An atom and a molecule in an isotropic environment, for instance, have Hamiltonians of that symmetry. But, as already mentioned in Sect. 2.1, conventional time reversal is broken by an external magnetic field. In identifying the canonical transformations of Hamiltonians from their symmetries in Sects. 2.5, 2.6, 2.7, and 2.8, extensive use was made of (2.9.2) but none,

28

2 Time Reversal and Unitary Symmetries

as the reader is invited to review, of (2.9.1). In fact, and indeed fortunately, the validity of (2.9.1) is not at all necessary for the above classification of Hamiltonians according to their group of canonical transformations. Interesting and experimentally realizable systems often have Hamiltonians that commute with some antiunitary operator obeying (2.9.2) but not (2.9.1). There is nothing strange or false about such a “nonconventional” time-reversal invariance: it associates another, independent solution, ψ (t) = T ψ(−t), with any solution ψ(t) of the Schr¨odinger equation, and is thus as good a time-reversal symmetry as the conventional one. An important example is the hydrogen atom in a constant magnetic field [4, 5]. Choosing that field as B = (0, 0, B) and the vector potential as A = B × x/2 and including spin-orbit interaction, one obtains the Hamiltonian H=

 e2 B 2  2 e2 eB p2 x + y 2 + f (r )L S. − − (L z + gSz ) + 2 2m r 2mc 8mc

(2.9.3)

Here L and S denote orbital angular momentum and spin, respectively, while the total angular momentum is J = L + S. This Hamiltonian is not invariant under conventional time reversal, T0 , but instead under T = eiπ Jx / T0 .

(2.9.4)

If spin is absent, T 2 = 1, whereas T 2 = −1 with spin. In the subspaces of constant Jz and J 2 , one has the orthogonal transformations as the canonical group in the first case (Sect. 2.2) and the unitary transformations in the second case (Sect. 2.7 and Problem 2.11). When a homogeneous electric field E is present in addition to the magnetic field, the operation T in (2.9.4) ceases to be a symmetry of H since it changes the electricdipole perturbation −ex · E. But T = RT0 is an antiunitary symmetry where the unitary operator R represents a reflection in the plane spanned by B and E. Note that the component of the angular momentum lying in this plane changes sign under that reflection since the angular momentum is a pseudo-vector. While the Zeeman term in H changes sign under both conventional time reversal and under the reflection in question, it is left invariant under the combined operation. The electric-dipole term as well as all remaining terms in H are symmetric with respect to both T0 and R such that [H, RT0 ] = 0 indeed results. As another example Seligman and Verbaarschot [6] proposed two coupled oscillators with the Hamiltonian H =

1 2



2  2 p1 − a x23 + 12 p2 + a x13

+ α1 x16 + α2 x26 − α12 (x1 − x2 )6 .

(2.9.5)

Here, too, T0 invariance is violated if a = 0. As long as α12 = 0, however, H is invariant under

2.10

Stroboscopic Maps for Periodically Driven Systems

T = eiπ L 2 / T0

29

(2.9.6)

and thus representable by a real matrix. The geometric symmetry T T0−1 acts as (x1 , p1 ) → (−x1 , − p1 ) and (x2 , p2 ) → (x2 , p2 ) and may be visualized as a rotation through π about the 2-axis if the two-dimensional space spanned by x1 and x2 is imagined embedded in a three-dimensional Cartesian space. However, when a = 0 and α12 = 0, the Hamiltonian (2.9.5) has no antiunitary symmetry left and therefore is a complex matrix. (Note that H is a complex operator in the position representation anyway.)

2.10 Stroboscopic Maps for Periodically Driven Systems Time-dependent perturbations, especially periodic ones, are characteristic of many situations of experimental interest. They are also appreciated by theorists inasmuch as they provide the simplest examples of classical nonintegrability: Systems with a single degree of freedom are classically integrable, if autonomous, but may be nonintegrable if subjected to periodic driving. Quantum mechanically, one must tackle a Schr¨odinger equation with an explicit time dependence in the Hamiltonian, ˙ iψ(t) = H (t)ψ(t).

(2.10.1)

The solution at t > 0 can be written with the help of a time-ordered exponential 

  t −i dt H (t ) U (t) = exp  0 +

(2.10.2)

where the “positive” time ordering requires 







A(t)B(t )

+

=

A(t)B(t ) if t > t

. B(t ) A(t) if t < t

(2.10.3)

Of special interest are cases with periodic driving, H (t + nτ ) = H (t), n = 0, ±1, ±2, . . . .

(2.10.4)

The evolution operator referring to one period τ, the so-called Floquet operator U (τ ) ≡ F,

(2.10.5)

is worthy of consideration since it yields a stroboscopic view of the dynamics, ψ(nτ ) = F n ψ(0).

(2.10.6)

30

2 Time Reversal and Unitary Symmetries

Equivalently, F may be looked upon as defining a quantum map, ψ ([n + 1]τ ) = Fψ(nτ ).

(2.10.7)

Such discrete-time maps are as important in quantum mechanics as their Newtonian analogues have proven in classical nonlinear dynamics. The Floquet operator, being unitary, has unimodular eigenvalues (involving eigenphases alias quasi-energies) and mutually orthogonal eigenvectors, FΦν = e−iφν Φν , 

Φμ |Φν = δμν .

(2.10.8)

I shall in fact be concerned only with normalizable eigenvectors. With the eigenvalue problem solved, the stroboscopic dynamics can be written out explicitly, ψ(nτ ) =



e−inφν Φν |ψ(0) Φν .

(2.10.9)

ν

Monochromatic perturbations are relatively easy to realize experimentally. Much easier to analyse are perturbations for which the temporal modulation takes the form of a periodic train of delta kicks, H (t) = H0 + λV

+∞ 

δ(t − nτ ).

(2.10.10)

n = −∞

The weight of the perturbation V in H (t) is measured by the parameter λ, which will be referred to as the kick strength. The Floquet operator transporting the state vector from immediately after one kick to immediately after the next reads F = e−iλV / e−iH0 τ/ .

(2.10.11)

The simple product form arises from the fact that only H0 is on between kicks, while H0 is ineffective “during” the infinitely intense delta kick. It may be well to conclude this section with a few examples. Of great interest with respect to ongoing experiments is the hydrogen atom exposed to a monochromatic electromagnetic field. Even the simplest Hamiltonian, H=

e2 p2 − − E z cos ωt, 2m r

(2.10.12)

defies exact solution. The classical motion is known to be strongly chaotic for sufficiently large values of the electric field E : A state that is initially bound (with respect to H0 = p 2 /2m − e2 /r ) then suffers rapid ionization. The quantum modifications of this chaos-enhanced ionization have been the subject of intense discussion. See [7] for the early efforts, and for a brief sketch of the present situation, see Sect. 7.1.

2.11

Time Reversal for Maps

31

A fairly complete understanding has been achieved for both the classical and quantum behavior of the kicked rotator [8], a system of quite some relevance for microwave ionization of hydrogen atoms. The Hamiltonian reads H (t) =

+∞  1 2 δ(t − nτ ). p + λ cos φ 2I n = −∞

(2.10.13)

The classical kick-to-kick description is Chirikov’s standard map [9]. Most of the chapter on quantum localization will be devoted to that prototypical system. Somewhat richer in their behavior are the kicked tops for which H0 and V in (2.10.10) and (2.10.11) are polynomials in the components of an angular momentum J. Due to the conservation of J 2 = 2 j( j + 1), j = 12 , 1, 32 , 2, . . . , kicked tops enjoy the privilege of a finite-dimensional Hilbert space. The special case H0 ∝ Jx , V ∝ Jz2

(2.10.14)

has recently been realized experimentally [10].

2.11 Time Reversal for Maps It is easy to find the condition which the Hamiltonian H (t) must satisfy so that a given solution ψ(t) of the Schr¨odinger equation ˙ iψ(t) = H (t)ψ(t)

(2.11.1)

˜ ψ(t) = T ψ(−t),

(2.11.2)

yields an independent solution,

where T is some antiunitary operator. By letting T act on both sides of the Schr¨odinger equation, − i

∂ T ψ(t) = T H (t)T −1 T ψ(t) ∂t

(2.11.3)

or, with t → −t, i

∂ T ψ(−t) = T H (−t)T −1 T ψ(−t). ∂t

(2.11.4)

For (2.11.4) to be identical to the original Schr¨odinger equation, H (t) must obey H (t) = T H (−t)T −1 , a condition reducing to that studied previously for autonomous dynamics.

(2.11.5)

32

2 Time Reversal and Unitary Symmetries

For periodically driven systems it is convenient to express the time-reversal symmetry (2.11.5) as a property of the Floquet operator. As a first step in searching for that property, we again employ the formal solution of (2.11.1). Distinguishing now between positive and negative times,  U+ (t)ψ(0) , t > 0 ψ(t) = U− (t)ψ(0) , t < 0

(2.11.6)

where U+ (t) is positively time-ordered as explained in (2.10.2) and (2.10.3) while the negative time order embodied in U− (t) is simply the opposite of the positive one. Now, I assume t > 0 and propose to consider ˜ ψ(t) = T ψ(−t) = T U− (−t)ψ(0) = T U− (−t)T −1 T ψ(0).

(2.11.7)

˜ must solve the If H (t) is time-reversal invariant in the sense of (2.11.5), this ψ(t) original Schr¨odinger equation (2.11.1) such that U+ (t) = T U− (−t)T −1 .

(2.11.8)

The latter identity is in fact equivalent to (2.11.5). The following discussion will be confined to τ -periodic driving, and we shall take the condition (2.11.8) for t = τ. The backward Floquet operator U− (−τ ) is then simply related to the forward one. To uncover that relation, we represent U− (−τ ) as a product of time evolution operators, each factor referring to a small time increment,

   i −τ

dt H (t ) U− (−τ ) = exp −  0 − = eiΔt H (−tn )/ eiΔt H (−tn−1 )/ . . . . . . eiΔt H (−t2 )/ eiΔt H (−t1 )/ .

(2.11.9)

As illustrated in Fig. 2.1, we choose equidistant intermediate times between t−n = −τ and tn = τ with the positive spacing ti+1 − ti = Δt = τ/n. The intervals ti+1 − ti are assumed to be so small that the Hamiltonian can be taken to be constant within each of them. Note that the positive sign appears in each of the n exponents in the second line of (2.11.9) since Δt is defined to be positive while the time integral in the negatively time-ordered exponential runs toward the

Fig. 2.1 Discretization of the time used to evaluate the time-ordered exponential in (2.11.9)

2.12

Canonical Transformations for Floquet Operators

33

left on the time axis. Now, we invoke the assumed periodicity of the Hamiltonian, H (−tn−i ) = H (ti ), to rewrite U− (−τ ) as U− (−τ ) = eiΔt H (0)/ eiΔt H (t1 )/ . . . . . . eiΔt H (tn−2 )/ eiΔt H (tn−1 )/ = U+ (τ )†

(2.11.10)

which is indeed the Hermitian adjoint of the forward Floquet operator. Now, we can revert to a simpler notation, U+ (τ ) = F, and write the time-reversal property (2.11.8) for t = τ together with (2.11.10) as a time-reversal “covariance” of the Floquet operator T F T −1 = F † = F −1 .

(2.11.11)

This is a very intuitive result indeed: The time-reversed Floquet operator T F T −1 is just the inverse of F. An interesting general statement can be made about periodically kicked systems with Hamiltonians of the structure (2.10.10). If H0 and V are both invariant under some time reversal T0 (conventional or not), the Hamiltonian H (t) is T0 -covariant in the sense of (2.11.5) provided the zero of time is chosen halfway between two successive kicks. The Floquet operator (2.10.11) is then not covariant in the sense (2.11.11) with respect to T0 , but it is with respect to T = eiH0 τ/ T0 .

(2.11.12)

The reader is invited, in Problem 2.13, to show that the Floquet operator defined so as to transport the wave vector by one period starting at a point halfway between two successive kicks is T0 -covariant. The above statement implies that for the periodically kicked rotator defined by (2.10.13), F has an antiunitary symmetry of the type (2.11.12). Similarly, the Floquet operator of the kicked top (2.10.14) is covariant with respect to T = eiH0 τ/ where K is complex conjugation in the standard representation of angular momenta in which Jx and Jz are real and Jy is imaginary.

2.12 Canonical Transformations for Floquet Operators The arguments presented in Sects. 2.4, 2.5, 2.7, and 2.8 for Hermitian Hamiltonians carry over immediately to unitary Floquet operators. We shall assume a finite number of dimensions throughout. First, irreducible N × N Floquet matrices without any T covariance have U (N ) as their group of canonical transformations. Indeed, any transformation from that group preserves eigenvalues, unitarity, and normalization of vectors. The proof is analogous to that sketched in Sect. 2.4.

34

2 Time Reversal and Unitary Symmetries

Next, when F is T covariant with T 2 = 1, one can find a T invariant basis in which F is symmetric. In analogy with the reasoning in (2.5.5), one takes matrix elements in T F † T −1 = F with respect to T invariant basis states,     ∗

 Fμν = ψμ T F † T −1  ψν = T ψμ T 2 F † T  ψν  ∗

 = ψμ  F † T  ψν = Fνμ .

(2.12.1)

It is worth recalling that time-reversal invariant Hamiltonians were also found, in Sect. 2.5, to be symmetric if T 2 = 1. Of course, for unitary matrices F = F˜ does not imply reality. The canonical group is now O(N ), as was the case for [H, T ] = 0, T 2 = 1. To see this, we assume that F = F˜ and that O is unitary, and require O F O † to be symmetric, ˜ † F O. ˜ O F O† = O

(2.12.2)

˜ and from the right with O gives Multiplication from the left with O ˜ OF = FO ˜ O. O

(2.12.3)

˜ must be unity, i.e., O Since F must be assumed irreducible, the product O O must be an orthogonal matrix. Finally, if F is T covariant with T 2 = −1, there is again Kramers’ degeneracy. To prove this, let F|φν  = e−iφν |φν , F † |φν  = eiφν |φν 

(2.12.4)

and let T act on the latter equation: e−iφν T |φν  = T F † T −1 T |φν  = F T |φν .

(2.12.5)

The orthogonality of |φν  and T |φν , following from T 2 = −1, has already been demonstrated in (2.6.3). The Hilbert space dimension must again be even. Which group of transformations is canonical depends, as for time-independent Hamiltonians, on whether or not F has geometric invariances. Barring any such invariances for the moment, again we employ the basis (2.8.1), thus giving the timereversal operator the structure T = Z K.

(2.12.6)

The restriction imposed on F by T covariance can be found by considering the 2 × 2 block

2.12

Canonical Transformations for Floquet Operators

 Fmn = =

m|F|n

m|F|T n

35



T m|F|n T m|F|T n (0) f mn 1

+ f mn · τ

(2.12.7)

which must equal the corresponding block of T F † T −1 . In analogy with (2.8.8),     f mn = T F † T −1 mn = Z K F † K Z −1 mn   = − Z F˜ Z mn = −τ2 ˜f nm τ2   (0) (0) = −τ2 f nm 1 + f nm · τ˜ τ2 = f nm 1 − f nm · τ .

(2.12.8)

Now, the restrictions in question can be read off as (0) (0) f mn = f nm

f mn = − f nm .

(2.12.9)

They are identical in appearance to (2.8.11) but, in contrast to the amplitudes (μ) (μ) h mn , the f mn are in general complex numbers. The pertinent group of canonical transformation is the symplectic group defined by (2.8.12) since S F S −1 is T covariant if F is. Indeed, reasoning in parallel to (2.8.13), T S F S −1 T −1 = Z K S F S −1 K Z −1 = Z S ∗ K F K S˜ Z −1 = S Z K F K Z −1 S −1 = ST F T −1 S −1  † = S F † S −1 = S F S −1 .

(2.12.10)

To complete the classification of Floquet operators by their groups of canonical transformations, it remains to allow for geometric symmetries in addition to Kramers’ degeneracy. Since there is no difficulty in transcribing the considerations of Sect. 2.7, one can state without proof that the group in question is U (N ) when there is one parity Rx with [T, Rx ] = 0, Rx2 = −1, while additional geometric symmetries may reduce the group to O(N ), where the convention for N is the same as in Sect. 2.7. We conclude this section with a few examples of Floquet operators from different universality classes, all for kicked tops. These operators are functions of the angular momentum components Jx , Jy , Jz and thus entail the conservation law J 2 = 2 j( j + 1) with integer or half-integer j. The latter quantum number also defines the dimension of the matrix representation of F as (2 j + 1). The simplest top capable of classical chaos, already mentioned in (2.10.14), has the Floquet operator [11, 12] F = e−iλJz /(2 j+1) e−i p Jx / . 2

2

(2.12.11)

36

2 Time Reversal and Unitary Symmetries

Its dimensionless coupling constants p and λ may be said to describe a linear rotation and a nonlinear torsion. (For a more detailed discussion, see Sect. 7.6.) The quantum number j appears in the first unitary factor in (2.12.11) to give to the exponents of the two factors the same weight in the semiclassical limit j  1. This simplest top belongs to the orthogonal universality class: Its F operator is covariant with respect to generalized time reversal T = ei p Jx / eiπ Jy / T0

(2.12.12)

where T0 is the conventional time reversal. By diagonalizing F, the level spacing distribution has been shown [11, 12] to obey the dictate of the latter symmetry, i.e., to display linear level repulsion (Chap. 3) under conditions of classical chaos. An example of the unitary universality class is provided by

F = e−iλ Jy /(2 j+1) e−iλJz /(2 j+1) e−i p Jx / . 2

2

2

2

(2.12.13)

Indeed, the quadratic level repulsion characteristic of this class (Chap. 3) is obvious from Fig. 1.2c that was obtained [13, 14] by diagonalizing F for j = 500, p = 1.7, λ = 10, λ = 0.5. Finally, the Floquet operator F = e−iV e−iH0 , H0 = λ0 Jz2 /j2 , V =

λ1 Jz4 /j 3 4





+ λ2 (Jx Jz + Jz Jx ) / + λ3 Jx Jy + Jy Jx / 2

(2.12.14) 2

is designed so as to have no conserved quantity beyond J 2 (i.e., in particular, no geometric symmetry) but a time-reversal covariance with respect to T = e−iH0 T0 . Now, since T 2 = +1 and T 2 = −1 for j integer and half integer, respectively, the top in question may belong to either the orthogonal or the symplectic class. These alternatives are most strikingly displayed in Fig. 1.3b, d. Both graphs were obtained [13, 14] for λ0 = λ1 = 2.5, λ2 = 5, λ3 = 7.5, values which correspond to global classical chaos. The only parameter that differs in the two cases is the angular momentum quantum number: j = 500 (orthogonal class) for graph b and j = 499.5 (symplectic class) for d. The difference in the degree of level repulsion is obvious (Chap. 3). Such a strong reaction of the degree of level repulsion to a change as small as one part per thousand is really rather remarkable. No quantity with a well-defined classical limit could respond so dramatically.

2.13 Beyond Dyson’s Threefold Way We have been concerned with the orthogonal, unitary, and symplectic symmetry classes of Hamiltonians or Floquet operators. Dyson [15] deduced that classification

2.13

Beyond Dyson’s Threefold Way

37

from group theoretical arguments about complex Hermitian (or unitary) matrices. Much of the present book builds on Dyson’s scheme. During the nineties of the last century, work on the low-energy Dirac spectrum in chromodynamics [16] and on low-energy excitations in disordered superconductors has highlighted universal behavior not fitting Dyson’s threefold way. In particular, Zirnbauer and his collegues [17–21] have argued that seven further symmetry classes exist and pointed to realizations in solid-state physics. The new classification corresponds to one given by Cartan for symmetric spaces.2 However, the correspondence rests, in its present form, on the assumption of effective single-Fermion theories; there is thus room for further work on interactions as well as Bosonic particles. More recently, the topic of nonstandard symmetry classes has reemerged in the relation of d dimensional topological insulators/superconductors to Anderson localization in d − 1 dimensions [22, 23]. A short discussion of the new classes is in order. My aim is to introduce the reader to the essence of the new ideas without even trying to do justice to advanced solid-state topics or the underlying mathematics. Most importantly, why does Dyson’s scheme need extension? The following answer will be fully appreciated only with the help of elementary notions of level statistics to be developed in the following two chapters. A spectrum comprising many (possibly infinitely many) energy (or quasi-energy) levels can be characterized by a local density δδ NE with δ E just large enough to make for but small fluctuations of the ratio under shifts of the location on the energy axis by a few levels. Now if the spectrum affords a much larger range ΔE within which the local ratio undergoes small fluctuations about the mean but no systematic change, one can call the specas the mean density; the trum homogeneous over the range ΔE and work with ΔN ΔE latter mean can still vary systematically on yet larger energy scales. For systems with homogeneous spectra the Dyson scheme is complete. On the other hand, a spectrum is non-homogeneous when near some distinguished point on the energy axis the local density δδ NE displays systematic changes. For systems with non-homogeneous spectra additional symmetries yield symmetry classes beyond Dyson’s scheme. The seven new classes have energy spectra symmetric about a point on the energy axis which can be chosen as E = 0. Near that spectral center, the local level densities display systematic variations. Such spectra are known e.g., for superconductivity and relativistic Fermions.

2 A symmetric space is a Riemannian manifold M with global invariance under a distance preserving geodesic inversion (sign change of all normal coordinates reckoned from any point on M). The curvature tensor is then constant. The scalar curvature can be positive, negative, or zero. The positive-curvature case deserves special interest since the pertinent compact symmetric spaces house the unitary quantum evolution operators. The set of evolution operators (or unitary matrices, in a suitable irreducible representation) can be shown to form a symmetric space, where the matrix inversion U → U −1 yields geodesic inversion w.r.t. the identity as a distance preserving transformation (isometry), with Tr (U −1 dU )2 as the metric. For a discussion of the ten symmetry classes in terms of symmetric spaces see [20].

38

2 Time Reversal and Unitary Symmetries

2.13.1 Normal-Superconducting Hybrid Structures Four of the new universality classes are realizable in normal-superconducting hybrid structures, like a normal conductor of the form of a billiard with superconductors attached at the boundary. Chaos must be provided either by randomly placed scatterers or by the geometry of the sample. A prominent effect distinguishing such hybrid structures from all-normal electronic billiards is Andreev scattering [24]: An electron leaving the normal conductor to enter a superconductor may there combine with another electron of (nearly) opposite velocity to form a Cooper pair. A hole with velocity (nearly) opposite to the lost electron must then enter the normal conductor and retrace the path of the lost electron, a small angular mismatch apart which is due to the small energy mismatch  of a quasiparticle relative to the Fermi energy E F . Roughly speaking, an electron of energy  is scattered into a hole of energy −. The hole picks up a scattering phase π/2 − φ where φ is the phase of the superconducting order parameter at the interface. Due to Andreev scattering a non-vanishing Cooper-pair amplitude forms within the normal conductor, close to the interface with each superconductor, as the following rough argument indicates. When the hole “created” by Andreev scattering retraces the path of the original electron back to the interface with the same superconductor and becomes retroreflected as an electron again the coherent succession of two electrons appears like a Cooper pair. An observable consequence is a gap, called Andreev gap in the excitation spectrum. The simplest description of the many-electron problem arises in the meanfield approximation which yields an effective single-particle theory. The pertinent second-quantized Bardeen-Cooper-Schrieffer (BCS) Hamiltonian involves annihilation operators cα and creation operators cα† . The index α accounts for, say, N orbital single-particle states as well as two spin states such that α = 1, 2, . . . 2N . Electrons being Fermions these operators obey the anticommutation rules †

†

cα cβ + cβ cα = δαβ .

(2.13.1)

The Hamiltonian then reads

H=

 αβ

 1 1 † h αβ cα† cβ + Δαβ cα† cβ + Δ∗αβ cβ cα ; 2 2

(2.13.2)

herein the matrix h accounts for normal motion due to kinetic energy, single-particle potential, and possibly magnetic fields; the order-parameter matrix Δ brings in superconduction and coupling of electrons with holes. Hermiticity of H and Fermi statistics restrict the 2N × 2N -matrices h and Δ as h αβ = h ∗βα ,

Δαβ = −Δβα .

(2.13.3)

2.13

Beyond Dyson’s Threefold Way

39

It is convenient to write the BCS Hamiltonian as row×matrix×column,   1 †  h Δ c H = c ,c + const , (2.13.4) c† −Δ∗ −h˜ 2 with const = 12 Tr h, so as to associate the BCS-Hamitonian H with a Hermitian 4N × 4N matrix   h Δ (2.13.5) H= −Δ∗ −h˜ known as the Bogolyubov-deGennes (BdG) Hamiltonian. The “physical space” spanned by the orbital and spin states is thus enlarged by a two-dimensional “particle-hole space.”3 The restrictions (2.13.3) take the form4 

H = −Σx H∗ Σx ,

Σx =

01 10

 .

(2.13.6)

It is immediately clear, then, that if ψ is an eigenvector of H and ω the associated eigenvalue, Σx ψ ∗ is an eigenvector with eigenvalue −ω. The announced symmetry of the spectrum about E = 0 is thus manifest. One might view the restriction (2.13.6) as a “particle-hole symmetry” (PHS) since it reflects the easily checked invariance of the BCS Hamiltonian (2.13.2) under the interchange c ↔ c† of creation and annihilation operators, combined with com   † plex conjugation; note Σx cc† = cc . Moreover, that PHS could be associated with an antiunitary operator of charge conjugation, C = Σx K ,

C2 = 1 ,

CHC −1 = −H

⇐⇒

CH + HC = 0 . (2.13.7)

I would like to warn the reader, however, that the foregoing restriction of the BdG Hamiltonian is not a symmetry in the usual sense since H and C do not commute but anticommute. Imaginary Hermitian matrices (which must be odd under transposition) also have spectra symmetric about zero. In fact, every BdG Hamiltonian becomes imaginary when conjugated with the unitary 4N × 4N matrix 1 U=√ 2



1 1 i −i

 (2.13.8)

3 No extra states are introduced here even though the BdG jargon does invite such misunderstanding; the 2N single-electron states acted upon by the matrix h may have their energies above or below the Fermi energy; BdG jargon terms them “particle states” and even indulges in speaking ˜ the BdG hole states are in fact identical copies of about “hole states” acted upon by the matrix −h; the BdG particle states.

A cleaner notation would be Σx = σx ⊗ 12 ⊗ 1 N with σx the familiar Pauli matrix operating in particle-hole space, the second factor refering to spin space, and the third factor refering to orbital space.

4

40

2 Time Reversal and Unitary Symmetries

as is readily verified by doing the matrix multiplications in HU = U HU −1 = −HU∗ = −H˜ U .

(2.13.9)

The isospectral representative HU of H is diagonalized by an S O(4N ) matrix g, gHU g −1 = diag(ω1 , ω2 , . . . , ω2N , −ω1 , −ω2 , . . . , −ω2N ). The BdH Hamiltonian itself is then diagonalized by gU . Symmetry class D. In the absence of any symmetries beyond the particle-hole symmetry, BdG Hamiltonians form the new symmetry class D. The group S O(4N ) is the pertinent group of canonical transformations since conjugation of the imaginary representative HU of H with any S O(4N ) matrix yields a new version of the Hamiltonian with imaginary matrix elements and the same spectrum. The set of Hamiltonians spanning the new symmetry class is most naturally characterized by looking at the real anti-Hermitian matrices5 X U = iHU = X U∗ which form the Lie algebra so(4N ); their exponentials are orthogonal matrices forming the Lie group S O(4N ). AntiHermitian representatives X = iH of Hamiltonians will remain with us throughout the discussion of the new symmetry classes. A further word on them is thus in order in the simplest context of class D where no symmetries reign, beyond the particle-hole symmetry as expressed in (2.13.6) and (2.13.7) for H or − X † = X = −Σx X˜ Σx .

(2.13.10)

The algebra formed by the X = iH is isomorphic to that formed by the X U = iHU = X U∗ , so(4N ). The name “D” for the present class is chosen in respect to Cartan. Symmetry class DIII. I proceed to BdG Hamiltonians enjoying time reversal invariance, [T, H] = 0. Since an effective single-electron theory is at issue the time reversal operator (2.3.9) must be employed, T = iσ y K ,

(2.13.11)

where the 2 × 2 matrix σ y operates in spin space; mustering more cleanliness of notation I write T = 12 ⊗ iσ y ⊗ 1 N K ≡ τ K . The time reversal invariance of the Hamiltonian can be noted as   iσ y 0 . (2.13.12) H = τ H∗ τ −1 , τ = 0 iσ y Kramers’ degeneracy arises as a further property of the spectrum since T 2 = −1. To characterize the class of Hamiltonians so restricted it is again convenient to argue with the antiHermitian matrices X = iH = −X † , simply because these form 5

Note that the imaginary Hermitian matrices do not form a closed algebra under commutation since the commutator of any two such is antiHermitian.

2.13

Beyond Dyson’s Threefold Way

41

a closed algebra under commutation; for class D, that algebra was just revealed as (isomorphic to) so(4N ). The restriction (2.13.6) of H due to Hermiticity and Fermi statistics and the property (2.13.12) due to time reversal invariance translate into the following restrictions of the representative X = −X † of H, −X † = X = −Σx X˜ Σx X = τ X˜ τ −1 .

and (2.13.13)

In search is the set P of solutions of the latter conditions within so(4N ). That set does not close under commutation. Indeed, for two members of P we have6 [X 1 , X 2 ] = τ [ X˜ 1 , X˜ 2 ]τ −1 = −τ [X 1 , X 2 ]T τ −1 , with the minus sign signalling disobedience of the commutator to (2.13.13). We can, however, easily identify an auxiliary set K, complementary to P in so(4N ), such that K does from a subalgebra of so(4N ). I define the members Y of K by replacing the condition (2.13.13) with the complementary one, Y = −τ Y˜ τ −1 . Indeed, the set K does close under commutation and therefore forms a subalgebra of so(4N ). The complementarity in play is owed to the fact that the condition (2.13.13) involves an involution, X → I (X ) with I (I (X )) = X . The so(4N ) matrices can be chosen such that each of them is either even or odd under I ; the even ones are the X ’s forming P while the odd ones are the Y ’s forming K. We may write so(4N ) = P + K and hold fast to K being a subalgebra of so(4N ). The equations for K can be rewritten as − Y † = Y = −Σx Y˜ Σx = +(Σx τ )Y (Σx τ )−1 .

(2.13.14)

The conjugation U −1 Y U ≡ YU with the unitary matrix   1 1 iσ y U=√ (2.13.15) 2 σ y −i   iσ y (in cleaner notation, U = √12 σ12y −i1 ⊗ 1 N ) turns the equations for K into 2 †

−1  − YU = YU = −Σx Y U Σx = +Σz YU Σz

(2.13.16)

where Σz = σz ⊗ 12 ⊗ 1 N with the Pauli matrix σz acting in particle-hole space. to commute with The set of solutions is now easy to ascertain. Since YU is required  Z pp 0 Σz it must be diagonal in particle-hole space, YU = 0 Z hh . The condition YU =   −Σx Y U Σx relates the two 2N ×2N blocks as Z hh = − Z pp , and the antiHermiticity of YU carries over to both blocks. Scratching off indices I write YU = diag(Z , − Z˜ ) and can recognize K as isomorphic to the Lie algebra of antiHermitian 2N × 2N 6 Matrix transposition is mostly denoted by a tilde but typographical reasons occasionally suggest to employ a right superscript “T ”.

42

2 Time Reversal and Unitary Symmetries

matrices, K  u(2N ). The space P of (the antiHermitian representatives X = iH of) time reversal invariant BdG Hamiltonians is obtained from so(4N ) by removing a u(2N ) algebra. Being the complement of u(2N ) in so(4N ), the space P can be seen as the tangent space of the quotient S O(4N )/U (2N ) of the corresponding Lie groups. The latter quotient is the symmetric space termed DIII by Cartan; hence the name DIII for the symmetry class of time reversal invariant BdG Hamiltonians. Symmetry class C. Next come BdG Hamiltonians without time reversal symmetry but with isotropy in spin space. The generators of spin rotations Jk (k = x, y, z) then commute with the Hamiltonian. To write out the familiar second-quantization form of these generators I must split the double index into an orbital and a spin part, α = qs with q = 1, 2, . . . , N and s =↑, ↓; the spin label ↑ indicates the eigenvalue + 2 for the z component of the spin angular momentum. The angular momenta take  † the form Jk = 2 qss cqs σkss cqs where σk , k = x, y, z are the Pauli matrices. In line with the “row×matrix×column” notation (2.13.5) for the BdG Hamiltonian I represent the angular momenta by the 4N × 4N matrices Jk =

 4



σk 0 0 − σk

 ⊗ 1N .

(2.13.17)

The spin isotropy for the antiHermitian representative X = iH of the BdG Hamiltonian, [X, Jk ] = 0, is easily seen to restrict the four 2N × 2N blocks of X in particle-hole space as X pp = ih = 12 ⊗ a , X hp = −iΔ∗ = −iσ y ⊗ c ,

X ph = iΔ = iσ y ⊗ b , X hh = −ih˜ = −12 ⊗ a˜ .

(2.13.18)

In terms of N × N (orbital) blocks the generator X reads ⎛ a ⎜0 X =⎜ ⎝0 c

0 a −c 0

0 −b −a˜ 0

⎞ b 0⎟ ⎟ 0⎠ −a˜

(2.13.19)

and decomposes into two commuting subblocks. One (the outer one) corresponds to spin-up particles and spin-down holes, the other (the inner one) to spin-down particles and spin-up holes. Because the two subblocks are uniquely related by b → −b, c → −c it suffices to focus on one of them, say X↑ =

a b  . c −a˜

(2.13.20)

It remains to reveal the restrictions (of AntiHermiticity and Fermi statistics)  (2.13.10) for the new subblock X ↑ . Since X ph = − X ph the equation X pp = iσ y ⊗ b ˜ Similar reasoning yields c = c˜ . AntiHermiticity requires a = −a † entails b = b. and c = −b† . All these conditions are summarized by

2.13

Beyond Dyson’s Threefold Way

43

†

− X ↑ = X ↑ = −Σ y X˜ ↑ Σ y ,

Σy = σy ⊗ 1N .

(2.13.21)

This is the defining equation of the Lie algebra sp(2N ) of which the X ↑ ’s under scrutiny thus turn out to be elements. The reader is kindly invited to crosscheck with the definition (2.8.12) of the symplectic group whose elements are the exponentials of the elements of the symplectic algebra. Of course, the canonical transformations for the present symmetry class are matrices from Sp(2N ) and diagonalization of X ↑ can be achieved by such a matrix. In line with Cartan’s notation for symmetric spaces the present symmetry class is called C. I had played with characterizing the conditions of Hermiticity and Fermi statistics for BdG Hamiltonians as the behavior HC + CH = 0 under charge conjugation, see (2.13.7). Indulging a bit further, I refine the definition of charge conjugation for the subblock H↑ = −iX ↑ . The above condition (2.13.10) demands H↑ C↑ + C↑ H↑ = 0 with C↑ = Σ y K and C↑2 = −1. A corresponding definition can be made for the spin-down subblock, such that the overall charge conjugation C = C↑ ⊗ C↓ squares to minus one for class C. Symmetry class CI. The final class of BdG Hamiltonians is distinguished by invariance under both spin rotation and time reversal. Subjecting the representation (2.13.19) of the generator X in terms of orbital blocks to the further restriction (2.13.13) due to time reversal invariance we easily find that X becomes a symmetric matrix, X = X˜ . That symmetry carries over to the two commuting subblocks, X ↑ = X˜↑ . The universality class under consideration is thus formed by the set P of symmetric matrices in sp(2N ). Calling K the subalgebra of antisymmetric matrices in sp(2N ) we have P = sp(2N ) − K. I propose to show that K is isomorphic to the Lie algebra u(N ). To that end I note that the solutions Y ∈ K of −Y † = Y = −Σ y Y˜ Σ y = −Y˜ have the form 12 ⊗ ReA + iσ y ⊗ ImA ≡ Y ( A) where A is an arbitrary antiHermitian N × N matrix, i.e., A ∈ u(N ). The function Y ( A), which maps a u(N ) matrix A to an antisymmetric matrix Y in sp(2N ), preserves the operation defining Lie algebras, commutation. Indeed, with two u(N ) matrices A1 and A2 we have Y ([A1 , A2 ]) = [Y ( A1 ), Y ( A2 )] and the latter commutator is again an antisymmetric sp(2N ) matrix The isomorphism K  u(N ) is thus established. The complement P of u(N ) in sp(2N ) can be regarded as the tangent space of the coset space Sp(2N )/U (N ) which is a symmetric space of type CI a` la Cartan, hence the name for the universality class of BdG Hamiltonians enjoying invariance under time reversal and spin rotation. No change relative to the class C arises for the antiunitary charge conjugation operator C; in particular, we still have C 2 = −1.

2.13.2 Systems with Chiral Symmetry The remaining three of the seven new symmetry classes have Hamiltonians (or, for relativistic electrons, Dirac operators) affording the block representation  H=

0 Z Z† 0

 ,

(2.13.22)

44

2 Time Reversal and Unitary Symmetries

due to a “symmetry” of the form H = −P H P −1 ,

P P† = 1 ,

P2 = 1 .

(2.13.23)

No conserverved quantity comes with that “symmetry” since P anticommutes with H . The latter anticommutativity entails an energy spectrum symmetric about zero, as the particle-hole symmetry does for BdG Hamiltonians. In contrast to the antiunitary charge conjugation operator (2.13.7) associated with the particle-hole symmetry, the operator P now involved is unitary. In complete analogy with the standard symmetry classes, the ensemble of chiral Hamiltonians without any further restrictions is called the chiral unitary class (in Cartan notation, AIII). By imposing time reversal invariance for half-integer spin, one gets the chiral symplectic class (CII, in Cartan notation). Finally, the presence ˜ of both time reversal invariance and full spin rotation invariance enforces H = H and defines the chiral orthogonal class (B DI, in Cartan notation). A solid-state realization of the chiral unitary class AIII is a disordered tightbinding model on a bipartite lattice with broken time reversal invariance, such as the random flux problem [25]. Moreover, the class AIII arises from BdG Hamiltonians with invariance under time reversal and spin rotation about the z axis in spin space [23]. For further solid-state applications of the chiral classes see [24]. Applications in chromodynamics have been discussed by Verbaarschot [16].

2.14 Problems 2.1 Consider a particle with the Hamiltonian H = ( p − (e/c) A)2 /2m + V (|x|) where the vector potential A represents a magnetic field B constant in space and time. Show that the motion is invariant under a nonconventional time reversal which is the product of conventional time reversal with a rotation by π about an axis perpendicular to B. Give the general condition for V (x) necessary for the given nonconventional T to commute with H . 2.2 Generalize the statement in Problem 2.1 to N particles with isotropic pair interactions. 2.3 Show that K x K −1 = x, K pK −1 = − p, and K L K −1 = −L, where L = x × p is the orbital angular momentum and K the complex conjugation defined with respect to the position representation. 2.4 (a) Show that antiunitary implies antilinearity. (b) Show that antilinearity and |K ψ|K φ|2 = |ψ|φ|2 together imply the antiunitarity of K . $ 2.5 Show that Uμν = Uνμ = d xμ|x ν|x, U † = U −1 , for K = U K˜ where K and K˜ are the complex conjugation operations in the continuous basis |x and the discrete basis |μ, respectively.

References

45

2.6 Show that for spin-1 particles, time reversal can be simply complex conjugation. 2.7 Show that the unitary matrix U in T = U K is symmetric or antisymmetric when T squares to unity or minus unity, respectively. 2.8 Show that φ|ψ = T φ|T ψ∗ for T = U K with U † = U −1 . 2.9 Use the associative law T T 2 = T 2 T for the antilinear operator of time reversal to show that the unimodular number α in T 2 = α must equal ±1. 2.10 Show that time-reversal invariance with T 2 = −1 together with full isotropy implies that the canonical transformations are given by the orthogonal transformations. 2.11 Find the group of canonical transformations for a Hamiltonian obeying [T, H ] = 0, T 2 = −1 and having cylindrical symmetry. 2.12 Show that the symplectic matrices S defined by S Z S˜ = Z form a group. 2.13 Let H0 and V commute with an antiunitary operator T. Show that T F T −1 = F † with F = e−H0 τ/2 e−ikV / e−iH0 τ/2 . 2.14 What would be the analogue of H (t) = T H (−t)T −1 if the Floquet operator were to commute with some T0 ? 2.15 Show that the eigenvectors of unitary operators are mutually orthogonal. 2.16 Show that U (N ) is canonical for Floquet operators without any T covariance. 2.17 Show that U (N ) ⊗ U (N ) is canonical for Floquet operators with T F T −1 = F † , T 2 = −1, [Rx , F] = 0, [T, Rx ] = 0, Rx2 = −1. 2.18 Show that O(N )⊗ O(N ) is canonical if, in addition to the symmetries in Problem 2.16, there is another parity R y commuting with F and T but anticommuting with Rx . 2.19 Give the group of canonical transformations for Floquet operators in situations of full isotropy.

References 1. E.P. Wigner: Group Theory and Its Applications to the Quantum Mechanics of Atomic Spectra (Academic, New York, 1959) 2. C.E. Porter (ed.): Statistical Theories of Spectra (Academic, New York, 1965)

46

2 Time Reversal and Unitary Symmetries

3. M.L. Mehta: Random Matrices (Academic, New York 1967; 2nd edition 1991; 3rd edition Elsevier 2004) 4. D. Delande, J.C. Gay: Phys. Rev. Lett. 57, 2006 (1986) 5. G. Wunner, U. Woelck, I. Zech, G. Zeller, T. Ertl, F. Geyer, W. Schweitzer, H. Ruder: Phys. Rev. Lett. 57, 3261 (1986) 6. T.H. Seligman, J.J.T. Verbaarschot: Phys. Lett. 108A, 183 (1985) 7. G. Casati, B.V. Chirikov, D.L. Shepelyansky, I. Guarneri: Phys. Rep. 154, 77 (1987) 8. G. Casati, B.V. Chirikov, F.M. Izrailev, J. Ford: In G. Casati, J. Ford (eds.) Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Lecture Notes in Physics, Vol. 93 (Springer, Berlin, Heidelberg, 1979) 9. B.V. Chirikov: Preprint no. 267, Inst. Nucl. Physics Novosibirsk (1969); Phys. Rep. 52, 263 (1979) 10. S. Chaudhury, A. Smith, B.E. Anderson, S. Ghose, P.S. Jessen: Nature 461, 768 (2009) 11. F. Haake, M. Ku´s, R. Scharf: Z. Physik B65, 381 (1987) 12. M. Ku´s, R. Scharf, F. Haake: Z. Physic B66, 129 (1987) 13. F. Haake, M. Ku´s, R. Scharf: In F. Ehlotzky (ed.) Fundamentals of Quantum Optics II, Lecture Notes in Physics Vol. 282 (Springer, Berlin, Heidelberg, 1987) 14. R. Scharf, B. Dietz, M. Ku´s, F. Haake, M.V. Berry: Europhys. Lett. 5, 383 (1988) 15. F. Dyson: J. Math. Phys. 3, 1199 (1962) 16. J. Verbaarschot: Phys. Rev. Lett. 72, 2531 (1994) 17. M.R. Zirnbauer: J. Math. Phys. 37, 4986 (1996) 18. A. Altland, M.R. Zirnbauer: Phys. Rev. B55, 1142 (1997) 19. A. Altland, B.D. Simons, M.R. Zirnbauer: Phys. Rep. 359 283 (2002) 20. M.R. Zirnbauer: arXiv:math-ph/0404058v1 25 Apr 2004 21. P. Heinzner, A. Huckleberry, M. Zirnbauer: Commun. Math. Phys. 257, 725 (2005) 22. A.P. Schnyder, S. Ryu, A. Furusaki, A.W.W. Ludwig: Phys. Rev. B 78, 195125 (2008) 23. A.P. Schnyder, S. Ryu, A. Furusaki, A.W.W. Ludwig: arXiv:0905v1 [cond-mat.mes-hall] 13 May 2009 24. B.D. Simons, A. Altland: Theories of Mesoscopic Physics CRM Series in Mathematical Physics (Springer, 2001) 25. A. Furusaki: Phys. Rev. Lett. 82, 604 (1999), and references therein

Chapter 3

Level Repulsion

3.1 Preliminaries In the previous chapter, I classified Hamiltonians H and Floquet operators F by their groups of canonical transformations. Now I propose to show that orthogonal, unitary, and symplectic canonical transformations correspond to level repulsion of, respectively, linear, quadratic, and quartic degree [1, 2]. The different canonical groups are thus interesting not only from a mathematical point of view but also have distinct measurable consequences. It is a fascinating feature of quantum mechanics that different behavior under time reversal actually becomes observable experimentally. Resistance of levels to crossings is a generic property of Hamiltonians and Floquet operators just as nonintegrability is typical for classical Hamiltonian systems with more than one degree of freedom. (In fact, as will become clear in Chap. 5, Hamiltonians with integrable classical limits and more than one degree of freedom do not display level repulsion.) Therefore, knowing that the degree of level repulsion is 1, 2, or 4 for some system implies nothing more than (1) some information about the symmetries and (2) that the system is classically nonintegrable. Conversely, the universality of spectral fluctuations calls for explanation. To reveal the phenomenon of repulsion, the levels of H or F must be divided into multiplets; each such multiplet has fixed values for all observables except H or

Fig. 3.1 Level crossings between different Zeeman multiplets

F. Haake, Quantum Signatures of Chaos, Springer Series in Synergetics, 3rd ed., C Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-05428-0 3, 

47

48

3 Level Repulsion

Fig. 3.2 Dependence of the quasi-energies of a kicked top on some control parameter λ. For λ < ∼ 3, the classical motion is predominantly regular, whereas classical chaos prevails for λ > 3. Note that revel repulsion is much more pronounced in the classically chaotic range

F from the corresponding complete set of conserved quantities. Levels belonging to different multiplets have no inhibition to cross when a parameter in H or F is varied. Such intermultiplet crossings are in fact well known, e.g., from the Zeeman levels pertaining to multiplets of different values of the total angular momentum quantum number j (Fig. 3.1). The levels within one multiplet typically avoid crossings, as illustrated in Fig. 3.2 for a kicked top. A simple explanation of level repulsion was given by von Neumann and Wigner in 1929 [3]. In fact, the present chapter will expound and generalize what is nowadays often referred to as the von Neumann–Wigner theorem.

3.2 Symmetric Versus Nonsymmetric H or F When two levels undergo a close encounter upon variation of a parameter in a Hamiltonian, their fate can be studied by nearly degenerate perturbation theory. Assuming that each of the two levels is nondegenerate, one may deal with a twodimensional Hilbert space spanned by approximants |1, |2 to the corresponding eigenvectors. By diagonalizing the 2 × 2 matrix  H=

 H11 H12 , ∗ H12 H22

(3.2.1)

one obtains the approximate eigenvalues E± =

1 2

(H11 + H22 ) ±

%

1 4

(H11 − H22 )2 + |H12 |2 .

(3.2.2)

An important difference between Hamiltonians with unitary and with orthogonal canonical transformations becomes manifest at this point. In the first case, the Hamiltonian (3.2.1) is complex. The discriminant in (3.2.2), D=

1 4

(H11 − H22 )2 + (Re {H12 })2 + (Im {H12 })2 ,

(3.2.3)

3.2

Symmetric Versus Nonsymmetric H or F

49

Fig. 3.3 Hyperbolic form of a typical avoided crossing of two levels

is thus the sum of three nonnegative terms. When orthogonal transformations are canonical, on the other hand, the matrix (3.2.1) is real symmetric, Im {H12 } = 0, whereupon the discriminant D has only two nonnegative contributions. Clearly, by varying a single parameter in the Hamiltonian, the discriminant and thus the level spacing |E + − E − | can in general be minimized but not made to vanish (Fig. 3.3). To make a level crossing, E + = E − , a generic possibility rather than an unlikely exception, three parameters must be controllable when unitary transformations apply, whereas two suffice in the orthogonal case. The resistance of levels to crossings should therefore be greater for complex Hermitian Hamiltonians than for real symmetric ones. The above argument can likewise be applied to Floquet operators. The most general unitary 2 × 2 matrix with unit determinant reads  F=

α + iβ

γ + iδ

−γ + iδ α − iβ

 , with

α 2 + β 2 + γ 2 + δ 2 = 1.

(3.2.4)

Apart from a common phase factor, which can be set equal to unity by a proper choice of the reference point of the eigenphases of F, the four parameters α, β, γ , δ are real. Three of these parameters, say β, γ , and δ, may be taken as independent. The unimodular eigenvalues of F are & e± = α ± i β 2 + γ 2 + δ 2 .

(3.2.5)

Again, the number of nonnegative additive contributions to the discriminant is generally three, unless the additional condition that F be symmetric, γ = 0, reduces that number to two. One might argue (correctly!) that due to the unitarity condition, α 2 + β 2 + γ 2 + 2 δ = 1, it suffices to control the single parameter 1 − α 2 . However, control over

50

3 Level Repulsion

β 2 + γ 2 + δ 2 allows one to achieve E + = E − in the Hamiltonian case as well. What counts in this argument is that the discriminants in (3.2.2) and (3.2.5) can be controlled by the same number of independent parameters. I shall refer to this number as the codimension of a level crossing. To summarize, both for Hamiltonian and Floquet operators, the number n of controllable parameters necessary to enforce a level crossing is three or two depending on whether H and F are unrestricted (beyond Hermiticity and unitarity, respectively) or are symmetric matrices. These two possibilities correspond to the alternatives of unitary and orthogonal canonical transformation groups:  n=

2 orthogonal . 3 unitary

(3.2.6)

It remains to be shown that the codimension n of a level crossing yields the degree of level repulsion as n − 1 (Sect. 3.4).

3.3 Kramers’ Degeneracy When there is an invariance of H or a covariance of F, T H T −1 = H or T F T −1 = F † ,

(3.3.1)

under some antiunitary time-reversal transformation that squares to minus unity, T 2 = −1,

(3.3.2)

each level is doubly degenerate. The close encounter of two levels must therefore be discussed for a four-dimensional Hilbert space [4]. Four approximants to the eigenvectors can be chosen and ordered as in (2.8.1), i.e., as |1, T |1, |2, T |2. The 4 × 4 matrix H or F can then be represented so that it consists of four 2 × 2 blocks h mn or f mn ; see (2.8.5) and (2.12.7). The symmetry (3.3.1) and /μ) (μ) (3.3.2) yields the restrictions (2.8.11) and (2.12.9) for the amplitudes h mn and f mn , respectively. Barring, for the moment, any geometric symmetry, there are no further restrictions. In a slightly less belligerent notation, the corresponding 4 × 4 Hamiltonian reads ⎛ ⎞ α+β 0 γ − iσ −ε − iδ ⎜ 0 α+β ε − iδ γ + iσ ⎟ ⎟ H =⎜ (3.3.3) ⎝ γ + iσ ε + iδ α−β 0 ⎠ −ε + iδ γ − iσ 0 α−β with six real parameters α, β, γ , δ, ε, σ. Rather than invoking (2.8.11), it is of course also possible to verify the structure (3.3.3) directly by using T 2 = −1 and the ordering of the four basis vectors given above. For instance,

3.3

Kramers’ Degeneracy

51

 1|H |T 1 = 1|T H T −1 |T 1 = 1|T H |1 = −T 1|H |1∗ = −1|H |T 1 = 0.

(3.3.4)

Since T 2 = −1 the quartic secular equation of (3.3.3) is biquadratic and yields the two double roots & (3.3.5) E ± = α ± β 2 + γ 2 + δ 2 + ε2 + σ 2 . Similarly, the 4 × 4 Floquet matrix F also has the structure (3.3.3), but by virtue of its unitarity, the five parameters iβ, iγ , iδ, iε, iσ become real, and α remains real; actually, α may be considered to be given in terms of the other five parameters since unitarity requires α 2 + (iβ)2 + (iγ )2 + (iδ)2 + (iε)2 + (iσ )2 = 1.

(3.3.6)

As in (3.2.4), I have chosen a phase factor common to all elements of F so as to make the two unimodular eigenvalues & e± = α ± i (iβ)2 + (iγ )2 + (iδ)2 + (iε)2 + (iσ )2

(3.3.7)

complex conjugates of one another. For both H and F discriminants appear in the eigenvalues (3.3.5) and (3.3.7) which have five nonnegative additive contributions. The codimension of a level crossing is thus n=5

symplectic.

(3.3.8)

As for the orthogonal and unitary cases, it will become clear presently that the value n = 5 is larger by one than the degree of the level repulsion characteristic of Hamiltonian and Floquet operators whose group of canonical transformations is symplectic. It is instructive to check that the number n reduces to 3 and 2 when geometric symmetries are imposed so as to break H or F down to block diagonal form with, respectively, U (2) and O(2) as canonical transformations of the 2 × 2 blocks on the diagonal. For instance, by assuming a parity R x with the properties (2.7.1), one may choose the four basis vectors as parity eigenstates such that Rx |1 = −i|1, Rx |T 2 = −i|T 2 Rx |2 = i|2, Rx |T 1 = i|T 1. It follows that 1|H |2 = 1|Rx H Rx−1 |2 = 1|Rx H Rx |2 = −1|H |2 = 0,

(3.3.9)

52

3 Level Repulsion

i.e., γ = σ = 0, but there are no further restrictions and so n = 3. Alternatively, by rearranging the order of the basis states to |1, |T 2, |2, |T 1, the matrix H indeed becomes block diagonal ⎛

α+β ⎜−ε + iδ H =⎜ ⎝ 0 0

−ε − iδ α−β 0 0

0 0 α−β ε − iδ

⎞ 0 0 ⎟ ⎟. ε + iδ ⎠ α+β

(3.3.10)

Both of the nonzero blocks have fixed parity. Time reversal connects the two blocks but has no consequences within a block. Clearly, then, U (2) is the canonical group, and n = 3 for each block. An additional parity R y obeying (2.7.7) allows one to endow the basis with R y |1 = |T 1, R y |T 1 = −|1 R y |2 = |T 2, R y |T 2 = −|2

(3.3.11)

whereupon 1|H |T 2 = −T 1|H |2, i.e., δ = 0. The matrix H becomes real, the canonical group of the 2 × 2 blocks is O(2), and the index n is reduced to 2.

3.4 Universality Classes of Level Repulsion It is intuitively clear that an accidental level crossing becomes progressively less likely, the larger the number of parameters in H or F necessary to enforce a degeneracy. This number of parameters, or the “codimension of a level crossing”, it was found in the preceding sections, takes on values characteristic of the group of canonical transformations, ⎧ ⎪ ⎨2 orthogonal n = 3 unitary ⎪ ⎩ 5 symplectic.

(3.4.1)

Now I shall show that the degree of level repulsion expressed by the level spacing distribution P(S) is determined by n according to P(S) ∝ S n−1 = S β for S → 0.

(3.4.2)

Linear, quadratic, and quartic level repulsion is thus typical for Hamiltonians and Floquet operators with, respectively, orthogonal, unitary, and symplectic canonical

3.4

Universality Classes of Level Repulsion

53

transformations. Following common practice, I have introduced the exponent of level repulsion as β = n − 1. The limiting behavior (3.4.2) follows from the following elementary argument [20, 21]. The spacing distribution for a given spectrum can be defined as P(S) = δ(S − ΔE)

(3.4.3)

where ΔE stands for a distance between neighboring levels and the angular brackets  . . .  mean an average over all ΔE. For S → 0, i.e., for S smaller than the average separation, the level pairs contributing to P(S) correspond to encounters sufficiently close for ΔE to be representable by nearly degenerate perturbation theory. In a unified notation, such spacings can be written as ΔE =

√

x2 =

%

x12 + . . . + xn2 .

(3.4.4)

Here ΔE may refer to energies, as in (3.2.2) and (3.3.5), or to quasi-energies. Of course, the level pairs contributing to P(S) for a fixed small value of S will generally possess different values of the n parameters x. The average over the level spectrum can therefore be understood as an average over x with a suitable weight W (x). The latter function may not be easy to obtain, but fortunately its precise form does not matter for the asymptotic behavior of  P(S) =

 √  d n x W (x)δ S − x 2 .

(3.4.5)

By simply rescaling the n integration variables x → Sx, P(S) = S

β



 √  d n x W (Sx)δ 1 − x 2 .

(3.4.6)

The power law (3.4.2) thus applies, provided that the weight W (x) is neither zero nor infinite at x = 0. A finite nonvanishing value of W (0) must, however, be considered generic since the dictates of symmetries are already accounted for in the codimension n = β + 1 of a level crossing. The full probability distribution P(S) in 0 ≤ S < ∞ is not as easily found as its asymptotic form for small S. It will be shown below that P(S) can be constructed on the basis of the theory of random matrices or by using methods from equilibrium statistical mechanics. Trying to establish the spacing distribution P(S) and the degree of level repulsion for some dynamical system is a sensible undertaking only if the number of levels is large: Only in that “semiclassical” situation can a smooth histogram be built up even for small S where P(S) itself is small; but then indeed P(S) becomes “selfaveraging” for a single spectrum.

54

3 Level Repulsion

3.5 Nonstandard Symmetry Classes The seven new symmetry classes introduced at the end of the previous chapter exhibit a somewhat richer variety of level repulsion. The reader will recall that due to either particle-hole symmetry (PHS) or chiral symmetry the spectra are symmetric about E = 0. Therefore, repulsion between nearest-neighbor pairs of levels can arise not only for (E i , E j ; j = i ± 1) but also between E i and its “image” −E i . As I shall show presently, power laws with integer exponents reign,  P(S) ∝

Sβ Sα

for for

S ∝ |E i − E j | S ∝ |E i | ,

i = j

(3.5.1)

and the exponents α, β are again fixed by the codimension of the pertinent level crossings. I shall confine the discussion to the four classes of Bogolyubov-deGennes Hamiltonians. For the three chiral classes see [5]. Class D. Repulsion between a level E i and its PHS image −E i has no analogue in the Dyson classes and thus deserves prime attention. It is actually absent for Hamiltonians from class D which have no symmetry beyond PHS, α=0

for class D .

(3.5.2)

To check this I need to recall that the particle-hole symmetry means H = − Σx H∗ Σx . Focussing on the smallest positive eigenvalue E, its image −E, and the pertinent eigenvectors ψ and Σx ψ ∗ , all referring to some “unperturbed” Hamiltonian, I consider a perturbed Hamiltonian H ∈ D. In the two dimensional space spanned by ψ, Σx ψ ∗ the perturbed H reads 

ψ|H|ψ

ψ|H|Σx ψ ∗ 



Σx ψ ∗ |H|ψ ∗  Σx ψ ∗ |H|Σx ψ ∗ 

=



 ψ|H|ψ

0

0

−ψ|H|ψ

,

(3.5.3)

due to the particle-hole symmetry. A crossing of the two levels ±E can be reached by controlling the single real parameter ψ|H|ψ, and the vanishing value of the pertinent repulsion exponent is ascertained. To determine the degree of repulsion for a pair (E i , ψi ; E j , ψ j ; i = j) I represent a perturbed Hamiltonian H by the 2 × 2 matrix   ψi |H|ψi  ψi |H|ψ j  ψ j |H|ψi  ψ j |H|ψ j 

(3.5.4)

which is unrestricted by PHS. As for Dyson’s unitary class I so obtain β=2

for class D .

(3.5.5)

3.5

Nonstandard Symmetry Classes

55

Class DIII. Time reversal invariance and Kramers degeneracy come into play. To get the exponent α I must deal with four states ψ, T ψ and Σx ψ ∗ , T Σx ψ ∗ , the first two pertaining to an unperturbed eigenvalue E i and the last two to the PHS image −E i . In that Hilbert space a perturbed Hamiltonian faithful to both time reversal invariance and particle-hole symmetry is easily seen to act as a real symmetric matrix determined by two real parameters, ⎛

a ⎜0 H=⎜ ⎝0 b

0 a −b 0

0 −b −a 0

⎞ b 0 ⎟ ⎟. 0 ⎠ −a

(3.5.6)

√ The two levels are thus shifted to E = ± a 2 + b2 . With the codimension of the level crossing thus established as 2 I infer the repulsion exponent α=1

for DIII .

(3.5.7)

On the other hand, the exponent β reflects the fate of two levels E i , E j , i = j under a perturbation. Of relevance now is not the most general perturbation allowed by PHS and TRI; rather, all couplings of the four states ψi , T ψi , ψ j , T ψ j to their four PHS partners can be dropped since these couplings are specific to the repulsion inside PHS pairs of levels as described by the exponent α. The remaining perturbed Hamiltonian matrix is thus block diagonal, and the two 4 × 4 blocks are related to one another by PHS. It therefore suffices to consider the block refering to the four states ψi , T ψi , ψ j , T ψ j . The restrictions of TRI inside such a block have already been studied above for Dyson’s symplectic class, see (3.3.3), and the exponent β of that class thus carries over, β=4

for DIII .

(3.5.8)

Class C. Bogolyubov-deGennes Hamiltonians enjoying spin isotropy as their only symmetry are at issue here. All levels thus have a trivial twofold spin degeneracy. As already ascertained in Eq. (2.13.19) of Sect. 2.13.1, the allowed Hamiltonians consist of two commuting blocks and can be denoted by ⎛

a ⎜0 H=⎜ ⎝0 b†

0 a −b† 0

0 −b −a˜ 0

⎞ b 0 ⎟ ⎟. 0 ⎠ −a˜

(3.5.9)

with symmetric b. To get the repulsion exponent α for PHS pairs ±E it suffices to consider orbital state, N = &1, and diagonalize the resulting outer 2 × 2   absingle . The eigenvalues are ± a 2 + |b|2 and exhibit codimension 3 of their block ba∗ −a crossing such that we confront

56

3 Level Repulsion

α=2

for C .

(3.5.10)

For the exponent β I can still work with the outer block of H but must allow for two orbital states, N = 2, in order to allow for a pair of levels not just differing in sign. On the other hand, all couplings between the resulting two PHS pairs, i.e., all BCS type pairing amplitudes in b, b† can be dispensed with. The influence of the remaining perturbation on the pairI am after can thus be determined from the a12  = 0 as E = 12 (a11 + a22 ) ± 12 [(a11 − a22 )2 + quadratic secular equation  a11a−E ∗ a22 −E 12

4|a12 |2 ]1/2 . Again, the crossing has codimension three and the repulsion exponent comes out as β=2

for C .

(3.5.11)

Class CI. Time reversal invariance as well as spin isotropy now reign. The most general Hamiltonian of class C, given in (3.5.9), therefore reduces to ⎛

a ⎜0 H=⎜ ⎝0 b

0 a −b 0

0 −b −a 0

⎞ b 0 ⎟ ⎟. 0 ⎠ −a

(3.5.12)

with real symmetric N × N matrices a and √ b. The quadratic secular equations met with for class C thus entail solutions ± a 2 + b2 for the PHS pair determining α 2 1/2 ] for the pair determining β, with excluand 12 (a11 + a22 ) ± 12 [(a11 − a22 )2 + 4a12 sively real matrix elements. The codimension of crossings is two in both cases and therefore I get the exponents α=β=1

for CI .

(3.5.13)

3.6 Experimental Observation of Level Repulsion Systematic statistical analyses of complex energy spectra first became popular among nuclear physicists half a century ago. Upon being sorted into histograms, the spacings between highly excited neighboring levels of nuclei revealed linear level repulsion and thus time-reversal invariance of the strong interaction mainly responsible for nuclear structure [1, 2, 6]. While the electroweak interaction does not enjoy that symmetry, it is far too weak to be detectable in spectral fluctuations on the scale of a mean level spacing in complex nuclei. There are rather fewer experimental verifications of linear level repulsion in electronic spectra. Evidence was found for NO2 molecules [7] and Rb atoms [8]. While the nuclear, atomic, and molecular data just mentioned are now generally accepted as quantum manifestations of chaos, the somewhat less difficult to observe linear repulsion between eigenfrequences of microwave resonators with sufficiently irregular shape [9–14] requires explanation in the framework of classical wave theory rather than quantum mechanics. In fact, the Helmholtz equation for any of the

3.7

Problems

57

components of the electromagnetic field within a resonator involves the very same differential operator, ∇ 2 +k2 with k denoting the wave vector, as does Schr¨odinger’s equation for a free particle in a container; as long as the boundary conditions at the walls do not mix different components of the electromagnetic field, there is a complete mathematical equivalence between the quantum and the classical wave problem. The chaos of which one sees the quantum or wave signatures is that of a “billiard” with the shape of the “box” in question, at least if the boundary conditions express specular reflection of the point particle that idealizes the billiard ball. The first of the microwave experiments just mentioned [9] was actually meant to simulate sound waves of air in a concert hall and was done in complete innocence of the jargon of present-day chaology; the simulation of sound by microwaves was preferred to “listening” since concert halls tend to have overlapping rather than separated resonances and are thus not too well suited for picking up spacing statistics. It is amusing to realize, though, that audio engineers do go for wave chaos when designing the boundaries of concert halls. Meanwhile, acoustic chaos has also been ascertained experimentally as linear repulsion of the elastomechanical eigenfrequencies of irregularly shaped quartz blocks [15, 16]. Inasmuch as the vibrating crystal is anisotropic and supports longitudinal as well as transverse sound waves, these experiments suggest that the distinction between chaotic and regular waves arises quite independently of the character of the medium and the detailed form of the pertinent wave equation. The common origin of quantum and wave chaos may be seen in the nonseparability of the wave equation, just as we may attribute chaos in classical mechanics to the nonseparability of the Hamilton–Jacobi equation. If we see the Hamilton–Jacobi equation as the short-wave limit of wave theories, we can consider nonseparability as the universal chaos criterion. Neither quadratic nor quartic level repulsion has been observed to date in nuclei, atoms, and molecules. It might be possible to break time-reversal invariance for Rydberg atoms with strongly inhomogeneous magnetic fields and thus realize the quadratic case. In the absence of such experiments on quantum systems, the recent observation of quadratic repulsion in microwave resonators with broken timereversal invariance was a most welcome achievement [17, 18]. For a comprehensive review of experimentally observed quantum manifestations of chaos, the reader is referred to St¨ockmann’s recent book [19].

3.7 Problems 3.1 Show that a unitary 4 × 4 matrix with Kramers’ degeneracy can be given in the form ⎛

α + iβ ⎜ 0 ⎜ F =⎝ −σ + iγ −δ − iε

0 α + iβ −δ + iε σ + iγ

σ + iγ δ + iε α − iβ 0

⎞ δ − iε −σ + iγ ⎟ ⎟ ⎠ 0 α − iβ

58

3 Level Repulsion

with the six real parameters α, β, . . . obeying α 2 + β 2 + γ 2 + δ 2 + ε2 + σ 2 = 1. Use basis vectors |1, |T 1, |2, |T 2, and T 2 = −1. 3.2 Show that the matrix F from Problem 3.1 has n = 3 if it is invariant under a parity Rx with [T, Rx ] = 0, Rx2 = −1. Moreover, show that n = 2 if a second parity holds with [T, R y ] = 0, Rx R y + R y Rx = 0, R 2y = −1. What is the structure of F if full isotropy holds? 3.3 Verify that nearly degenerate perturbation theory typically gives a hyperbolic form to a level crossing. 3.4 In the orthogonal case, adjacent energy levels can be steered to crossing in a two-dimensional parameter space. Show that the two energy surfaces are connected at the degeneracy like the two sheets of a double cone (Berry’s diabolo [20, 21]).

References 1. C.E. Porter (ed.) Statistical Theories of Spectra (Academic, New York, 1965) 2. M.L. Mehta: Random Matrices (Academic, New York 1967; 2nd edition 1991; 3rd edition Elsevier 2004) 3. J. von Neumann, E.P. Wigner: Phys. Z. 30, 467 (1929) 4. R. Scharf, B. Dietz, M. Ku´s, F. Haake, M.V. Berry: Europhys. Lett. 5, 383 (1988) 5. M.A. Stephanov, J.J.M. Verbaarschot, T. Wettig: arXiv:hep-ph/0509286v1 6. O. Bohigas, R.U. Haq, A. Pandey: In K.H. B¨ochhoff (ed.) Nuclear Data for Science and Technology (Reidel, Dordrecht, 1983) 7. Th. Zimmermann, H. K¨oppel, L.S. Cederbaum, C. Persch, W. Demtr¨oder: Phys. Rev. Lett. 61, 3 (1988); G. Persch, E. Mehdizadeh, W. Demtr¨oder, T. Zimmermann, L.S. Cederbaum: Ber. Bunsenges. Phys. Chem. 92, 312 (1988) 8. H. Held, J. Schlichter, G. Raithel, H. Walther: Europhys. Lett. 43, 392 (1998) 9. M.R. Schroeder: J. Audio Eng. Soc. 35, 307 (1987) 10. H.-J. St¨ockmann, J. Stein: Phys. Rev. Lett. 64, 2215 (1990) 11. H. Alt, H.-D. Gr¨af, H.L. Harney, R. Hofferbert, H. Lengeler, A. Richter, P. Schart, H.A. Weidenm¨uller: Phys. Rev. Lett. 74, 62 (1995) 12. H. Alt, H.-D. Gr¨af, R. Hofferbert, C. Rangacharyulu, H. Rehfeld, A. Richter, P. Schart, A. Wirzba: Phys. Rev. E 54, 2303 (1996) 13. H. Alt, C. Dembowski, H.-D. Gr¨af, R. Hofferbert, H. Rehfeld, A. Richter, R. Schuhmann, T. Weiland: Phys. Rev. Lett. 79, 1026 (1997) 14. A. Richter: In D.A. Hejhal, J. Friedman, M.C. Gutzwiller, A.M. Odlyzko (eds.) Emerging Applications of Number Theory IMA volume 109, p. 109 (Springer, New York, 1998) 15. C. Ellegaard, T. Guhr, K. Lindemann, H.Q. Lorensen, J. Nyg˚ard, M. Oxborrow: Phys. Rev. Lett. 75, 1546 (1995) 16. C. Ellegaard, T. Guhr, K. Lindemann, J. Nyg˚ard, M. Oxborrow: Phys. Rev. Lett. 77, 4918 (1996) 17. U. Stoffregen, J. Stein, H.-J. St¨ockmann, M. Ku´s, F. Haake: Phys. Rev. Lett. 74, 2666 (1995) 18. P. So, S.M. Anlage, E. Ott, R.N. Oerter: Phys. Rev. Lett. 74, 2662 (1995) 19. H.-J. St¨ockmann: Quantum Chaos, An Introduction (Cambridge University Press, Cambridge, 1999)

References

59

20. M.V. Berry: In G. Iooss, R.H.G. Helleman, R. Stora (eds.) Les Houches, Session XXXVI, 1981, Chaotic Behavior of Deterministic Systems (North-Holland, Amsterdam, 1983) 21. T.A. Brody, J. Floris, J.B. French, P.A. Mello, A. Pandey, S.S.M. Wong: Rev. Mod. Phys. 53, 385 (1981) Appendix B

Chapter 4

Random-Matrix Theory

4.1 Preliminaries A wealth of empirical and numerical evidence suggests universality for local fluctuations in quantum energy or quasi-energy spectra of systems that display global chaos in their classical phase spaces. Exceptions apart, all such Hamiltonian matrices of sufficiently large dimension yield the same spectral fluctuations provided they have the same group of canonical transformations (see Chap. 2). In particular, the level spacing distribution P(S) generally takes the form characteristic of the universality class defined by the canonical group. Most notable among the exceptions barred by the term “untypical” are systems with “localization” that will be discussed in Chap. 7. Conversely, “generic” classically integrable systems with at least two degrees of freedom tend to display universal local fluctuations of yet another type, to be considered in Chap. 5. The aforementioned universality is the starting point for the theory of random matrices (RMT). After early success in reproducing universal features in spectra of highly excited nuclei, that theory was boosted into even higher esteem when the connection of “integrable” and “chaotic” with different types of universal spectral fluctuations was spelled out by Bohigas, Giannoni, and Schmit [1], with important hints due to Berry and Tabor [2], McDonald and Kaufman [3], Casati, Valz-Gris, and Guarneri [4], and Berry [5]. The classic version of random-matrix theory deals with three Gaussian ensembles of Hermitian matrices, one for each group of canonical transformations. Any member of an ensemble can serve as a model of a Hamiltonian. Similarly, there are three ensembles of random unitary matrices to represent Floquet or scattering matrices. “Poissonian” ensembles of diagonal matrices with independent, random, diagonal elements are often used to model integrable Hamiltonians. Even systems with localization have recently been accommodated in their own “universality class” of banded random matrices that is to be touched upon in Chap. 11. Random-matrix theory phenomenologically represents spectral fluctuations such as those expressed in the level spacing distribution or in correlation functions of the density of levels by suitable ensemble averages. The immense usefulness of RMT lies in the fact that it yields closed-from results for many spectral characteristics. The

F. Haake, Quantum Signatures of Chaos, Springer Series in Synergetics, 3rd ed., C Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-05428-0 4, 

61

62

4 Random-Matrix Theory

extent to which an individual Hamiltonian or Floquet operator can be expected to be faithful to the RMT averages is open to discussion. A partial answer to that question is provided by a certain ergodicity property of the various ensembles. Explanations of the success of random-matrix theory will be presented in Chap. 6 (level dynamics) and Chap. 10 (periodic-orbit theory). This chapter accounts for classic elementary material. In particular, the seven new symmetry classes introduced by Zirnbauer and Altland [6] will be given short shrift. However, I shall include a brief introduction to and some applications of Grassmann analysis. The latter thread will be spun further toward “superanalysis” in Chap. 11. For more extensive treatments and greater mathematical rigor, the reader may consult [7, 8]. An up-to-date review can be found in [9]. For the sake of notational convenience, I set  = 1 throughout this chapter.

4.2 Gaussian Ensembles of Hermitian Matrices The construction of the Gaussian ensembles will be illustrated by considering real symmetric 2 × 2 matrices with O(2) as their group of canonical transformations (If reflections are to be excluded, the group would be S O(2)). What we are seeking is a probability density P(H ) for the three independent matrix elements H11 , H22 , H12 normalized as 

+∞ −∞

d H11 d H22 d H12 P(H ) = 1 .

(4.2.1)

Two requirements suffice to determine P(H ). First, P(H ) must be invariant under any canonical, i.e., orthogonal transformation of the two-dimensional basis, ˜ , O ˜ = O −1 . P(H ) = P(H ) , H = O H O

(4.2.2)

Second, the three independent matrix elements must be uncorrelated. The function P(H ) must therefore be the product of three densities, one for each element, P(H ) = P11 (H11 )P22 (H22 )P12 (H12 ) .

(4.2.3)

The latter assumption can be reinterpreted as one of minimum-knowledge input or of maximum disorder. To exploit (4.2.2) and (4.2.3), it suffices to consider an infinitesimal change of basis,  O=

1 −Θ Θ 1

 ,

(4.2.4)

4.2

Gaussian Ensembles of Hermitian Matrices

63

˜ gives for which H = O H O

H11 = H11 − 2Θ H12

H22 = H22 + 2Θ H12

H12 = H12 + Θ(H11 − H22 ) .

(4.2.5)

Factorization and the invariance of P(H ) yield



d ln P11 d ln P22 − 2H12 d H11 d H22  d ln P12 −(H11 − H22 ) . d H12

P(H ) = P(H ) 1 − Θ 2H12

(4.2.6)

Since the infinitesimal angle Θ is arbitrary, its coefficient in (4.2.6) must vanish, 2 1 d ln P12 − H12 d H12 H11 − H22



d ln P22 d ln P11 − d H11 d H22

 =0.

(4.2.7)

This gives three differential equations, one for each of the three independent functions Pi j (Hi j ) since each Pi j has its own exclusive argument Hi j . The solutions are Gaussians and have the product   2  2 2 P(H ) = C exp −A H11 − B (H11 + H22 ) . + H22 + 2H12

(4.2.8)

Of the three integration constants, B can be made to vanish by appropriately choosing the zero of energy, A fixes the unit of energy, and C is determined by normalization. Without loss of generality, then, P(H ) can be written as P(H ) = Ce−A Tr H . 2

(4.2.9)

The discussion of complex Hermitian Hamiltonians with U (2) as the group of canonical transformations proceeds analogously. There are four real parameters H11 , H22 , Re {H12 }, Im {H12 } to be dealt with now, and they are all assumed statistically independent. The probability density P(H ) thus factorizes as in (4.2.3) but with P12 ∗ as a function of Re {H12 }, Im {H12 } or, equivalently, of H12 and H12 . The normalization (4.2.1) is modified such that P12 is integrated over the whole complex H12 plane, 

+∞

−∞

d H11 d H22 d 2 H12 P(H ) = 1

(4.2.10)

where d 2 H12 = dRe H12 dIm H12 . The three functions P11 (H11 ), P22 (H22 ), and ∗ ) are determined by demanding that P(H ) is invariant under unitary P12 (H12 , H12

64

4 Random-Matrix Theory

transformations of the matrix H . Up to an inconsequential phase factor, the general infinitesimal change of basis is represented by the 2 × 2 matrix U = 1 − iε · σ

(4.2.11)

with ε an infinitesimal vector and σ the triple of Pauli matrices. This U shifts the Hamiltonian by d H = −i [ε · σ , H ] .

(4.2.12)

Now, the analogue of (4.2.7) reads 

   ∂ ln P12 d ln P22 d ln P11 H12 − + + (H11 − H22 ) ∗ d H11 d H22 ∂ H12





+2εz H12

∂ ln P12 − c.c. = 0 . ∂ H12

εx + iε y

(4.2.13)

Again, three differential equations can be extracted from this identity and the solutions are all Gaussians. If the zero of energy is chosen appropriately, the probability density P(H ) once more takes the form (4.2.9). Let us turn finally to Hamiltonians with Kramers’ degeneracy and with no geometric invariance. The smallest Hilbert space is now four dimensional. The 4 × 4 Hamiltonian is most conveniently written in quaternion notation,  H=

h 11 h 12 h 21 h 22

 ,

(4.2.14)

where each h i j is a 2 × 2 block, representable as a superposition of unity and the triplet τ = −iσ [see (2.8.7)]. Due to (2.8.9) and (2.8.11), H is determined by six (μ) (0) real amplitudes h (0) 11 , h 22 , h 12 , all of which are taken to be independent random numbers with a probability density of the form    + μ  μ (0) 3 P(H ) = P11 h (0) μ = 0 P12 h 12 , 11 P22 h 22 1=

$

(0) dh (0) 11 dh 22

+3

(4.2.15)

(μ)

μ = 0 dh 12 P(H ).

To find the six respective probability densities, it suffices to require invariance of P(H ) when H is subjected to an arbitrary infinitesimal symplectic transformation. Such a change of basis is represented by S=

  1−ξ ·τ α −α 1 + ξ · τ

(4.2.16)

4.2

Gaussian Ensembles of Hermitian Matrices

65

with α an infinitesimal angle and ξ an infinitesimal real vector; of course, (4.2.16) is meant in quaternion notation, and the matrices τ = −iσ are as defined in (2.8.6). The 2 × 2 blocks of the Hamiltonian acquire the increments dh 11 = −dh 22 = 2αh (0) 12   (0) (0) dh 12 = α h 22 − h 11 + ξ · h12 − 2h (0) 12 ξ · τ .

(4.2.17)

Now, we invoke P(H + d H ) = P(H ) for arbitrary α and ξ and thus obtain  2h (0) 12

d ln P11 dh (0) 11 h (i) 12

−

d ln P22 dh (0) 22

(0) d ln P12

dh (0) 12



(0)   (0) d ln P12 − h =0, + h (0) 22 11 dh (0) 12

− h (0) 12

(i) d ln P12

dh (i) 12

= 0 for i = 1, 2, 3 .

(4.2.18)

The reader should note that the first of these identities has the same structure as (4.2.7) and (4.2.13). Since each of the six functions we are seeking has its own exclusive argument, the identities (4.2.18) imply six separate differential equations. As in the cases considered previously, the solutions are all Gaussians. With a proper choice for the zero of energy, their product1

  3   2 2   2 (μ) (0) P(H ) = C exp −2A h (0) + h + 2 h 12 11 22

(4.2.19)

μ=0

again gives the density P(H ) in the form (4.2.9). To summarize, three different ensembles of random matrices follow from demanding (1) invariance of P(H ) under the three possible groups of canonical transformations and (2) complete statistical independence of all matrix elements. In view of the Gaussian form of P(H ) and the three groups of canonical transformations, these ensembles are called Gaussian orthogonal, Gaussian unitary, and Gaussian symplectic. Although P(H ) has been constructed here for the smallest possible dimensions, the result P(H ) = C e−A Tr H , 2

(4.2.20)

holds true independently of the dimensionality of H [7, 8]. Of course, in both P(H ) and in the integration measure, the appropriate number of independent real 1 The reader’s attention is drawn to the overall factor 2 in the exponent of (4.2.19); it stems from Kramers’ degeneracy. When dealing with the GSE, some authors choose to redefine the trace operation as one half the usual trace, so as to account for only a single eigenvalue in each Kramers’ doublet; correspondingly, these authors take the determinant as the square root of the usual one. Nowhere in this book will such tampering be permitted.

66

4 Random-Matrix Theory

parameters must be accounted for; that number is N (N + 1)/2 and N 2 , respectively, for the real symmetric and complex Hermitian N × N matrices and N (2N − 1) for the quaternion real 2N × 2N matrices. The invariance of the ensembles under the appropriate canonical transformations is not fully established before the invariance of the differential volume element in matrix space is shown, see Sect. 4.6.

4.3 Eigenvalue Distributions for Dyson’s Ensembles The probability density (4.2.20) for the matrix elements of a random Hamiltonian H implies reduced probability densities for the eigenvalues of H. To perform the reduction, it is necessary to replace the Hi j as independent variables with another set of equal dimension which contains the eigenvalues as a subset; the remaining variables, over which to integrate, are parameters specifying the particular canonical transformation that diagonalizes H. The procedure will be illustrated again for the smallest possible number of dimensions. The easiest case to deal with is that of real symmetric 2 × 2 Hamiltonians. The two eigenvalues read E± =

1 2

(H11 + H22 ) ±

1 2



2 (H11 − H22 )2 + 4H12

1/2

.

(4.3.1)

The simplest orthogonal transformation diagonalizing H, 

cos Θ − sin Θ O= sin Θ cos Θ

 ,

(4.3.2)

involves a single angle Θ. The reader should recall that the infinitesimal version of ˜ one can read off (4.3.2) was employed in Sect. 4.2. From H = O diag (E + , E − ) O, the relationships between the matrix elements Hi j and the quantities E ± , Θ : H11 = E + cos2 Θ + E − sin2 Θ H22 = E + sin2 Θ + E − cos2 Θ H12 = (E + − E − ) cos Θ sin Θ .

(4.3.3)

The Jacobian of this transformation is easily evaluated, J = det

∂(H11 , H22 , H12 ) = E+ − E− , ∂(E + , E − , Θ)

(4.3.4)

whereupon the reduced probability density of the eigenvalues for the Gaussian orthogonal ensemble of 2 × 2 Hamiltonians takes the form P (E + , E − ) = C |E + − E − | e−A(E+ +E− ) . 2

2

(4.3.5)

4.3

Eigenvalue Distributions for Dyson’s Ensembles

67

For complex Hermitian 2 × 2 Hamiltonians, the eigenvalues E ± are also given 2 → |H12 |2 . Diagonalization may be achieved by the unitary by (4.3.1) but with H12 transformation   cos Θ −e−iφ sin Θ . (4.3.6) U= eiφ sin Θ cos Θ The reader might note that this is not the most general unitary 2 × 2 matrix (see Problem 4.2). Together with the two eigenvalues E ± , the two angles Θ and φ suffice to represent the matrix elements of H. From U † diag (E + , E − )U = H , H11 = E + cos2 Θ + E − sin2 Θ H22 = E + sin2 Θ + E − cos2 Θ

(4.3.7)

∗ H12 = H21 = (E + − E − ) eiφ cos Θ sin Θ

and thus the Jacobian J = det

∂(H11 , H22 , Re {H12 }, Im {H12 }) ∂(E + , E − , Θ, φ)

= (E + − E − )2 cos Θ sin Θ .

(4.3.8)

By integrating out the angles Θ and φ, one finally arrives at the eigenvalue distribution of the Gaussian unitary ensemble of random 2 × 2 Hamiltonians P (E + , E − ) = const (E + − E − )2 e−A(E+ +E− ) . 2

2

(4.3.9)

The procedure in the symplectic case is completely analogous: The 4 × 4 matrix H can be written in the quaternion form (4.2.14), and the pair of doubly degenerate eigenvalues is given by E± =

1 2



h (0) 11

−

h (0) 22



±

1 2



h (0) 11

−

h (0) 22

2

+4

3  

h (μ) 12

2 1/2

.

(4.3.10)

μ=0

Now, a symplectic matrix is needed for diagonalization. A convenient choice is the generalization of the infinitesimal transformation (4.2.16) to a finite one, 

e−ξ ·τ cos α sin α S= − sin α eξ ·τ cos α

 .

(4.3.11)

Remarks similar to those above are appropriate at this point. The choice (4.3.11) is not the most general symplectic 4 × 4 matrix. However, it suits the present purposes because the four real parameters α, ξ provided by S, plus the two eigenvalues

68

4 Random-Matrix Theory

E ± , are equal in number to the independent matrix elements of H. From S † diag (E + , E − )S = H , we obtain the relation 2 2 h (0) 11 = E + cos α + E − sin α 2 2 h (0) 22 = E + sin α + E − cos α

(4.3.12)

h (0) 12 = − (E + − E − ) cos α sin α cos ξ h12 = ξˆ (E + − E − ) cos α sin α sin ξ ,

where ξ = ξ ξˆ , ξ = |ξ |. The Jacobian of the transformation (4.3.11) from the six (μ) matrix elements h mn to the six parameters E ± , α, ξ is easily evaluated, J = det

(μ) (0) ∂(h (0) 11 , h 22 , h 12 ) ∝ (E + − E − )4 . ∂(E + , E − , α, ξ )

(4.3.13)

The coefficient unspecified in the latter proportionality is independent of the eigenvalues E ± and thus irrelevant for the reduced distribution P(E + , E − ) = const (E + − E − )4 e−A(E+ +E− ) . 2

2

(4.3.14)

Comparing (4.3.5), (4.3.9), and (4.3.14) one finds that the codimension n of a level crossing is related to the exponents of the powers in front of the Gaussians as β = n − 1. The arguments of this section can be generalized to higher dimensions of the matrix H, and the general result for the joint distribution of eigenenergies E μ is [7, 8] ⎛ ⎞ 1, ... N N   n−1  E μ − E ν  exp ⎝−A P(E) = const E μ2 ⎠ (4.3.15) μ 1 ,   sin πe 2 1 − cos 2πe 2 YCUE (e) = s(e) = = . πe 2π 2 e2

(4.14.16)

Both the form factor K CUE (τ ) and the cluster function YCUE (e) are depicted in Fig. 4.4, together with their variants for the orthogonal and symplectic classes. The saturation of the form factor at |tn |2 = N for n ≥ N is a peculiarity of the CUE. Easy to appreciate intuitively, basically as a consequence of the central limit theorem, is asymptotic saturation as n → ∞, such as will  also be encountered N e−inφi sees the below for the COE and CSE. We must realize that the trace tn = i=1 phases nφi only modulo 2π . Accidental close proximity now arises between pairs of phases whose counterparts in the original quasi-energy spectrum (i.e., for n = 1) are far apart and thus only weakly correlated, apart from a certain tendency to form a “lattice” with “sites” roughly spaced by 2π/N . The phase factors e−inφi will become ever more weakly correlated as n grows; their sum t N must asymptotically become the end point of an N -step random walk in the complex plane starting at the origin, with each step √of unit length but random direction. The central limit theorem would entail tn → N and irrespectively so of the provenance of the quasi-energies. In particular, this reasoning should (and does) also apply to quasi-energy spectra of

Fig. 4.4 Form factors K (τ ) and cluster functions Y (e) for the unitary, orthogonal, and symplectic ensembles according to (4.14.16), (4.14.29), and (4.14.36)

8 The (N → ∞) limit K (τ ) of the form factor is related to the Fourier transform Y˜ (τ ) of the cluster function Y (e) by K (τ ) = δ(τ ) + 1 − Y˜ (τ ).

106

4 Random-Matrix Theory

dynamical systems. The rigidity of spectra alluded to under the heading “lattice” does not stand in the way of the central limit theorem here but rather accentuates the onset of applicability only for n > N : The “lattice sites” are preferred with an uncertainty /N where  is a number of order unity; that uncertainty is blown up to one of order 2π, i.e., complete uncertainty, for n > N . The unit slope of the form factor at small “times” τ will be revealed as due to classical ergodicity in Sect. 8.7. It is not counterintuitive that the break between the linear small-time behavior and the large-time saturation takes place at the “Heisenberg” time τ H = 1 alias n H = N , the time scale on which the discreteness of the quasi-energy spectrum becomes resolvable. After all, n H = N is the only time scale around in the random-matrix ensembles under study.9 In view of Chap. 10, a comment on the cluster function YCUE (e) is also indicated. The last member of (4.14.16) is split into a nonoscillatory and an oscillatory term both of which decay as 1/e2 as e → ∞. The former is due to the corner of |τ | in the form factor at τ = 0 and the latter to the corner at the Heisenberg time τ H = 1, as the reader is invited to check by solving Problem 4.13.

4.14.4 Form Factor for the COE Modifying the foregoing calculation for the circular orthogonal ensemble provides an opportunity to explain the classic method of “integration over alternate variables” [7]. Again, we start from the characteristic function (4.14.5)  P˜ n1N (k)

= (1/N1 )

- .  ,  , −ik cos(nφ ) −iφ −iφ k l i e d φ −e  e N

kφ1 >φ2 ... φ N >0

i

dφ1 . . . dφ N {. . .}

(4.14.17)

where due to symmetry the hypercube of integration {0 < φi < 2π } could be traded against the hypertriangle 2π > φ1 > φ2 . . . > φ N > 0, making up by a factor N !. Within the hypertriangle, we write the product of moduli as , kφ1 >φ2 ... φ N >0

×

,

1...N  ,  −iφk −iφl 4  e dφ1 . . . dφ N −e

e−ik cos(nφi ) ,

kφ1 >φ2 ... φ2N >0

×

N ,

e−ik cos(nφi )

i=1

dφ1 . . . dφ2N

1...2N , k 2 ,

YCSE (e) = s(2e)2 − I (2e)D(2e)

(4.14.36)

with I (e) the integral of s(e) = (π e)−1 sin(πe) given in (4.10.16). A peculiarity of the CSE is the logarithmic singularity of the form factor at |τ | = 1. Saturation at the plateau unity sets in at τ H = 2 which time I choose to call the Heisenberg time here; the distinction of both τ H and τ H /2 may be seen as due to Kramers’ degeneracy. At small and large e, the cluster function reads  YCSE (e) =

4

e) 1 − (2π + ... , 135 sin 2π|e| π cos 2π e − − 2 2π |e| + 1+(π/2) (2π e)2

3+cos 4πe 2(2πe)4

|e|  1 (4.14.37) . . . , e|  1 .

Plots of K CSE (τ ) and YCSE (τ ) are shown in Fig. 4.4. When comparing the form factor K CSE (τ ) or the cluster function YCSE (e) with results for dynamic systems, one should not forget about Kramers’ degeneracy, i.e., the factor 1/2 in (4.14.2) for the symplectic case. One must also keep in mind that the time n/N → τ is measured in units of half the Heisenberg time here, whereas the unit of time for the other ensembles is the Heisenberg time itself. Again, the nonoscillatory terms in YCSE (e) are contributed by the corner of 12 |τ | in the form factor; the terms oscillating with frequencies 2π and 4π stem, respectively, from the logarithmic singularity at τ = 1 and the jump of the third derivative of K CS (τ ) at τ = 2; see Problem 4.13.

4.15 Newton’s Relations 4.15.1 Traces Versus Secular Coefficients Before digging further into more up-to-date issues in random-matrix theory, we had better recall some well-known facts from algebra. At the top of the list is the Hamilton–Cayley theorem which in a cavalier wording says that every matrix obeys its own secular equation. We need that theorem only for diagonalizable matrices like unitary and Hermitian ones, and for these the proof is trivial: Consider the secular polynomial of an N × N matrix F with eigenvalues λi , P(λ) = det (λ − F) =

N , i=1

(λ − λi ) =

N  n=0

(λ)n A N −n ;

(4.15.1)

4.15

Newton’s Relations

113

Note A0 = 1 and A N = det(−F) = (−1) N det F. I shall refer to the coefficients An as the secular coefficients. By substituting the matrix F for the variable λ, a new N × N matrix U † P(F)U results which is diagonalized by the same unitary matrix U as is F itself. In other words, if U † FU = diag(λ1 , λ2 , . . . , λ N ), then U † P(F)U is also diagonal with diagonal elements P(λi ), but these latter vanish by the definition of the secular polynomial, and thus the diagonal matrices U † P(F)U and P(F) vanish as well. Upon taking the trace of P(F), we get Tr P(F) =

N 

tn A N −n = 0 ,

(4.15.2)

n=0

i.e., a linear relationship between the first N traces tn = TrF n of the matrix F. All other traces can then be expressed in terms of the first N ones, as becomes clear by multiplying P(F) by arbitrary powers of F and then taking the trace. Next, I turn to a little treasure left to posterity by Isaac Newton, a relationship between traces and secular coefficients.10 A variation of well known a theme (see also the next section) is to be played, the relationship between moments and cumulants of random variables. To briefly explain the latter, let us consider a function M(x) of a single real variable x with M(0) = 1 and compare the Taylor expansions of that function and its logarithm, M(x) = 1 + C(x) ≡ ln M(x) =

∞  1 Mn x n , n! n=1

∞  1 Cn x n . n! n=1

(4.15.3)

If M(x) is the characteristic function of some random quantity, then it is said to generate the moments Mn and its logarithm C(x) generates the cumulants Cn . By repeatedly differentiating the identity M(x) = eC(x) with respect to x and then setting x = 0, one immediately gets the linear implicit relationship between moments and cumulants11  n   n−1 Mn = Cm Mn−m . m−1 m=1

(4.15.4)

10 Newton was concerned with relationships between different symmetric functions of N variables [37]. 11 Note that in (4.15.3), (4.15.4), (4.15.5), in contrast to the proceeding subsection, I depart from Mehta’s sign convention; readers not willing to put up with such wicked confusion may return to the random-matrix path of virtue by Cn → (−1)n−1 Cn .

114

4 Random-Matrix Theory

This may be solved either to express cumulants in terms of moments or vice versa. The first few moments are M1 = C 1 M2 = C2 + (C1 )2

(4.15.5)

M3 = C3 + 3C2 C1 + (C1 )3 M4 = C4 + 4C3 C1 + 3(C2 )2 + 6C2 (C1 )2 + (C1 )4 . It will be instructive for the reader to ponder about “equivalence” and differences between the foregoing moment–cumulant relationships and (4.17.6) below. But on to Newton’s formulae! They are just the above, read a little differently. Writing the secular polynomial (4.15.1) in the form 

∞  1 −n tn λ det(λ − F) = exp Tr ln(λ − F) = λ N exp − n n=1

 (4.15.6)

one recognizes that the secular coefficients an correspond to moments and the traces tn to cumulants; the obvious differences in prefactors are easily accounted for and make the recursion relation (4.15.4) chameleon into the desired identities − n An =

n 

tm An−m .

(4.15.7)

m=1

A little recursive hocus-pocus solves Newton’s relations for the secular coefficients in terms of the traces [38] ,   t1  t (−1)n  2 ··· An = n!  t  n−1  tn

1 t1 ··· tn−2 tn−1

0 2 ··· tn−3 tn−2

 ··· 0 0  ··· 0 0  · · · · · · · · ·  ; · · · t1 n − 1  · · · t2 t 1 

(4.15.8)

by expanding the determinant along the last row it is indead easy to check that the foregoing expression solves the recursion relation. Conversely, the traces can be expressed in terms of the secular coefficients as   A1 1 0   A2 A1 1  A A2 A1 tn = (−1)n  3 ... ... ...  An−1 An−2 An−3   n An (n − 1)An−1 (n − 2)An−2

... ... ... ... ... ...

0 0 0 ... A1 2A2

0 0 0 ... 1 A1

      .     

(4.15.9)

4.15

Newton’s Relations

115

It is quite remarkable that neither Newton’s recursion relations themseves nor their foregoing solutions explicitly contain the size N of the matrix F. For a given value of N the traces tn are defined for arbitrary non-negative integer values of (the exponent) n, while the An are defined in the range 0 ≤ n ≤ N . We may extend the definition of the secular coefficients beyond the limit N as An = 0 for n > N , and then the solutions (4.15.8) and (4.15.9) remain valid for all non-negative integers n. The foregoing determinants for An and tn can of course be multiplied out. A compact form arises when we consider all partitions of the integern into sums of non-negative integers l with integer multiplicities vl , namely n = l=0,1,... lvl . We vector” v = {v1 , v2 , . . .} and define two propermay collect the vl to a “multiplicity  ties of that vector as L(v) = l=0,1,... lvl and V (v) = l=0,1,... vl . The solution of Newton’s formulae can then be written as {L(v)=n} 

, t vl l , vl v ! l l l≥1

{L(v)=n} 

, Avl l . v ! l v v l≥1 (4.15.10) The solutions of Newton’s relations can play an important role even outside random-matrix theory. In particular, if the traces tn with n = 1, . . . , N of a matrix F are known, Newton’s formulae allow us to access the spectrum since they yield the full secular polynomial from the first N traces. The interested reader might enjoy expressing the cumulant Cn explicitly in terms of the moments M1 , . . . Mn and vice versa, starting from the solutions (4.15.10) of Newton’s equations (problem 4.16). An =

(−1)V (v)

tn = n

(−1)V (v) (V (v) − 1)!

4.15.2 Solving Newton’s Relations Here comes a quick derivation of the solutions (4.15.10). It is convenient to employ a one-sided discrete Fourier transform, ˜ A(ϕ) =

∞ 

An einϕ = 1 +

n=0

N 

An einϕ ,

t˜(ϕ) =

n=1

∞ 

tn einϕ

(4.15.11)

n=0

with the inverses  An = 0

2π

dϕ ˜ A(ϕ)e−inϕ , 2π



2π

tn = 0

dϕ t˜(ϕ)e−inϕ . (4.15.12) 2π

˜ The Fourier transforms A(ϕ), t˜(ϕ) are 2π-periodic functions of the real phase ϕ. They are respective kins to the spectral determinant and the resolvent of the matrix F as ˜ A(ϕ) = det(1 − eiϕ F) ,

t˜(ϕ) = Tr

1 . 1 − eiϕ F

(4.15.13)

116

4 Random-Matrix Theory

Spectral determinant and resolvent of some matrix are related by the identity det M = eTr ln M which for our M = 1 − eiϕ F entails t˜(ϕ) = N + i

∂ ˜ ln A(ϕ) . ∂ϕ

(4.15.14)

It is a pleasant surprise to recognize (the one-sided Fourier transform of) Newton’s formulae (4.15.7) in the latter identity. Indeed, the Fourier transform turns ˜ the convolution in the r. h. .s of (4.15.7) into the product t˜(ϕ) A(ϕ). I proceed to the explicit solution of Newton’s formulae by reverting the Fourier transform in (4.15.14), 

2π

tn = N δn,0 − n 0

dϕ −inϕ ˜ ln A(ϕ) . e 2π

(4.15.15)

˜ The ϕ-integral is easily done after expanding the logarithm of A(ϕ) =  Nremaining inϕ 1 + n=1 An e ≡ 1 + S, ˜ ln A(ϕ) = ln(1 + S) =

∞  (−1)m S m , m m=1

(4.15.16)

and invoking the polynomial expansion Sm =

m=v1  +v2 +...v N v

=

m=V (v) v

m! Av1 Av2 . . . AvNN ei(v1 +2v2 +...N v N ) ϕ v1 !v2 ! . . . v N ! 1 2

m! Av1 Av2 . . . AvNN eiL(v)ϕ . v1 !v2 ! . . . v N ! 1 2

(4.15.17)

The ϕ-integral annuls all terms except those with m = L(v). The non-vanishing contributions immediately yield (4.15.10). Conversely, I get the secular coefficients in terms of the traces by first solving the ˜ Fourier transform of Newton’s relation (4.15.14) for A(ϕ) in terms of t˜(ϕ), ˜ A(ϕ) = e−

∞

tk k=1 k

eikϕ

;

this is again the familiar matrix identity det(1 − eiϕ F) = eTr ln(1−e Fourier transform now yields  An = 0

2π

 ∞   2π  tk dϕ dϕ −Q(ϕ) ikϕ . exp − e e ≡ 2π k 2π 0 k=1

(4.15.18) iϕ

F)

. The inverse

(4.15.19)

4.16

Selfinversiveness and Riemann–Siegel Lookalike

After expanding the exponential exp (−Q) =

117 ∞  (−1)m Q m 0

nomial expansion of Q m , Qm =

{V  (v)=m} v

m!

m!

and invoking the poly-

vk , 1  tk eikϕ v ! k k≥1 k

(4.15.20)

the secular coefficient becomes expressed in terms of the traces as in (4.15.10).

4.16 Selfinversiveness and Riemann–Siegel Lookalike If the matrix F is unitary, a further simple property of the secular coefficients, commonly called self-inversiveness, [39–42] arises, A N −n = A N A∗n .

(4.16.1)

Self-inversiveness is checked as follows: − F) = (λ) N det F det(F †  N det(λ −1 N † −1 n ∗ − λ ) = λ A N det(F − λ ) = A N n=0 (λ) An . It reduces the number of independent real parameters in the secular polynomial to N , i.e., the number of eigenphases φi . The roots of a self-inversive polynomial are either unimodular or come in pairs such that λi λi∗ = 1, hence the name [39] (see Problem 4.14); the unitarity of F is, of course, a stronger property than self-inversiveness since it requires all eigenvalues to be unimodular. I finally turn to a consequence of self-inversiveness for the secular polynomials ˜ The spectrum of a unitary F has N of unitary matrices det(e−iϕ − F) = e−iN ϕ A(ϕ). eigenphases which determine all secular coefficients and all the traces tn = TrF n . Conversely, the spectrum may be determined by N real parameters taken from the sets of A’s and t’s. Convenient choices for these parameters differ slightly for even and odd values of N . For even N , I may choose the N /2 secular coefficients An with 1 ≤ n ≤ N /2 as independent parameters and get the remaining ones (n > N /2) from the former through the self-inversiveness (4.16.1); note that A0 = 1 carries no information; note further that the “last” coefficient, A N = det(−F) is then fixed as A N = A N /2 /A∗N /2 . For odd N , on the other hand, I may pick the An with 1 ≤ n ≤ [ N2 ] and A N as representatives12 ; since A N is a phase factor containg just one real parameter we thus again have 2[ N2 ] + 1 = N independent real parameters determining the remaining secular coefficients through self-inversiveness.

12

The symbol [ N2 ] denotes N /2 for even N and the next smaller integer for odd N

118

4 Random-Matrix Theory

I now choose N odd for the remainder of the present Section; the reader is invited to write out the following for case of even N . The secular polynomial can be expressed as ⎛

N

˜ A(ϕ) =

[2] 

An einϕ + A N eiN ϕ ⎝

n=0

= ≡

& &

⎞∗

N

[2] 

An einϕ ⎠

n=0 iN ϕ/2

%

AN e



N

A∗N e−iN ϕ/2

[2] 

An einϕ + c.c.

(4.16.2)

n=0

A N eiN ϕ/2 ζ (ϕ)

with yet another variant of the secular determinant, the zeta function

ζ (ϕ) =

%

N

A∗N e−iN ϕ/2

[2] 

An einϕ + c.c.

for N odd .

(4.16.3)

n=0

That zeta function is obviously real for real ϕ. All three variants of the sec˜ and ζ (ϕ) are thus expressed in terms of ular determinant, det(e−iϕ − F), A(ϕ), the first [N /2] secular coefficients An and the last one, A N = det(−U ). I leave ˜ for to the reader the task to write out the zeta function and its relation to A even N . In the foregoing zeta function the secular coefficients An with 1 ≤ n ≤ [N /2] can be expressed in terms of the traces tn using (4.15.10). The resulting expression, nothing but a variant of self-inversiveness, the so-called functional equation, plays an important role in the semiclassical quantization [43, 44], as we shall see in Sect. 10.5; in the semiclassical context, the identity in question is also called the “Riemann-Siegel lookalike”, that name referring to a certain analogy to a well known property of Riemann’s zeta function.

4.17 Higher Correlations of the Level Density 4.17.1 Correlation and Cumulant Functions I normalized  the mean density of levels ρ(φ) as a probability density, choosing N δ(φ − φi ). Common currency is also R1 (φ) = N ρ(φ), and this ρ(φ) = N −1 i=1 quantity actually has a better right to the name mean level density. We likewise met with several variants of the two-point correlation function which it is well to put into systematic view here. First, there is the density–density correlator ρ(φ)ρ(φ ) and its connected version S(φ, φ ) = ρ(φ)ρ(φ ) − ρ(φ)ρ(φ ). By purging the density–density correlator of “self-correlation” terms (coinciding

4.17

Higher Correlations of the Level Density

119

indices) and again changing normalization, one arrives at the second-order correlation function customarily defined as R2 (φ, φ ) =

1...N 

δ(φ − φi )δ(φ − φ j )

(4.17.1)

i= j

= N 2 ρ(φ)ρ(φ ) − N δ(φ − φ )ρ(φ) . Finally, by subtracting the product of two mean densities from R2 , one gets the connected second-order correlation function (cumulant or cluster function) C2 (φ, φ ) = −R2 (φ, φ ) + R1 (φ)R1 (φ ) .

(4.17.2)

Inasmuch as the limit of infinite dimension N is of interest, one refers (quasi) energies to their mean spacing 1/R1 (φ) and introduces the rescaled variable e = R1 (φ)φ ;

(4.17.3)

to secure meaningful limits for the two-point correlation functions, one proceeds to divide the product of two mean densities. Thus, Dyson’s asymptotic two-point cluster function, for instance, is arrived at as Y2 (e, e ) = Y (e − e ) = lim C2 (φ, φ )[R1 (φ)R1 (φ )]−1 N →∞

(4.17.4)

with the arguments of C2 , R1 expressed in terms of the rescaled (quasi)energy e. All of the above formulae hold for energy as well as quasi-energy spectra and the respective Gaussian and circular ensembles of random matrices. If the discussion is restricted to unitary matrices and their circular ensembles, as anticipated by the N N , e = 2π φ etc.; then, Dyson’s asymptotic twonotation, of course, R1 (φ) = 2π point cluster function results from the original second-order cumulant function as , e 2π )( 2π )2 → Y (e − e ). C2 (e 2π N N N Having sorted the various one- and two-point functions, we are prepared to face their higher order generalizations. The nth order correlation function generalizing R2 may be obtained by integrating the joint probability density PN of all N levels over N − n variables, Rn (φ , . . . , φ ) = 1

=

N! (N − n)!

n

 0

= 

δ(φ 1 − φi1 ) . . . δ(φ n − φin )

i 1 ...i n 2π



2π

dφn+1 . . . 0

(4.17.5)

dφ N PN (φ 1 , . . . , φ n , φn+1 , . . . , φ N ) .

120

4 Random-Matrix Theory

I shall not write out the general expression relating correlation functions to their connected versions, the cumulant or cluster functions [7], since we shall need only those up to order four; these read13 R1 (φ) = C1 (φ) , R2 (φ 1 , φ 2 ) = −C2 (φ 1 , φ 2 ) + C1 (φ 1 )C1 (φ 2 ) ,

(4.17.6)

R3 (φ 1 , φ 2 , φ 3 ) = C3 (φ 1 , φ 2 , φ 3 ) + C1 (φ 1 )C1 (φ 2 )C1 (φ 3 )   − C2 (φ 1 , φ 2 )C1 (φ 3 ) + two permutations R4 (φ 1 , φ 2 , φ 3 , φ 4 ) = −C4 (φ 1 , φ 2 , φ 3 , φ 4 )   + C1 (φ 1 )C3 (φ 2 , φ 3 , φ 4 ) + three permutations   + C2 (φ 1 , φ 2 )C2 (φ 3 , φ 4 ) + two permutations   − C2 (φ 1 , φ 2 )C1 (φ 3 )C1 (φ 4 ) + five permutations +C1 (φ 1 )C1 (φ 2 )C1 (φ 3 )C1 (φ 4 ) . Asymptotic cumulants are obtained by rescaling in the fashion of (4.17.3) and (4.17.4), Cn (φ 1 , . . . , φ n ) N →∞ R1 (φ 1 ) . . . Rn (φ n )   n  2π 1 2π n 2π . ,...,e = lim Cn e N →∞ N N N

Yn (e1 , . . . , en ) = lim

(4.17.7)

The first in the previous chain of equations holds for the Gaussian and circular ensembles, whereas the second is specialized to the circular case. At any rate, Yn depends on the n variables ei only through n − 1 independent differences. The cluster functions Cn (φ 1 , . . . , φ n ) are known [7] for the classic ensembles; needless to say, those with n > 1 vanish for the Poissonian ensembles thus signaling the absence of any correlations between levels. I shall not derive the Cn here but urge the reader to consult Chaps. 5 and 10 of Mehta’s book [7] and enjoy the unsurpassable beauty of the orthogonal–polynomial method used there. For the CUE and the GUE, the asymptotic versions Yn read Yn (e1 , . . . , en ) =



s(e12 )s(e34 ) . . . s(en1 )

(4.17.8)

with ei j = |ei − e j |; the sum runs over the (n − 1)! distinct cyclic permutations of the indices (1, 2, . . . , n), and s(e) is the familiar s(e) = (π e)−1 sin(π e). In particular, Y2 (e1 , e2 ) = Y (e12 ) = s(e12 )s(e21 ) = s(e12 )2 , and Y3 (e1 , e2 , e3 ) = s(e12 )s(e23 )s(e31 ) + s(e13 )s(e32 )s(e21 ) = 2s(e12 )s(e23 )s(e31 ).

13 Like most authors in random-matrix theory, Mehta [7] included, I keep here to a sign tradition differing from the otherwise more widespread one by the factor(−1)n−1 for the cumulant Cn .

4.17

Higher Correlations of the Level Density

121

N 1 N n A conclusion of relevance for what follows is that ( 2π )n Rn ( 2π e , . . . , 2π e ) N i remains finite in the limit N → ∞ when the {e } are fixed.

4.17.2 Ergodicity of the Two-Point Correlator Taking up Pandey’s thread of Sect. 4.10, I propose here to show in which sense every large random unitary matrix can be expected to display universal spectral fluctuations. For the sake of brevity, I shall confine myself to general $ e unitary matrices drawn from the CUE and prove the ergodicity of the integral 0 de Y (e ) of the two-point cluster function Y (e) which is the CUE average of the quantity

T (e, φ) = e −

1...N 



i= j

2π N

2 

e

de



0



φ+Δφ

φ

   dφ 

2π

δ φ − φi δ φ + e − φj Δφ N (4.17.9)

in the limit N → ∞. The φ-integral amounts to a local spectral average of which the angular brackets around the quantity T are meant as a reminder. The CUE average of the product of the two delta functions in (4.17.9) is $independent of φ, whereupon the e φ-integral becomes trivial and lim N →∞ T (e) = 0 de Y (e ) results. The quantity T (e) is well suited for studying the ergodicity of the two-point cluster function since the two integrals over φ and e smooth the product of delta functions such that one deals with a well-behaved object, even before any ensemble average. The integrated two-point cluster function in question is proven ergodic by showing that the ensemble variance of T  vanishes as the matrix size N and subsequently the mean number of levels ΔN = Δφ N /(2π) go to infinity. That variance reads VarT (φ, e) =

   2π 4   i= j k=l

N

e

de de



φ+Δφ

φ

0

dφ dφ

Δφ Δφ

(4.17.10)

3 2π

2π

e − φ j )δ(φ

− φk )δ(φ

+ e − φl ) × δ(φ − φi )δ(φ + N N 4 2π 2π

−δ(φ − φi )δ(φ + e − φ j ) δ(φ

− φk )δ(φ

+ e − φl ) . N N The sum over double pairs i = j, k = l in the foregoing variance has contributions from genuine quadruplets of levels, as well as from triplets and doublets, due to partial coincidences of indices,  i= j k=l

=

=  i jkl

+

=  i=k, j,l

+

=  i=l, j,k

+

=  j=k,i,l

+

=  j=l,i,k

+

=  i=k, j=l

+

=  i=l, j=k

,

(4.17.11)

122

4 Random-Matrix Theory

where the summands are as in (4.17.10). As a first step, I shall show now that the genuine triplets and doublets sum to a contribution of order 1/ΔN and can be dropped. The first of the two genuine-pair sums reads = 

(. . . ) =

i=k, j=l

  2π 4   i= j

N

e 0

de de

 φ

φ+Δφ

dφ dφ

Δφ Δφ

(4.17.12)

3 N 2π

e − φj) × δ(φ − φ

) δ(e − e

)δ(φ − φi )δ(φ + 2π N 4 2π

2π

−δ(φ − φi )δ(φ + e − φ j ) δ(φ

− φi )δ(φ

+ e − φl ) . N N Now, the definition of the second-order correlation function R2 (φ, φ ) comes in handy, as well as its property R2 (φ, φ ) = R2 (0, φ − φ ) following from the homogeneity of the CUE. The sum in work simplifies to = 



2π N



(. . . ) =

i=k, j=l

4 (4.17.13) e

× 0

 N 2π

1 de R2 (0, e)− 2π Δφ N N (N − 1)



e 0

 2π 2 de R2 (0, e) . N

)2 R2 ( 2π e) has the finite limit 1 − Y (e) as N → ∞ and ΔN = Recalling that ( 2π N N Δφ N /2π, we immediately see that after the limit N → ∞, only a term ∝ 1/ΔN remains which vanishes as ΔN → ∞. The second genuine-pair sum in (4.17.11), similarly checked, vanishes in the same sense. The first of the four genuine-triplet sums in (4.17.11) can be expressed in terms of the correlation functions R2 and R3 as 

 2π 4 (. . . ) = (4.17.14) N i=k, j,l 

 e  2π 2π

N −2  e

2π 2 1 de de

R3 (0, de R2 (0, e, e )− e) . × Δφ N N N (N − 1) 0 N 0 = 

N 1 N n )n Rn ( 2π e , . . . , 2π e ) again leaves an overall weight 1/ΔN The finiteness of ( 2π N such that this, as well as the other three genuine-triple sums, vanishes in the ordered double limit announced. The remaining sum over genuine quadruplets cannot be discarded so simply; it can be expressed in terms of R4 and R2 such that

4.17

Higher Correlations of the Level Density

123



  e   φ+Δφ dφ dφ

2π 4



VarT (φ, e) = de de (4.17.15) N Δφ Δφ 0 φ 3 2π

2π

× R4 (0, e , φ − φ , φ

− φ + e ) N N 2π

2π

1 4 (N − 2)(N − 3) R2 (0, e )R2 (0, e ) + O( ) . − N (N − 1) N N N −3) may be replaced by unity, imagining the irrelevant Here the fraction (NN−2)(N (N −1) 1 O( N ) term suitably modified. Of the two phase integrals over φ and φ

, one may be done right away since the integrand depends only on the difference φ − φ

;  and ΔN = Δφ N /2π, introducing φ

− φ = 2π N

 4   e  ΔN 1 2π de de

d (ΔN − ) (4.17.16) VarT (φ, e) = N (ΔN )2 0 0

2π 2π 2π 2π

2π

1 2π

× R4 (0, e, , + e ) − R2 (0, e )R2 (0, e ) + O( ) N N N N N N N  + same with  → − . At this point, the limit N → ∞ with fixed ΔN can be taken, whereupon we must turn to studying the behavior of the remaining triple integral for large ΔN . Expressing all correlation functions in terms of asymptotic cumulant functions according to (4.17.6) and (4.17.7), some further terms cancel, and we are left with N →∞



e

1 de de (ΔN )2





VarT (φ, e) −→ 0 3 × − Y4 (0, e , ,  + e

)



ΔN

d (ΔN − )

(4.17.17)

0

+Y3 (e , ,  + e

) + Y3 (0, ,  + e

) + Y3 (0, e ,  + e

) + Y3 (0, e , ) +Y2 (0, )Y2 (0,  + e

− e ) + Y2 (0,  + e

)Y2 (0,  − e ) −Y2 (0, ) − Y2 (0,  + e

) − Y2 (0,  − e ) − Y2 (0,  + e

− e ) 4 + same with  → − . Upon inserting the CUE result (4.17.8) for the asymptotic cluster functions, we conclude from the symmetry of the integrand under exchange of e and e

and the homogeneity of the cluster functions that the terms labelled “same with  → −” equal the previous ones in the curly brackets. Moreover, by patiently checking term by term, we find at least two factors s with their arguments containing  in each summand. It follows that the whole integrand falls off with growing  as (ΔN − )/ 2 such that the integral over  yields a sum of terms decaying asymptotically 1 ln ΔN , (ΔN , or faster. with ΔN like ΔN )2

124

4 Random-Matrix Theory

The announced ergodicity of the two-point cluster function with respect to the CUE is now established. It is well to keep in mind that the limit N → ∞ was taken first and eventually also ΔN → ∞ while keeping the rescaled phase variable e finite. Only the limiting form Y (e) is revealed as ergodic for finite arguments e; “finite e” means that given by the mean spacing 2π/N on the finest quasi-energy scale; on finite quasi-energy scales corresponding to infinite e, no ergodicity holds and no universality of the two-point cumulant can be expected. Critical readers might have wondered why I worked with a local spectral average rather than a global one (which would have replaced Δφ by 2π and correspondingly ΔN by N ) in the integrated form factor T (e); they are kindly invited to spot the point in the foregoing reasoning where an attempted merger of the two ordered limits into a single one would have to be regretted.

4.17.3 Ergodicity of the Form Factor The title of this subsection may sound like too much of a mouthful. Indeed, (if N is even) the sequence of N /2 traces tn = TrF n of a unitary N × N matrix F with n = 1, . . . , N /2 uniquely determines the spectrum. By following the n dependence of the form factor |tn |2 of a single matrix F, one thus sets the eye on “system specific” rather than only universal properties. There is no reason to expect much similarity between the form factor |tn |2 of a single matrix and its ensemble average |tn |2 . However, just as a local spectral average is involved in establishing the ergodicity of, say, the two-point cluster function of the level density we may subject the form factor of a single matrix F to a local “time average,” i.e., a smoothing of the n-dependence over some interval Δn. Moreover, so as not to confuse fluctuations and systematic n dependence, it is reasonable to consider the normalized form factor |tn |2 /|tn |2 and its local time average

|tn |2 / |tn |2  =

1 Δn



n+Δn/2

|tn |2 / |tn |2 .

(4.17.18)

n =n−Δn/2

If it can be shown that this smoothed normalized version of the form factor has a small ensemble variance within the appropriate circular ensemble, we can indeed expect some degree of universality in that quantity for a single Floquet matrix F, provided, of course, that the pertinent classical dynamics is globally chaotic. We shall in fact establish such ergodicity [46] by showing that the ensemble variance  Var|tn | / |tn 2

|2 

=

1 Δn

2



n+Δn/2 n ,n

=n−Δn/2



|tn |2 |tn

|2 |tn |2 |tn

|2

 −1

(4.17.19)

4.17

Higher Correlations of the Level Density

125

vanishes in the limit N → ∞, Δn → ∞, again restricting ourselves to the CUE. The yet simpler Poissonian case is recommended to the reader as Problem 4.15. Obviously, we need to calculate the fourth moments |tn |4 and |tn |2 |tn |2 . The interested reader who has labored through the calculation of the mean squared traces will have no difficulties in extending the Taylor expansion of the characteristic function (4.14.5) to the next nonvanishing term (∝ |k|4 ) which yields ⎧ 4 N ⎪ ⎪ ⎪ ⎨2n 2 |tn |4 = |t−n |4 = ⎪ N + 2n(n − 1) ⎪ ⎪ ⎩ 2N (N − 12 )  2 n>0 = 2 |tn |2 + |t2n |2 − 2|tn |2 .

for n = 0 for 1 ≤ n ≤ N /2 for N /2 ≤ n ≤ N for N ≤ n (4.17.20)

This is consistent, in the limit of large dimension N and for n > 0, with a Gaussian distribution of the tn with vanishing mean and variance |tn |2 given by (4.14.14), a first hint at a more general result to be established below. More precisely speaking, μ the moments tn (tn∗ )ν up to order μ+ν = 4 display strictly Gaussian relationships for 0 < n ≤ N /2 while for larger orders, such behavior prevails to within corrections of relative order 1/N . 2

At any rate, the ensuing standard deviation (|tn |4 − |tn |2 )1/2 has no tendency to become small as the dimension N increases and defines an n dependent “band” around the variance |tn |2 within which Gaussian statistics would suggest that the |tn |2 of an individual matrix lies with probability e−1/2 − e−3/2 ≈ 0.38. This quantifies the above remarks about the nonuniversality of the n dependence of the form factor. To establish the cross-correlation |tm |2 |tn |2 , we may extend the definition (4.14.5) of the characteristic function to two different traces  5  −imφ −inφ ˜P(km , kn ) = exp − i i i + kn e ) + c.c. (km e 2 i

(4.17.21)

and analogously to (4.14.11) arrive at a Toeplitz determinant, ˜ m , kn ) = det M = exp Tr ln M , P(k (4.17.22)

 2π  dφ iφ(μ−ν) i Mμν = exp − (km e−imφ + kn e−inφ ) + c.c. e 2π 2 0 with μ, ν = 1, . . . , N . I shall have to say something about such Toeplitz determinants in the following subsection. For now we can be content with a single term in its Taylor expansion,

126

4 Random-Matrix Theory

∗ ∗ 4 2 2 ˜ ∂ 4 P/∂k m ∂km ∂kn ∂kn |km =kn =0 = (−i/2) |tm | |tn | . With a little patience and the guidance of Sec. 4.14.2, the reader will check for m, n > 0 that

|tm |2 |tn |2 = |tm |2 |tn |2 + |tm+n |2 + |tm−n |2 − 2|tmax(m,n) |2 = N 2 + (N − m)(N − n)Θ(N − m)Θ(N − n) −N (N − m)Θ(N − m) − N (N − n)Θ(N − n) −(N − m − n)Θ(N − m − n) − (N − |m − n|)Θ(N − |m − n|) +2[N − max(m, n)]Θ[N − max(m, n)] .

(4.17.23)

Thus, we are led to the following measure of the relative cross-correlation between two traces: ⎧ N −m−n ⎪ ⎨ mn |tm n −m+n − 1 = − NNmin(N ,n) ⎪ |tm |2 |tn |2 ⎩ 0 |2 |t

|2

for N − n ≤ m ≤ N for N < m ≤ N + n otherwise

(4.17.24)

where without loss of generality I have assumed that m > n > 0. Statistical independence of the two traces would entail vanishing relative cross-correlation, and that is in fact the behavior prevailing in most ranges of m, n. Otherwise, where the cross-correlation does not vanish, it at least turns out small, i.e., of order 1/N ; more precisely, −1/N ≤ |tm |2 |tn |2 /(|tm |2 |tn |2 ) − 1 ≤ 0. We may conclude from the above investigations that the traces tn of CUE matrices behave, at least as far as moments of orders 1 to 4 are concerned, as if they were independent Gaussian random variables to within corrections of relative weight 1/N . That same statement holds true for matrices from the COE and the CSE; in the latter case the precision is only O(1/ ln N ) when n is near half the Heisenberg = N ; the interested reader is referred to [46] for these cases and to [45] time n CSE H for higher order moments in the case of the CUE. Equipped with |tm |2 |tn |2 and |tm |4 , we can estimate the CUE variance (4.17.19) of the time-averaged form factor. Since the cross-correlation (4.17.24) of two traces is nonpositive for m = n, we get an upper limit for Var|tn |2 / |tn |2  by dropping the “off-diagonal” terms with n = n

in (4.17.19). For the diagonal terms with n > 0, we get an estimate from the second line of (4.17.20), |tn |4 − (|tn |2 )2 ≤ (|tn |2 )2 since 2|tn |2 ≥ |t2n |2 . It follows that Var|tn |2 / |tn |2  ≤

1 Δn

Δn→∞

−→ 0 .

(4.17.25)

Thus, the (time-averaged) form factor turns out to be every bit as ergodic as is, e.g., the (locally spectrally averaged) two-point cluster function, its unruly fluctuations from one n to the next notwithstanding.

4.17

Higher Correlations of the Level Density

127

4.17.4 Joint Density of Traces of Large CUE Matrices A powerful theorem about Toeplitz determinants, due originally to Szeg¨o and Kac and extended by Hartwig and Fischer [47], helps to find the marginal and joint distributions of the traces tn of CUE matrices with finite “times” n in the limit, as the dimension N goes to infinity [42]. In that limit, the finite-time traces will turn out to be statistically independent and to have Gaussian distributions with the means and variances already determined above, tn = 0, |tn |2 = n. Once more I invoke the ubiquitous theorem (4.12.4) for the CUE mean of a symmetric function of all eigenphases, again for that function a product. The integral over the phase φi can then be pulled into the ith row of the determinant in (4.12.4), whereupon the average becomes the Toeplitz determinant N ,

f (φm ) = det( fl−m ) ≡ T ({ f }) ,

m=1

l, m = 1, 2, . . . , N ,

(4.17.26)

the elements of which are the Fourier coefficients 

2π

fm = 0

dφ imφ e f (φ) 2π

(4.17.27)

of the function f (φ). The Hartwig–Fischer theorem assigns to the determinant the asymptotic large-N form  T ({ f }) = exp Nl0 +

∞ 

 nln l−n

(4.17.28)

n=0

 imφ with the ln the Fourier coefficients of ln f (φ), i.e., ln f (φ) = ∞ . The n=−∞ l n e function f (φ) must meet the following four conditions for the theorem to hold: (1)  2 |n|| f | < ∞, f (φ) = 0 for 0 ≤ φ < 2π, (2) arg f (2π ) = arg f (0), (3) ∞ n n=−∞ and (4) ∞ n=−∞ | f n | < ∞. For a first application, I consider the marginal distribution of the nth trace tn whose Fourier transform is the characteristic function (4.14.11) and involves   i f (φ) = exp − (ke−inφ + k ∗ einφ ) . 2

(4.17.29)

Provided that n remains finite as N → ∞, that function fulfills all conditions of the theorem (see below for an explicit check), and the only nonvanishing Fourier ∗ = −ik/2. The asymptotic form of the coefficients of its logarithm are ln = l−n

128

4 Random-Matrix Theory

Toeplitz determinant thus reads e−nk /4 and yields the density of the nth trace as its Fourier transform as the Gaussian already anticipated several times, 2

P(tn ) =

1 −|tn |2 /n . e πn

(4.17.30)

The foregoing reasoning is easily extended to the joint density of the first n traces  N imφ + ˜ 1 , . . . , kn ) e l ) whose Fourier transform P(k P(t1 , . . . , tn ) = nm=1 δ 2 (tm − l=1 is once more of the form (4.17.26) with the function f (φ) from (4.17.29) generalized to the sum  n i  −imφ ∗ imφ (km e + km e ) ] . (4.17.31) f (φ) = exp − 2 m=1 ∗ The nonvanishing Fourier coefficients of ln f (φ) are now lm = l−m = −ikm /2 with m = 1, . . . , n and entail a limiting form of the characteristic function which is just the product of the marginal ones met above. The joint density then comes out as the announced product of marginal distributions

 n   1 exp − |tm |2 /m . P(t1 , . . . , tn ) = n!π n m=1

(4.17.32)

The result obviously generalizes to the joint density of an arbitrary set of finitetime traces. The previously resulting hints of statistical independence and Gaussian behavior of all finite-time traces in the limit N → ∞ are thus substantiated. To appreciate the importance of the restriction to finite n and to avoid a bad mathematical conscience, it is well to verify the aforementioned conditions of the Hartwig–Fischer theorem on the function f (φ) in (4.17.31). The first two of them are clearly fulfilled since i ln f (φ) is real, continuous, and (2π)-periodic. The third and fourth conditions are met since the derivative f (φ) is square integrable (provided n remains finite!), 

2π 0

dφ| f (φ)|2 =

∞  m=−∞

m 2 | f m |2 = π

n 

m 2 |km |2 < ∞ .

(4.17.33)

m=1

2 Since check condition ∞ |m| ≤ |m|2 in the foregoing Parseval inequality, we  (3), m=−∞ |m|| f m | < ∞. Finally, Cauchy’s inequality yields ∞ m=−∞ | f m | =    | f 0 | + m=0 | m1 ||m f m | ≤ | f 0 | + ( m=0 | m1 |2 )1/2 ( m=0 |m f m |2 )1/2 ; due to Parsefal’s  inequality and the convergence of m=0 1/m 2 , we find condition (4) met as well. For finite dimension N , the independence as well as the Gaussian character of the traces are only approximate; both properties tend to get lost as the sum of the orders (times) of the traces involved in a set increases. This is quite intuitive a finding

4.18

Correlations of Secular Coefficients

129

since already all traces for a single unitary matrix are uniquely determined by the N eigenphases such that only N /2 traces are linearly independent.

4.18 Correlations of Secular Coefficients I propose here to study the autocorrelation of the secular polynomial of random matrices from the circular ensembles [41, 42, 48]. These correlations provide a useful reference for the secular polynomials of Floquet or scattering matrices of dynamic systems and their semiclassical treatment. In particular, we shall be led to an interesting quantum distinction between regular and chaotic motion. A starting point is the secular polynomial of an N × N unitary matrix F, det(λ − F) =

N N  , (λ)n A N −n = (λ − e−iφi ) , n=0

(4.18.1)

i=1

which enjoys the self-inversiveness discussed in the preceding subsection. An autocorrelation function of the secular polynomial for any of the circular ensembles may be defined as β

PN (ψ, χ ) =

,

(e−iφi − e−iψ )(eiφi − eiχ ) ,

(4.18.2)

i=1...N

with the overbar indicating an average over the circular ensemble characterized by the level repulsion exponent β. For the orthogonal and unitary circular ensembles, this correlation function may be identified with det(F − e−iψ )(F † − eiχ ). In the symplectic case, however, the definition (4.18.2) accounts only for one level 2  N n = per Kramers’ doublet such that we have det(F − λ) = n=0 (−λ) a N −n + N −iφi 1 4 2 − λ) and thus PN (ψ, χ ) = [det(F − e−iψ )(F † − eiχ )] 2 . i=1 (e Due to the homogeneity of the circular ensembles, the correlation function in question depends on the two phases ψ and χ only through the single unimodular variable x = ei(χ−ψ) ; thus it may be written in terms of the auxiliary function f (φ, x) = (e−iφ − x)(eiφ − 1)

(4.18.3)

as β

PN (x) =

, i

f (φi , x) =

N  n=0

x n | A n |2 .

(4.18.4)

130

4 Random-Matrix Theory

In writing the last member of the foregoing equation, I have once more invoked the homogeneity of the circular ensembles to conclude that Am A∗n = δmn .

(4.18.5)

Obviously, our correlation function may serve as a generating function for the β ensemble variances of the secular coefficients, |An |2 = n!1 d n PN (x)/d x n |x=0 . β The correlation function PN (x) and the variances | An |2 are most easily calculated in the Poissonian case β = 0,  PN0 (x)

2π

= 0

|An |2 =

N

dφ f (φ, x) 2π

= (1 + x) N ,

  N . n

(4.18.6)

The treatment of the CUE, COE, and CSE proceeds in parallel to the calculation of the characteristic function of the traces in the preceding section. As in (4.14.5), we must average a product, now with factors f (φi , x) rather than exp(−i cos nφi ). For the CUE, that replacement changes (4.14.11) into  PN2 (x) = det

2π 0



dφ i(m−n)φ f (φ, x) e 2π

 = det (1 + x)δ(m − n) − δ(m − n + 1) − xδ(m − n − 1) =

N 

xn .

(4.18.7)

n=0

The CUE variance of the secular coefficient thus comes out independent of n, |An |2 = 1 ,

(4.18.8)

and that independence is in marked contrast to the binomial behavior (4.18.6) found for the Poissonian case. Similarly, for the orthogonal and symplectic ensembles, we are led to Pfaffians of β antisymmetric matrices Aβ , PN (x) = Pf( Aβ (x))/Pf(Aβ (0)). In the orthogonal case where for simplicity I assume even N , one finds (4.14.23) replaced with

4.18

Correlations of Secular Coefficients

A1mm (x)

131

 2π 1



= dφdφ sign(φ − φ ) f (φ, x) f (φ , x)ei(mφ+m φ ) 4π i 0    (1 + x)2 1 1 = +x − δm,−m

m m + 1 m + 1 1  1  − (1 + x) − δm,−m +1 + x δm,−m −1 m m 1 x2 + δm,−m +2 + δm,−m −2 m−1 m+1

(4.18.9)

where again the underlined index runs in unit steps through the half-integers in 0 < |m| ≤ (N − 1)/2. The analogue of (4.14.33) in the symplectic case reads A4mm (x)



= (m − m )



2π

dφ i(m+m )φ f (φ, x) e 2π

(4.18.10) 0  = (m − m ) (1 + x)δ(m + m ) − δm,−m −1 − xδm,−m +1

with 0 < |m| ≤ (2N − 1)/2. The evaluation of the Pfaffian is a lot easier in the latter case than in the former since A1 is14 “pentadiagonal” but A4 is only “tridiagonal.” Moreover, the 2N × 2N matrix A4 falls into four N × N blocks, and the two diagonal ones vanish such that the Pfaffian Pf( A4 (x)) equals, up to the sign, the determinant of either off-diagonal N × N block. One reads off the following recursion relation for the correlation function PN4 (x) = Pf( A4 (x))/Pf(A4 (0)) in search, PN4 (x) = (1 + x)PN4 −1 (x) − x

(2N − 2)2 P 4 (x) ; (2N − 1)(2N − 3) N −2

(4.18.11)

this in turn is easily converted into a recursion relation for the variances , ) 2 (N −1) 2 | A(N | + n | = | An

1 (N −1) 2 (2N − 2)2 |An−1 | − | A(N −2) |2 . n n(2N − 1)(2N − 3) n−1

(4.18.12)

) ) With A(N = 1 for all naturals N , one finds the solution (with A(N n typographi0 cally stripped back down to An )

 2  −1 N 2N | A n |2 = n 2n

6 N 1

≈

N . πn(N − n)

(4.18.13)

14 To remove the following quotation marks, think of the rows of the determinant swapped pairwise, the first with the last, the second with the last but one, etc.

132

4 Random-Matrix Theory

Finally mustering a good measure of courage, I turn to the orthogonal case, i.e., the Pfaffian of the “pentadiagonal” antisymmetric matrix A1m,m (x) given in (4.18.10). A little convenience is gained by actually removing those quotation marks in the way indicated in the footnote, i.e., by looking at the matrix Am,m ≡ A1m,−m ; recall that the underlined indices take the half-integer values −M, −M + 1, . . . , M with M = (N − 1)/2; the determinants of A and A1 can differ & at most in their sign 1 (x) = det A1 (x)/ det A1 (0) = which is irrelevant for our correlation function P N √ det A(x)/ det A(0). The key to the latter square root is to observe that the N × N matrix Am,m (x) fails to be the product BC of the two tridiagonal matrices 1+x x 1 δm,m + δm,m −1 + δm,m +1 m m+1 m−1 = −(1 + x)δm,m + xδm,m −1 + δm,m +1

Bm m = − Cm m

(4.18.14)

only in the two elements in the upper left and the lower right corner. As a remedy, I momentarily extend the matrix dimension to N + 2, bordering the matrices A, B, C by frames of single rows and columns as ⎛

⎞ 1 1 ... 1 1 1 1 ⎜ (−M − 1)−1 ⎟ 0 ⎜ ⎟ ⎜ ⎟ 0 0 ⎜ ⎟ ⎟ .. .. B =⎜ ⎜ ⎟, . B . ⎜ ⎟ ⎜ ⎟ 0 0 ⎜ ⎟ −1 ⎠ ⎝ x(M + 1) 0 0 0 0 ... 0 0 1 ⎛

−M − 1 x 0 . . . 0 0 ⎜ 0 ⎜ ⎜ 0 ⎜ .. C =⎜ ⎜ C . ⎜ ⎜ 0 ⎜ ⎝ 0 0 0 0 ... 0 1

⎞ −M − 1 −M ⎟ ⎟ −M + 1 ⎟ ⎟ ⎟ .. ⎟, . ⎟ M −1 ⎟ ⎟ M ⎠ M +1

(4.18.15)

(4.18.16)

⎛

⎞ −M − 1 0 0 . . . 0 0 0 ⎜ 1 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ 0 ⎜ ⎟ .. ⎟ . .. A =⎜ ⎜ ⎟ . A . ⎜ ⎟ ⎜ ⎟ 0 0 ⎜ ⎟ ⎝ 0 ⎠ 0 0 0 0 ... 0 1 M + 1

(4.18.17)

4.18

Correlations of Secular Coefficients

133

A = B C . The The bordered matrix A now enjoys the product +structure in full, M+1 C B and det A = various determinants obviously obey det = m=−M−1 m det −(M + 1)2 det A such that upon drawing the square root of det A , we get 6 PN1 (x)

=

det A(x) = (−1)? det A(0)



2 N +1

2 det C .

(4.18.18)

It remains to evaluate the (N + 2) × (N + 2) determinant det C . Since the first column contains just a single non-zero element, we reduce the dimension to N + 1, ⎛

−(1 + x) x ⎜ 1 −(1 + x) ⎜ ⎜ 0 1 ⎜ det C ⎜ .. .. =⎜ . . −(M + 1) ⎜ ⎜ 0 0 ⎜ ⎝ 0 0 0 0

... ... ... ... ... ... ...

⎞ −M −M + 1 ⎟ ⎟ −M + 2 ⎟ ⎟ ⎟ .. ⎟. . ⎟ x M −1 ⎟ ⎟ M ⎠ −(1 + x) 1 M +1 0 0 0 .. .

Here the original N × N matrix C appears in the upper left corner. Starting with the uppermost one, we now add to each row the sum of all of its lower nneighbors such that in the elements of the new last column the sums f (n) = m=−M−1 m appear and the determinant reads ⎛

−x ⎜ 1 ⎜ ⎜ 0 ⎜ C det ⎜ = ⎜ ... M +1 ⎜ ⎜ 0 ⎜ ⎝ 0 0

0 −x 1 .. .

... 0 ... 0 ... 0 . . . . ..

f (M) f (M − 1) f (M − 2) .. .

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ 0 . . . 0 f (−M + 1) ⎟ ⎟ f (−M) ⎠ 0 . . . −x 0 . . . 1 f (−M − 1)

We finally annul the secondary diagonal with unity as elements: Beginning at the top, we add to each row the x1 -fold of its immediate upper neighbor. The arising determinant equals the product of its diagonal elements, det C = (M + 2M+1 1)(−x)2M+1 n=0 ( x1 )n f (−M − 1 + n). Recalling M = (N − 1)/2 with N even, we get from (4.18.18) the correlation function in search as

PN1 (x) =

 n(N − n) xn 1 + N +1 n=0

N 

(4.18.19)

134

4 Random-Matrix Theory

and read off the mean squared secular coefficients for the COE, |An |2 = 1 +

n(N − n) . N +1

(4.18.20)

Let us summarize and appreciate the results for the four circular ensembles. The mean squared secular coefficients (4.18.6), (4.18.8), (4.18.13), and (4.18.20) are special cases of β

|An |2 =

  N Γ (n + 2/β)Γ (N − n + 2/β) n Γ (2/β)Γ (N + 2/β)

(4.18.21)

for the appropriate values of the repulsion exponent β. Actually, the latter general formula has been shown [42] to be valid for arbitrary real β, provided the joint density of eigenvalues (4.11.2) is so extended. β

With the family relationship between the |An |2 pointed out, the differences should be commented on as well. For fixed n = 0, N , we face a growth of that function with decreasing β. That trend is not surprising; an increase of the degree of level repulsion implies a tendency toward equidistant levels. For β → ∞, one must expect a perfectly rigid spectrum15 according to the secular equation λ N − 1 = 0, i.e., An = 0 except for n = 0, N . Conversely, the Poissonian statistics arising for β → 0 entails the weakest spectral stiffness and thus the largest mean squared   secular coefficients. For a large dimension N , the Poissonian limit | A0n |2 = Nn β

is exponentially larger than the three other distinguished | An |2 with β = 1, 2, 4. The difference in question may in fact be seen as one of the quantum criteria to distinguish regular (β = 0) from chaotic dynamics (β = 1, 2, 4), as will become clear in the following section where a detailed comparison of random matrices and Floquet matrices of dynamical systems will be presented; see in particular Fig. 4.9. β The correlation function PN (x) itself is due some attention. To give it a nice appearance, we return to (4.18.4) and realize that due to the symmetry of the |an |2 β under n ↔ N − n, we may switch from PN (x) to a function [48] C β (η) =

N

β

PN (e−i2πη/N )eiπη β PN (1)

=

n=0

|an |2 e−i2πη(n/N −1/2) N 2 n=0 |an |

(4.18.22)

β

which is real for real η; note the normalization to C N (0) = 1. Using our results for the variances | An |2 and doing sums as integrals,

Indeed, generalizing the joint distribution of eigenvalues (4.3.15) to arbitrary real β = n − 1, 2 one is led to a Wigner distribution Pβ (S) = AS β e−B S with A and B fixed by normalization and S = 1 which for β → ∞ approaches the delta function δ(S − 1).

15

4.19

Fidelity of Kicked Tops to Random-Matrix Theory

⎧  N ⎪ 2πη ⎪ ≈ exp − cos ⎪ N ⎪ ⎪   ⎪ 2 ⎨3 sin πη ∂ 1 + π12 ∂η 2 πη C β (η) = 2 ⎪ sin πη ⎪ ⎪ πη ⎪ ⎪ ⎪ ⎩ J0 (π η)

2π 2 η2 N

135

 for

β=0

for

β=1

for

β=2

for

β=4

.

(4.18.23)

Checking for the small-η behavior, we find that the decay of the correlation function proceeds faster, the larger the level-repulsion exponent β, and actually much more slowly in the Poissonian case than in any of the three cases corresponding to classical chaos.

4.19 Fidelity of Kicked Tops to Random-Matrix Theory Periodically kicked tops are unsurpassed in their faithfulness to random-matrix theory, and that fact will be put into view here. As already mentioned several times, tops can be designed so as to be members of any universality class. The Floquet operators employed for the unitary class read    τ y Jy2 τz Jz2 − iαz Jz exp −i − iα y Jy F = exp −i 2j + 1 2j + 1 



 τx Jx2 × exp −i − iαx Jx . 2j + 1

(4.19.1)

Indeed, if all torsion constants τi and rotational angles αi are non-zero and of order unity, no time-reversal invariance (nor any geometric symmetry) reigns even approximately and the classical dynamics is globally chaotic. To secure an antiunitary symmetry as time-reversal invariance while keeping global classical chaos, we just have to erase torsion and rotation with respect to one axis; the resulting Floquet operator then pertains to the orthogonal class, as discussed in Sect. 2.12. Finally, simple symplectic tops are attained by choosing a representation of half-integer j for 

τ1 Jz2 τ3 (Jx Jy + Jy Jx ) τ2 (Jx Jz + Jz Jx ) −i −i F = exp −i 2j + 1 2(2 j + 1) 2(2 j + 1)  2  τ4 Jz , × exp −i 2j + 1



(4.19.2)

136

4 Random-Matrix Theory

P(s)

I(s)

s Fig. 4.5 Spacing distributions P(S) and their integrals I (S) for kicked tops with the Floquet operators (4.19.1) and (4.19.2), with rotational angles and torsion constants as given in the text. Hilbert space dimension 2 j + 1 = 2,001 for the Poissonian (β = 0), orthogonal (β = 1), and unitary (β = 2) cases; in the symplectic case (β = 4), distributions for all half-integer j between 49.5 and 99.5 were superimposed while keeping the coupling constants τi and thus the classical dynamics unchanged

since no geometric symmetries are left while the appearance of only two unitary factors secures an antiunitary symmetry.16 Figure 4.5 shows the spacing distributions P(S) and their integrals I (S) = $S

0 d S P(S ) for tops with the Floquet operators just listed. The graphs pertaining to the unitary (τz = 10, τ y = 0, τx = 4, αz = α y = 1, αx = 1.1), orthogonal (τz = 10, αz = 1, α y = 1, all others zero), and Poissonian (τz = 10, αz = 1, all others zero) cases were obtained for j = 1 000. The symplectic case is a little more obstinate for large matrix dimensions for which reason the pertinent graph was constructed by averaging over the distributions obtained from all half-integers j between 49.5 and 99.5, keeping the control parameters fixed to τ1 = 10, τ2 = 1, τ3 = 4, τ4 = 2.1. In all cases, agreement with the predictions of random-matrix theory is good. Incidentally, for the databases involved, the Wigner surmises (4.5.2) suffice as representatives of random-matrix theory. 16 The reader might (and should!) wonder whether the “orthogonal” version of (4.19.1) goes “symplectic” for half-integer j. It does not. The reason is T 2 = +1 for all j in that case since two components of an angular momentum can simultaneously be given real representations.

4.19

Fidelity of Kicked Tops to Random-Matrix Theory

137

When trying to build the two-point cluster function from a set of 2 j + 1 quasienergies, one must first worry about how to generate a smooth function from the sum over products of two delta functions provided by the naive definition17  y(φ, e) = 1 −

2π N

2  i= j

δ(φ − φi )δ(φ +

2π e − φj) . N

(4.19.3)

We encountered a similar problem in proving the ergodicity of the two-point cluster function for the CUE in Sect. 4.17.1. The delta functions there were smoothed by subjecting y(φ, e) to both a local spectral average and an integral over e (see (4.17.9)). I resort here to the same strategy with one slight modification: Inasmuch as the spectrum to be examined has a density of levels fluctuating about the uniform N , we make optimal use of the data by employing a global rather than only mean 2π local spectral average18 and consider the integral  I (e) = 0

e

de

 0

2π

 dφ 1   e2π Θ y(φ, e ) = e − + φi − φ j . 2π N i= j N

(4.19.4)

For this “self-averaging” quantity, Fig. 4.6 indeed reveals fine agreement between tops and the random-matrix predictions for the Poissonian, unitary, orthogonal, and symplectic universality classes. A glance at Fig. 4.7 suggests more delicacies in the contrast between the form factor |tn |2 of a single top and the mean over the pertinent circular ensemble, here the CUE. As already argued in Sect. 4.17.11, the form factor derived from a single spectrum performs a rather unruly dance about the ensemble mean in its n dependence. No disobedience to random-matrix theory is to be lamented, however, since that theory itself predicts such fluctuations [46]. The shortdashed lines in the graphs of part (a) of Fig. 4.7 define a band of one standard deviation −1 %  2 |tn |4 − |tn |2 Δ(n, β) = |tn |2

(4.19.5)

around the CUE mean, the latter shown as the full curve. Inasmuch as the underlying matrix dimension is large, the asymptotic Gaussian behavior found in Sect. 4.17.3

17 In general, two integrations produce a piecewise constant curve with upward and downward steps of order N1 . One of these integrations will always be over the center (quasi-)energy, the range either the whole spectrum (if the mean density is uniform) or over a window large enough to contain a large number N of levels but small enough for the mean density not to vary within. The second integration might, in variation of the procedure chosen below, over the (quasi-)energy offset with a window size small compared to the mean spacing but large enough for the product of the two integration ranges in question to contain many pairs of levels. 18

While the technicalities of the proof of ergodicity demanded two independent, large parameters, N and ΔN , no such needs arise here.

138

4 Random-Matrix Theory

I(e)

I(c)

e

e

Fig. 4.6 Integrated and spectrally averaged two-point cluster function according to (4.19.4) for kicked tops (rugged curves) with Hilbert space dimensions and coupling constants as for Fig. 4.5. The full curves represent the predictions of the pertinent circular ensembles

can be expected to prevail. Thus, we may expect the fraction e− 2 − e− 2 ≈ 0.3834 of all points |tn |2 to lie within that band and actually find the fraction 0.3966 for the top. Similarly, the abscissa and the longdashed curve surround a band of two standard deviations which should contain 86.47% of all points and does 87.0%. To 1

3

Fig. 4.7 (a) Time dependence of form factor of unitary kicked top (dots) with Hilbert space dimension and control parameters as for (4.5). The full curve depicts the CUE form factor (4.14.16). In good agreement with random-matrix theory, about 40% (87%) of all dots lie within a band of one (two) standard deviation(s) (4.19.5) around the CUE mean, the respective bands are delimited by the shortdashed (longdashed) lines (b) Histogram for the normalized form factor τ = |tn |2 /|tn |2CUE for unitary kicked top with j = 500 and coupling constants as for (4.5). The full curve depicts the prediction of random-matrix theory based on the Gaussian distribution (4.17.29) of the traces

4.19

Fidelity of Kicked Tops to Random-Matrix Theory

139

Fig. 4.8 (a) Time dependence of local time average of normalized form factor |tn |2 /|tn |2CUE  according to (4.17.18) for unitary kicked top with j = 500 and coupling constants as for (4.5); the time window is Δn = 20 ≈ (2 j + 1)/50 (b) The full curve gives the upper bound 1/Δn of CUE variance of |tn |2 /|tn |2CUE  according to (4.17.25) vs. the time window Δn; the dots depict the standard deviation from the mean of |tn |2 /|tn |2CUE  for the same unitary top as in (a), determined from clouds as in (a).

further corroborate the obedience of the top to random-matrix theory, we display,  1/2 in part (b) of Fig. 4.7, a histogram for the distribution of |tn |2 / |tn |2CUE which indeed closely follows the full curve provided by the Gaussian distribution of the traces. Figure 4.8 presents a check on the ergodicity of the form factor discussed in Sect. 4.17.11. Part (a) of the figure pertains to the “unitary” top specified above (here with j = 500) and shows a local time average |tn |2 /|tn |2  taken over a time window Δn = 20 ≈ N /50. The fluctuations from one n to the next are much smoothed compared to the nonaveraged form factor in Fig. 4.7; they define a band whose width should be compared with the CUE variance Var|tn |2 /|tn |2  estimated in (4.17.25). In fact, the CUE bound 1/Δn of (4.17.25) is respected by the top: Part (b) of Fig. 4.8 shows that bound as the full curve and the numerically found width of the fluctuations of the local time average for various values of the time window Δn as dots. The statistics of secular coefficients of Floquet matrices remain to be scrutinized. A simple indicator is the n dependence of the coefficients an whose mean squared moduli were evaluated in Sect. 4.19. Remarks similar to those above on the n dependence of the traces tn apply here: A single Floquet matrix produces a sequence of secular coefficients fluctuating so wildly that the underlying systematic n dependence becomes visible only after some careful smoothing. As done for the traces above, one could average the |an |2 over some interval Δn or over different dimensions N and plot versus n/N , while keeping rotational angles and torsion constants fixed. Instead, Fig. 4.9 is based on averages over small volume elements of control parameter space, while the dimension N ∝ 1/ remains fixed. All four universality classes are seen to yield n dependences of |an |2 not dissimilar to the averages over the pertinent circular ensemble. In particular, the integrable case has its |an |2 so overwhelmingly larger than any of the other three classes that we may indeed see

140

4 Random-Matrix Theory

Fig. 4.9 Mean squared moduli of secular coefficients |an |2 for the circular ensembles (dashes) and for kicked tops (dots) vs n. The averages for the tops are based on 20,000 unprejudiced random points in control parameter space from the intervals τz ∈ [10, 15], αz ∈ [0.6, 1.2], α y ∈ [0.6, 1.2] for the orthogonal top; for the unitary top, αx = 1.1, α y = αz = 1, τ y = 0, andτx , τz ∈ [10, 25]; and for the symplectic top, τ1 ∈ [15, 20], τ4 ∈ [10, 15], τ2 = 1, τ3 = 4. All cases involve spectra of 51 eigenvalues

that difference as one of the quantum indicators of the alternative regular/chaotic. The term “not dissimilar” needs to be qualified: The differences in the circularensemble averages are appreciably larger for the secular coefficients than for any of the other quantities shown above. Worse yet, the differences change when the underlying volume element in control parameter space is shifted. The reason for this strong nonuniversality of the secular coefficients can be understood on the basis of the semiclassical periodic-orbit theory to be discussed in Chap. 10. Periodic orbits of period n will turn out to determine the nth trace tn and through Newton’s formulae (4.15.8) the secular coefficients am with all m ≥ n; system specific properties show up predominantly in short periodic orbits with periods not much larger than unity which do not affect the large-m traces tm but can and indeed do affect the large-m coefficients am . To summarize, no greater faithfulness to random-matrix theory could be expected than that met in tops. The importance of these systems as standard models of both classical and quantum chaotic behavior will be further underscored in Chap. 7 by showing that they comprise, in a certain limit, the prototypical system with localization, the kicked rotator. A rough characterization of that limit at which we shall arrive there is that a top will begin to show quantum localization as soon as an appropriate localization length is steered to values smaller than the dimension 2 j + 1.

4.20

Problems

141

4.20 Problems 4.1 Show that S in (4.2.16) is indeed a symplectic matrix. 4.2 Show that the most general unitary 2 × 2 matrix can be specified with the help of four real parameters α, ε as U = exp (iα +iε ·σ ), and characterize the specialization to (4.3.6). In which geometric sense does (4.3.6) generalize (4.2.11) to a finite (rather than infinitesimal) change of basis? 4.3 Show the invariance of the differential volume element in matrix space for the GSE. Proceed analogously to the case of the GUE and the GOE, write the GSE matrix H for N = 2 as a 6 × 6 matrix, and calculate the Jacobian for symplectic transformations in a way allowing generalization to arbitrary N . 4.4 Present an argument demonstrating that Wigner’s semicircle lawholds for all three of the Gaussian ensembles. 4.5 Calculate the local mean level density for the spin Hamiltonian H = −k j/2 + p Jz +k Jz2 /2 j with − j ≤ Jz ≤ + j and j  1, where k and p are arbitrary coupling constants. Compare with Wigner’s semicircle law. 4.6 Calculate the mean value and the variance of the squared moduli of the components y of the eigenvectors of random matrices for all three ensembles. 4.7 Calculate the ensemble mean for the density of levels defined in (4.10.3) by using the eigenvalue distribution of the CUE. +  4.8 Calculate the ensemble mean of (1/N ) i1 ... N 1j(...=i)N Θ(|φ j − φi | − s) using (4.12.4). What meaning does the resulting probability density have? 4.9 Generalize (4.12.7) to P(S) = (∂ 2 E/∂ S 2 )/2 for a constant but nonunit mean density of levels . 4.10 Show that a real antisymmetric matrix a of even dimension can be brought to block diagonal form with the blocks (4.13.18) along the diagonal by a real orthogonal transformation L, L L˜ = 1, L = L ∗ . Hint: Why are the eigenvalues imaginary and why must they come in pairs ±iai ? What can be said about the two eigenvectors pertaining to a pair ±iai of eigenvalues? Reshuffle the diagonalizing unitary matrix by suitably combining columns so as to make it orthogonal. What can finally be done to enforce det L = 1? Why is det a a perfect square and positive? What changes if the dimension is odd? 4.11 Prove the representation (4.13.20) for a Pfaffian.

142

4 Random-Matrix Theory

4.12 Evaluate the normalization constant N4 as defined in (4.11.2) of the joint density of eigenphases of the CSE. Proceed as done for the COE in Sect. 4.14.4 by keeping track of all proportionality factors, powers of i apart. The result is N4 = π N N ! (2N − 1)!! 4.13 Leisure permitting, the reader will want to check the Fourier transforms from the form factor to the cluster function in (4.14.16), (4.14.30), and (4.14.36); only the orthogonal case requires ambition. A more worthwhile exercise, quasi-mandatory for those intending to acquire later a semiclassical understanding of random-matrix type behavior of generic dynamic systems, is to prove the following: (1) If the (2n)th derivative K (2n)  (τ ) of an even real function K (τ ) has a delta-function singularity, K (2n) (τ ) = a δ(τ − τ0 ) + δ(τ + τ0 ) + . . ., that singularity contributes to the Fourier $∞ transform as −∞ dτ K (τ ) cos 2π eτ = (−1)n 2a(2πe)−2n cos 2πeτ0 + . . .. (2) A log  arithmic singularity in K (2n) (τ ) = a ln |τ − τ0 | + ln |τ + τ0 | + . . . goes hand in $∞ hand with −∞ dτ K (τ ) cos 2πeτ = (−1)n+1 a(2π)−2n |e|−(2n+1) cos 2πeτ0 + . . .. Use this to explain the leading nonoscillatory and oscillatory terms in the cluster functions, checking, in particular, correctness of the coefficients. 4.14 Show that the roots of a self-inversive polynomial are either unimodular or come in pairs such that λi λi∗ = 1. This is an easy task for N = 2, and you may be satisfied with that. 4.15 Evaluate the variance of the time-averaged form factor (4.17.19) for the Poissonian ensemble, and thus show that the inequality (4.17.25) holds here as well. 4.16 Solve the moment–cumulant relations (4.15.4) to express the general cumulant Cn in terms of the moments M1 , . . . Mn and vice versa, after the model of the solutions (4.15.10) of Newton’s relations.

References 1. O. Bohigas, M.J. Giannoni, C. Schmit: Phys. Rev. Lett. 52, 1 (1984); J. de Phys. Lett. 45, 1015 (1984) 2. M.V. Berry, M. Tabor: Proc. R. Soc. Lond. A356, 375 (1977) 3. S.W. McDonald, A.N. Kaufman: Phys. Rev. Lett. 42, 1189 (1979) 4. G. Casati, F. Valz-Gris, I. Guarneri: Lett. Nouvo Cimento 28, 279 (1980) 5. M.V. Berry: Ann. Phys. (USA) 131, 163 (1981) 6. M. Zirnbauer A. Altland, M.R. Zirnbauer: Phys. Rev. B55, 1142 (1997) 7. M.L. Mehta: Random Matrices (Academic, New York 1967; 2nd edition 1991; 3rd edition Elsevier 2004) 8. C.E. Porter (ed.): Statistical Theories of Spectra (Academic, New York, 1965) 9. T. Guhr, A. M¨uller-Groeling, H.A. Weidenm¨uller: Phys. Rep. 299, 189 (1998) 10. M.A. Stephanov, J.J.M. Verbaarschot, T. Wettig: arxiv:hep-ph/0509286v1 11. E.P. Wigner: Proceedings of the 4th Canadian Mathematical Congress, Toronto, 1959, p. 174 12. M. Ku´s, J. Mostowski, F. Haake: J. Phys. A: Math. Gen. 21, L 1073–1077 (1988) 13. F. Haake, K. Zyczkowski: Phys. Rev. A42, 1013 (1990)

References 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.

143

M. Feingold, A. Peres: Phys. Rev. A34, 591 (1986) B. Mehlig, K. M¨uller, B. Eckhardt: Phys. Rev. E59, 5272 (1999) C.E. Porter, R.G. Thomas: Phys. Rev. 104, 483 (1956) T.A. Brody, J. Floris, J.B. French, P.A. Mello, A. Pandey, S.S.M. Wong: Rev. Mod. Phys. 53, 385 (1981) M.V. Berry: J. Phys. A10, 2083 (1977) V.N. Prigodin: Phys. Rev. Lett. 75, 2392 (1995) H.-J. St¨ockmann: Quantum Chaos, An Introduction (Cambridge University Press, Cambridge, 1999) A. Pandey: Ann. Phys. 119, 170–191 (1979) M.L. Mehta: Commun. Math. Phys. 20, 245 (1971) F.J. Dyson: J. Math. Phys. (N.Y.) 3, 166 (1962) F.J. Dyson: Commun. Math. Phys. 19, 235 (1970) F.J. Dyson: J. Math. Phys. 3, No. 1, 140, 157, 166 (1962) B. Dietz, F. Haake: Z. Phys. B80, 153 (1990) F.J. Dyson: Commun. Math. Phys. 47, 171–183 (1976) B. Riemann: Monatsberichte d. Preuss. Akad. d. Wissensch., Berlin 671 (1859) H.M. Edwards: Riemann’s Zeta Function (Academic, New York, 1974) H.L. Montgomery: Proc. Symp. Pure Math. 24, 181 (1973) A.M. Odlyzko: Math. Comp. 48, 273 (1987) M.V. Berry, J.P. Keating: In J.P. Keating, I.V. Lerner (eds.) Supersymmetry and Trace Formulae: Chaos and Disorder (Plenum, New York, 1998) C. Itzykson, J.-B. Zuber: Quantum Field Theory (McGraw-Hill, New York, 1890) F.A. Berezin: Method of Second Quantization (Academic, New York, 1966) F.A. Berezin: Introduction to Superanalysis (Reidel, Dordrecht, 1987) K. Efetov: Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997) I. Newton: Universal Arithmetic (1707); see e.g., W.W. Rouse Ball: A Short Account of the History of Mathematics, 4th edition (1908) or ww.maths.tcd.ie/pub/HistMath/ People/Newton/RouseBall/RB Newton.html A. Mostowski, M. Stark: Introduction to Higher Algebra (Pergamon, Oxford, 1964) M. Marden: Geometry of Polynomials (American Mathematical Society, Providence, 1966) E. Bogomolny, O. Bohigas, P. Leboeuf: Phys. Rev. Lett. 68, 2726 (1992); J. Math. Phys. 85, 639 (1996) M. Ku´s, F. Haake, B. Eckhardt: Z. Physik B92, 221 (1993) ˙ F. Haake, M. Ku´s, H.-J. Sommers, H. Schomerus, K. Zyckowski, J. Phys. A: Math. Gen. 29, 3641 (1996) M.V. Berry, J.P. Keating: J. Phys. A23, 4839 (1990) E. Bogomolny: Comments At. Mol. Phys. 25, 67 (1990); Nonlinearity 5, 805 (1992) H.-J. Sommers, F. Haake, J. Weber: J. Phys. A: Math. Gen. 31, 4395 (1998) F. Haake, H.-J. Sommers, J. Weber: J. Phys. A: Math. Gen. 32, 6903 (1999) R.E. Hartwig, M.E. Fischer: Arch. Rat. Mech: Anal. 32, 190 (1969) U. Smilansky: Physica D109, 153 (1997)

Chapter 5

Level Clustering

5.1 Preliminaries Here I propose to cosnsider classical autonomous systems with f degrees of freedom and 2 f pairs of canonical variables pi , qi . We shall meet with invariant tori, caustics, and Maslov indices and proceed to the semiclassical torus quantization a` la Einstein, Brillouin and Keller (EBK) and its modern variant, a periodic-orbit theory. The latter will allow us to understand why there is no repulsion but rather clustering of levels for generic integrable systems with two or more degrees of freedom; the density of level spacings therefore usually takes the form of a single exponential, P(S) = e−S . Single-freedom systems, if autonomous, are always integrable and do not respect any general rule for their spacing statistics; we shall postpone a discussion of their behavior to the subsequent chapter. Readers with more curiosity about EBK than can be satisfied here are referred to Percival’s review [1] or the recent book by Brack and Bhaduri [2]. A semiclassical treatment of classically chaotic dynamics will follow in Chap. 10.

5.2 Invariant Tori of Classically Integrable Systems The dynamics in question have f independent constants of the motion Ci ( p, q) with vanishing Poisson brackets:  ∂Ci ∂C j  ∂Ci ∂C j − =0. Ci , C j = ∂ p ∂q ∂q ∂ p

(5.2.1)

Each phase-space trajectory p(t), q(t) thus lies on an f -dimensional surface, the embedded in the 2 f -dimensional phase space. If that surface is smooth, compact, and simply connected, it can be given the shape of a torus by a suitable canonical transformation. A nontrivial example is provided by a point mass moving in a plane under the influence of an attractive central potential V (r ). The energy E and the angular momentum L are two independent constants of the motion and define a twodimensional torus in the four-dimensional phase space.

F. Haake, Quantum Signatures of Chaos, Springer Series in Synergetics, 3rd ed., C Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-05428-0 5, 

145

146

5 Level Clustering

A phase-space trajectory starting on such a torus will keep winding around it forever. In general, trajectories will not be closed; only for the harmonic [V (r ) ∼ r 2 ] and the Kepler [V (r ) ∼ 1/r ] potentials are all trajectories closed. For other potentials, periodic orbits are possible but constitute, as will become clear presently, a subset of measure zero in the set of all trajectories. When projected onto the two-dimensional configuration space, every trajectory of the present model remains between two circles, which are concentric with respect to the center of force. These circles, called caustics, are the singularities of the projection of the 2-torus onto the configuration plane. When touching a caustic the trajectory has a momentarily vanishing radial momentum, i.e., a turning point of its radial libration. For a circular “billiard table,” every configuration-space orbit also remains in between two concentric circles of which only the inner one is a caustic, however; the outer one is a boundary at which the particle bounces off rather than coming to rest momentarily. Note that no caustic is ever encountered in a rotation for which an angle increases indefinitely, where all other coordinates, if any, remain constant. Classical motion on an f -torus embedded in a (2 f )-dimensional phase space is conveniently described in terms of f pairs of action and angle variables Ii , Θi . The actions characterize the f -torus via 7 1 p dq > 0 , (5.2.2) Ii = 2π Γi where Γi is one of the f independent irreducible loops around the torus. Being properties of invariant tori, the Ii are constants of the motion. By a suitable canonical transformation, the actions Ii become new momenta with angles Θi as their conjugate coordinates. The Hamiltonian, a constant of the motion itself, can be expressed as a function of the f actions and does not depend on the angles. Hamilton’s equations thus give ˙ i = ∂ H ≡ ωi , Θ ∂ Ii Θi (t) = ωi t + Θi (0) .

(5.2.3)

The frequencies ωi are the angular velocities with which the phase space point travels around the torus. The angle Θi changes by 2π as the loop Γi orbits once. When the frequencies ωi are incommensurate, the trajectory starting from Θ(0) will tend to fill the torus densely as time elapses. We might even speak of ergodic motion in that case since time averages (of suitable quantities) will tend to ensemble averages with uniform probability density on the torus. Periodic orbits result, on the other hand, when the ωi are related rationally, i.e., when all ωi are integral multiples of a certain fundamental frequency ω0 , ω = Mω0 , Mi integer .

(5.2.4)

Such an orbit closes in on itself after after the fundamental period 2π/ω0 , having completed M1 trips around the loop Γ1 , M2 around Γ2 etc. The corresponding torus

5.3

Einstein–Brillouin–Keller Approximation

147

is called rational. Every rational torus accommodates an f -parameter continuum of closed orbits related to one another by shifts of the initial angles Θ(0). Rational tori are as exceptional among the tori in phase space as the rational numbers among the reals.

5.3 Einstein–Brillouin–Keller Approximation To solve the Schr¨odinger equation of a classically integrable system in the semiclassical limit, one may use the ansatz [1–7] ψ(q) = a(q)eiS(q)/ .

(5.3.1)

By demanding that ψ(q) be single valued in the q, one obtains a quantum condition for each of the f actions,   Ii = m i + 14 αi  ,

(5.3.2)

with nonnegative integer quantum numbers m i and the so-called Maslov indices αi . The ith Maslov index αi equals the number of caustics encountered along the ith irreducible loop Γi around the classical torus with respect to which the action Ii was defined. The quantization prescription (5.3.2) is often referred to as semiclassical or torus quantization. The quantum condition on the action originates from the phase change in the semiclassical wave function along the loop Γi , ΔS = 

7 Γi

p dq . 

(5.3.3)

Another contribution to the total change of phase may come from the amplitude a(q), which can be shown to diverge at caustics. Such a singularity means, of course, that the ansatz (5.3.1) breaks down near a caustic. It was a brilliant idea of Maslov to switch to momentum space when q approaches a caustic and to replace (5.3.1) ˜ p). Since a caustic in q-space is not a caustic in p-space, with a similar ansatz for ψ( ˜ p) where the semiclassical form (5.3.1) of ψ(q) runs no singularity threatens ψ( ˜ p) near the q-space into trouble. Replacing the latter by the Fourier transform of ψ( caustic (see Sect. 10.4.2), one finds the total phase increment picked up along Γi as 2π Ii π ΔS = − αi .   2

(5.3.4)

That change must be an integral multiple of 2π for the wave to close in on itself uniquely, i.e., not to destroy itself by destructive interference. By expressing the Hamiltonian H (I) in terms of the quantized actions (5.3.2), one obtains the semiclassical approximation for the energy levels

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5 Level Clustering

E m = H (I m ) = H



 m + 14 α  .

(5.3.5)

In integrable systems, these semiclassical levels usually give an excellent approximation for sufficiently high degrees of excitation. Their accuracy is often surprisingly good for low energies, too, and in some exceptional cases even the whole spectrum is reproduced rigorously. The f -dimensional harmonic oscillator, for instance, has the Hamiltonian   f   1 2 1 2 2 ωi Ii . p + mωi qi = H= 2m 1 2 i i =1

(5.3.6)

Since a libration encounters two turning points in every period, one must set αi = 2 and thus obtain Em =



  ωi m i + 12 , m i = 0, 1, 2, . . .

(5.3.7)

i

which is indeed the exact result. Quite amusingly, the contribution of the Maslov indices gives the correct zero-point energy. As a second example, the reader is invited to check that the Hamiltonian of the hydrogen atom can be expressed in terms of the radial, polar, and azimuthal actions as H =−

me4 . 2(Ir + IΘ + Iφ )2

(5.3.8)

The radial and polar motions are librations and the azimuthal motion is a rotation, so one has αr = αΘ = 2 and αφ = 0 and thus the semiclassical energies Em = −

me4 , n = mr + m Θ + m φ + 1 22 n 2

(5.3.9)

which is again exact. Finally, a particle in an f -dimensional box is described by H=

f 1  2 p , 0 ≤ qi ≤ ai . 2m i = 1 i

(5.3.10)

The squared momenta pi2 are f independent constants of the motion and the actions read 1 Ii = 2π



ai 0

 dq| pi | +

0

ai

 | pi | dq (−| pi |) = ai . π

(5.3.11)

5.4

Level Crossings for Integrable Systems

149

Reexpressed in the actions, the Hamiltonian H=

π 2  Ii2 2m i ai2

(5.3.12)

yields the semiclassical levels Em =

π 2 2  m i2 2m i ai2

which again coincide with the exact ones, except that the quantum numbers m 1 start from 1 instead of 0 [2]. For further examples and a more systematic treatment, the reader is referred to [2].

5.4 Level Crossings for Integrable Systems It is easy to see that the levels obtained by torus quantization generally are not inhibited from crossing when f ≥ 2; the codimension of a crossing is n = 1. To this end, one imagines that the Hamiltonian and thus the asymptotic eigenvalues    E m (k) = H  m + 14 α , k ,

(5.4.1)

depend on a single parameter k. In an f -dimensional space with axes m 1 , m 2 , . . . , m f , the quantum numbers in (5.4.1) are represented by the points of a primitive cubic lattice where all components of the lattice vectors are nonnegative integers. In this space, the function H of f continuous variables m defines a continuous family of energy surfaces. An allowed energy eigenvalue arises for every such energy surface intersecting one of the discrete lattice points, say m∗ . In general, an energy surface intersecting m∗ will not intersect any other lattice point. There is a whole bundle of energy surfaces through m∗ , labelled by the parameter k, however, and by properly choosing k, a special surface can, in general, be picked which does intersect a second lattice point. The restriction “in general” of the foregoing statement excludes funny exceptions like the harmonic oscillator for which the energy surface is a hyperplane. Barring such exceptional surfaces, one concludes that a level crossing can be enforced by controlling a single parameter in the Hamiltonian. It follows from the arguments of Sect. 3.4 that the spacing density typically approaches a non-zero constant at zero spacing, P(S) → P(0) = 0 for S → 0 .

(5.4.2)

This is in contrast to the power-law behavior typical of nonintegrable systems.

150

5 Level Clustering

The foregoing reasoning does not apply to a single ladder or “multiplet” of levels arising if we let one of the f quantum numbers run while keeping all others fixed or, equivalently, if we have f = 1 to begin with. The parametric changes possible for such single multiplets will be discussed in Sect. 6.5.

5.5 Poissonian Level Sequences Before proceeding to show that the generic integrable system has an exponential distribution of its level spacings, it will be useful to introduce some probabilistic concepts. In particular, I intend to demonstrate here that exponentially distributed level spacings allow an interpretation of the sequence of levels as a random sequence without any correlation between the individual “events.” Let us assume that the spectrum is already unfolded to unit mean spacing of neighboring levels everywhere, S¯ = 1. Then, we employ, as an auxiliary concept, the conditional probability g(S)d S of finding a level in the interval [E + S, E + S + d S] given one at E. By virtue of the assumed homogeneity of the level distribution, g(S) does not depend on E. The probability density P(S) of finding the nearest neighbor of the level at E in dE at E + S is evidently the product of g(S)dS with the probability that there is no other level between E and E + S, 

∞

P(S) = g(S)

dS P(S ) .

(5.5.1)

S

Differentiating with respect to S, ∂P ∂g = ∂S ∂S



∞ S

  ∂g dS P(S ) − g P = g −1 −g P . ∂S

(5.5.2)

This differential equation is solved by P(S) = const g(S)e−

$S 0

d S g(S )

.

(5.5.3)

For a Poissonian level sequence, now, there are no correlations at all between the individual levels. Thus, the conditional probability g(S) is a constant, and P(S) = C exp (−AS). Normalizing and scaling so that S¯ = 1, one obtains the exponential distribution P(S) = e−S .

(5.5.4)

As a by-product of the above reasoning, the result is that the conditional probability density g(S) behaves like a power g(S) → S β

(5.5.5)

5.6

Superposition of Independent Spectra

151

with β = 0, 1, 2, 4 for, respectively, integrable systems and nonintegrable systems with orthogonal, unitary, and symplectic canonical transformations.

5.6 Superposition of Independent Spectra In this section I shall present an important limit theorem which will help us to understand level spacings of integrable systems: L independent spectra, each with its own homogeneous density and spacing distribution, superpose to a spectrum with exponentially distributed spacings in the limit L → ∞. The arguments presented below go back in essence to Rosenzweig and Porter [8] and Berry and Robnik [9]. For the present purpose, it is again convenient to employ the probability E(S) that an interval of length S is empty of levels. Assuming that the mean distance is unity, the spacing distribution P(S) can be obtained from (4.12.7) as the second derivative of the gap probability E(S), P(S) =

∂2 E . ∂ S2

(5.6.1)

Now consider L independent ladders of levels, the μth of which has the constant mean spacing 1/μ and the spacing distribution Pμ (S) normalized as $∞ 0

d S Pμ (S) = 1

0

d S S Pμ (S) =

$∞

1 μ

.

(5.6.2)

The probability of finding no level of the μth ladder in an interval of length S is, in accordance with (5.6.1), 



∞

E μ (S) = μ

∞

dx S



∞

= μ

dy Pμ (y) x

d x(x − S)Pμ (x) .

(5.6.3)

S

It follows that E μ (S) falls off monotonically from E μ (0) = 1 to E μ (∞) = 0, and the “initial” slope is E μ (0) = −μ , E μ (S) = 1 − μ S + . . . .

(5.6.4)

152

5 Level Clustering

Due to the assumed independence of the ladders, the joint probability that there is no level from any ladder in an interval of length S is just the product E(S) =

L ,

E μ (S) = exp



ln E μ (S) .

(5.6.5)

μ

μ=1

Now, we assume that L  1 and that the μ all have roughly equal magnitudes, μ ∼ 1/L , normalizing for convenience such that the total density is unity, L 

μ = 1 .

(5.6.6)

μ=1

The joint probability E(S) falls off more rapidly by a factor of the order L than the typical single-ladder probability E μ (S). Therefore, it is legitimate to replace the E μ (S) on the right-hand side of (5.6.5) by the first two terms of their Taylor series (5.6.4). The exponential form for E(S) results in the limit L → ∞, E(S) → e−S .

(5.6.7)

It is hard to resist speculating, by naive appeal to the limit theorem just established, that the semiclassical spectrum (5.3.5) must have, for f ≥ 2 and generic functions H (I), an exponential spacing distribution. After all, the semiclassical levels tend to follow one another quite erratically with respect to the direction of the vector m, as is obvious from Fig. 5.1. There is, no doubt, an element of truth in such naive reasoning. However, the only reasonably sound derivation of the exponential

Fig. 5.1 Adjacent energy eigenvalues of integrable systems may lie far apart in quantum number space if the number of degrees of freedom is larger than one

5.7

Periodic Orbits and the Semiclassical Density of Levels

153

spacing distribution for integrable systems known at present is lengthy and technical and does not use the limit theorem (5.6.7).

5.7 Periodic Orbits and the Semiclassical Density of Levels The starting point for deriving the spacing distribution is the density of semiclassical energy levels 

(E) =

     δ E − H  m + 14 α

(5.7.1)

{m i >0}

which will be shown to be representable as a sum of contributions from all classical periodic orbits. The presentation will closely follow Berry and Tabor’s original one [10, 11]. Note that the density (E) is normalized here such that its integral over an energy interval ΔE gives the number of levels contained in the interval (rather than the fraction of the total number of atoms). To pursue the goal indicated it is convenient to rewrite (5.7.1) as  d f I δ [E − H (I)]

(E) =



I>0

m

  α  δf I − m+ . 4

(5.7.2)

The previous restriction on the f quantum numbers m > 0 is dropped here in favor of the restriction I > 0 on the f -fold integral. Next, we employ Poisson’s summation formula, +∞ 

ei2π M x =

+∞ 

δ(x − m) ,

(5.7.3)

m = −∞

M = −∞

to replace the f -dimensional train of delta functions by an f -fold sum over exponentials, (E) =

+∞ 1  −iπ α·M/2 e  f M = −∞  × d f I δ [E − H (I)] ei2π M·I/ ,

(5.7.4)

I>0

and each of the f summation variables Mi runs through all integers. The M = 0 term in the series representation (5.7.4), (E) ¯ ≡

1 f

 d f I δ [E − H (I)] , I>0

(5.7.5)

154

5 Level Clustering

is the venerable Thomas–Fermi result already alluded to in Sect. 4.7 which is also known as Weyl’s law. [To see the equivalence of (4.7.4) and (5.7.5) note that − f = h − f (2π ) f and that H (I) is independent of the f angle variables Θ conjugate to the actions I.] As already indicated in the symbol (E), ¯ the Thomas–Fermi level density (5.7.5) may be considered a local spectral average of (E), that does not reflect local fluctuations in the level sequence such as local clustering or repulsion. An important and intuitive interpretation of (5.7.5) becomes accessible through its integral, the average level staircase σ¯ (E) =



1 f

d f I Θ [E − H (I)] ,

(5.7.6)

I>0

which counts the number of levels below E as the phase-space volume “below” the corresponding classical energy surface divided by  f ; each quantum state is thus assigned the phase-space volume  f . The M = 0 terms in (5.7.4) describe local fluctuations in the spectrum with scales ever finer as |M| increases. Defining 

e−iπα·M/2  M

(5.7.7)

d f I δ [E − H (I)] ei2π M·I/ .

(5.7.8)

Δ(E) = (E) − (E) ¯ =

M=0

one has the Mth density fluctuation M =

1 f

 I>0

To evaluate the I integral, we introduce a new set of orthogonal coordinates as integration variables, ξ0 , ξ1 , . . . , ξ f −1 , the “zeroth” one of which measures the perpendicular distance from the energy surface, and the remaining f − 1 variables parametrize the energy surface (Fig. 5.2). The integral then takes the form 

d f −1 ξ ei2π M·I(ξ )/

M =

 dξ0

δ(ξ0 ) |∂ H/∂ξ0 |

(5.7.9)

where it is understood that d f −1 ξ contains the Jacobian of the transformation of the integration variables and the vector ξ lies in the energy surface so as to have the f − 1 components ξ1 , . . . , ξ f −1 . By noting that 

∂H ∂ξ0

 ξ0 =0

= |∇ I H (I)| H = E = |ω (I(ξ ))|

(5.7.10)

is the length of the f -dimensional frequency vector, one obtains  M =

d f −1 ξ

ei2π M·I(ξ )/ . |ω(I(ξ ))|

(5.7.11)

5.7

Periodic Orbits and the Semiclassical Density of Levels

Fig. 5.2 Energy surface in action space

155

I3 ξ0 ξ1 I

ξ2 I2

I1

Without sacrificing consistency with the semiclassical approximation for the energy levels, the remaining integral over the energy surface can be evaluated in the limit  → 0 in which the phase 2π M · I(ξ )/ is a rapidly oscillating function of ξ . The integral will thus be negligibly small unless there is a point ξ M on the energy surface for which the phase in question is stationary, M·

∂I = 0 for ξ = ξ M , i = 1, 2, . . . , f − 1 . ∂ξi

(5.7.12)

Since the f − 1 vectors ∂ I/∂ξi are all tangential to the energy surface, the stationary-phase condition (5.7.12) requires the lattice vector M to be perpendicular to the energy surface. An equally significant consequence of the vector ξ lying in the energy surface is ∂I ∂I ∂H = ∇I H · =ω· =0, ∂ξi ∂ξi ∂ξi

(5.7.13)

i.e., the orthogonality of the frequency vector to the energy surface. It follows from (5.7.12) and (5.7.13) that the lattice vector M and the frequency vector ω must be parallel at the points of stationarity of the phase 2π I · M/, ω1 : ω : . . . : ω f = M 1 : M 2 : . . . M f .

(5.7.14)

Because the Mi are integers, one concludes that the ωi must be commensurate and the corresponding tori, defined by I M = I(ξ M ), are rational. Obviously, a nonnegligible Mth density fluctuation  M corresponds to periodic orbits on the rational torus I M . The lattice vector M defines the topology of the closed orbits inasmuch as an orbit obeying (5.7.14) closes in on itself after M1 periods 2π of Θ1 , M2 periods of Θ2 , etc. Figure 5.3 illustrates closed orbits of simple topologies (Mr , Mφ ) for a two-dimensional potential well, Mr counting the number

156

5 Level Clustering

Fig. 5.3 Closed orbits of simple topologies in a two-dimensional configuration space

of librations of the radial coordinate and Mφ the number of revolutions around the center before the orbit closes. When a lattice vector M induces a nonnegligible density fluctation  M so do all its integer multiples since these also fulfill the stationary-phase condition (5.7.12). Therefore, it is useful to introduce the primitive version μ of M such that the components μi are relatively prime, together with its multiples M = qμ, where q is a positive integer. The density fluctuation qμ corresponds to a closed orbit which is just the primitive orbit μ traversed q times. The total action along a closed orbit M = qμ, S(M) = 2π M · I M ,

(5.7.15)

is obviously q times the action along the primitive orbit μ, S(M) = q S(μ) . It remains to write the stationary-phase approximation to  M . The general formula for the stationary-phase approximation of a multiple integral reads  d n xei f (x) Φ(x) 6  (2π)n s = Φ(x s )ei f (x )+iβπ/4 2 s |det(∂ f (x)/∂ xi ∂ x j )| x=x xs

(5.7.16)

where x s are the points of stationary phase, defined by ∂ f /∂ xis = 0, and β(x s ) is the difference in the number of positive and negative eigenvalues of the n × n matrix ∂ 2 f /∂ xi ∂ x j .

5.7

Periodic Orbits and the Semiclassical Density of Levels

157

The determinant in (5.7.16) deserves special comment. According to (5.7.11), in this case it is ( f − 1) × ( f − 1) and reads  det

∂2 I 2π M·  ∂ξi ∂ξ j



 =  ≡

2πq 

 f −1

2πq|μ| 

  ∂2 I det μ · ∂ξi ∂ξ j

 f −1

  K Iμ ,

(5.7.17)

where     ∂2 I ˆ· K I μ = det μ ∂ξi ∂ξ j

(5.7.18)

is the scalar curvature of the energy surface at the point I M . To appreciate this interpretation of K , let us consider f = 2; the vector d I/dξ1 is tangential to the energy surface and d(d I/dξ1 ) = (d 2 I/dξ12 )dξ1 is the increment of that tangent ˆ · d 2 I /dξ12 , the stronger the curvature of vector along dξ1 ; the larger the value of μ the energy surface (Fig. 5.4). For f = 3, it is easy to see that K is the Gaussian curvature of the then two-dimensional energy surface. The final result for the Mth density fluctuation now reads  M =

 q|μ|

( f −1)/2

exp{i[q S(μ)/ + πβ(μ)/4]} . |ω(I μ )| · |K (I μ )|1/2

(5.7.19)

When summing up according to (5.7.7), one must realize that (5.7.14) implies that the Mi = qμi all have the same sign because all ωi are nonnegative. Moreover, from (5.7.11) ∗ ,  M = −M

(5.7.20)

d(dI/d ξ1)

I(ξ + dξ) I(ξ)

Fig. 5.4 Curvature of the energy surface

158

5 Level Clustering

and thus the total density fluctuation takes the form Δ(E) = 2−( f +1)/2



     −1 1/2   |μ|( f −1)/2 ω I μ K I μ 

μ>0

×

∞  q =1

q ( f −1)/2 cos



q S(μ) qμ · απ β(μ)π − +  2 4

 . (5.7.21)

This remarkable formula expresses the semiclassical level density fluctuations in terms of purely classical quantities, the latter related to periodic orbits. Berry and Tabor [10] followed the synthesis of the level density for the Morse potential in two dimensions according to (5.7.21) by accounting for more and more closed orbits. Starting from the smooth Thomas–Fermi background (5.7.5), ever finer variations of (E) become visible as more closed orbits are included. In the limit, because all such orbits are allowed to contribute, (E) develops a delta-function peak at each energy level E i . It is quite remarkable that the periodic-orbit result (5.7.21) breaks down for the harmonic oscillator since the energy surfaces for this “pathological” system are planes with zero curvature. In fact, the harmonic oscillator was already excluded in Sect. 5.4 for the same reason.

5.8 Level Density Fluctuations for Integrable Systems It is convenient to start by rescaling the semiclassical energy levels E m = H (I = m)

(5.8.1)

to uniform mean density (Note that I have dropped the Maslov indices in (5.8.1); they may be imagined eliminated by a shift of the origin in m space; in the semiclassical limit, m i  αi , they are quite unimportant anyway). The unfolding functions (4.8.10) and (4.8.11) are less suitable for the present purpose than one designed by Berry and Tabor [11] which endows the rescaled levels em with the homogeneity property eβm = β f em , β > 0 .

(5.8.2)

The construction of the em proceeds in two steps. First, one replaces Planck’s constant  with a continuous variable h and follows the energy levels E m (h) = H (mh) as h varies. Their intersections with a fixed reference energy E define

5.8

Level Density Fluctuations for Integrable Systems

159

Fig. 5.5 Reshuffling of energy levels in the Berry–Tabor rescaling

a sequence of discrete values h m . The mapping of the original energy levels E m () onto the numbers h m will in general be nonlinear (Fig. 5.5) and may even reshuffle the ordering. A second nonlinear mapping gives the rescaled levels as  em =

1 hm



f  d I Θ [E − H (I)] = f

I>0

 hm

f σ¯ (E)

(5.8.3)

where σ¯ (E) is the Thomas–Fermi or Weyl form of the average level staircase, normalized such that σ¯ (E)/N → 1 for E → ∞. The homogeneity (5.8.2) follows trivially from E = H (mh m ) = H (βmh m /β). It also follows that h m ≥  and em ≤ σ¯ (E) for E m ≤ E.

(5.8.4)

The rescaled levels em have the density (e) =



δ(e − em )

(5.8.5)

m>0

the local spectral average of which is obtained when the summation over m is replaced with an f -fold integration,  d f mδ(e − em ) ,

(e) ¯ =

(5.8.6)

m>0

as was explained in Sect. 5.7. [The reader may recall the derivation of (5.7.5) from (5.7.2) with the help of Poisson’s summation formula.] By virtue of the homogeneity

160

5 Level Clustering

property (5.8.2), the average density is a constant. Indeed, by changing the integration variables in (5.8.6) as m = α 1/ f x with arbitrary positive α, 

 (e) ¯ =α  =

x>0

d f xδ (e − eα1/ f x ) = α f

d xδ

e

x>0



− e x = ¯

α

d f xδ (e − αe x ) x>0

e α

.

(5.8.7)

The constant , ¯ however, must have the value unity since the rescaled average staircase  e de (e ¯ ) = e ¯ (5.8.8) 0

grows from 0 to σ¯ (E) as e grows from 0 to σ¯ (E), so that σ¯ (E) = ¯ σ¯ (E) gives ¯ = 1 .

(5.8.9)

The remainder of the argument parallels that given in Sect. 5.7. Poisson’s summation formula is invoked again to write the density fluctuations as Δ(e) = (e) − 1 =



M ,

(5.8.10)

M=0

 M =

d f mδ(1 − em )ei2π M·me

1/ f

.

(5.8.11)

In the Mth density fluctuation, the “standard” energy surface em = 1 occurs in the delta function, and the current energy e is elevated, with the help of the homogeneity (5.8.2), into the phase factor. Again introducing orthogonal coordinates ξ = (ξ, . . . , ξ f −1 ) on the energy surface and using the stationary-phase approximation, M

  exp [i2π M · me1/ f − iπ ( f − 1)/4]  = ,  2 1/2 ( f −1)/2 f |∇m em | · |det(M · (∂ m/∂ξi ∂ξ j )| e m=m M

(5.8.12)

provided there is a point m M on the energy surface at which the phase is stationary M·

∂m =0; ∂ξi

(5.8.13)

otherwise, if there is no solution m M to (5.8.13), the Mth density fluctuation is negligible.

5.8

Level Density Fluctuations for Integrable Systems

161

In a spectral region near some energy e0 , the density fluctuation  M displays oscillations of the form  M (e) = A M (e0 ) exp {i [K M (e0 )(e − e0 ) + Φ M (e0 )]}

(5.8.14)

with an amplitude A M (e0 ) = |∇m em |

−1

.   − 12 2   m ∂ −( f −1)/2 f det M  e 0  ∂ξ ∂ξ  i

j

,

(5.8.15)

m = mM

a “wave number” K M (e0 ) =

2π M · m M 1−1/ f

f e0

,

(5.8.16)

and a phase 1/ f

Φ M (e0 ) = 2π M · m M e0

−

π ( f − 1) . 4

For a fixed large energy e0 , the phase Φ M varies more rapidly (by a factor e0 ) than the wave number K M when one of the components of the vector M changes by unity. Consecutive phase jumps, taken modulo 2π, will tend to fill the interval [0, 2π] randomly. Therefore, the total density fluctuation Δ = (e) − 1 may be expected to depend quite erratically on e − e0 ; correspondingly, the density–density correlation function S(e0 + τ, e0 ) =

1 Δσ



+Δσ/2 −Δσ/2

dσ Δ(e0 + τ + σ )Δ(e0 + σ )

(5.8.17)

should decay rapidly with increasing τ. The reader may recall that S(e , e

) was already studied in Sect. 4.10, with a slightly different normalization and with an ensemble average instead of the local spectral average in (5.8.17). The averaging interval Δσ must be large enough to contain many levels but small enough to allow approximation (5.8.14). To check on the expected behavior of S, we insert (5.8.10, 5.8.14) into (5.7.17). Only the “diagonal” terms in the double sum over two vectors M, M survive. For these, M = −M , K −M = K M , Φ−M = −Φ M .

(5.8.18)

It follows that S(e0 + τ, e0 ) =

 M

A2M eiK M τ .

(5.8.19)

162

5 Level Clustering

For convenience, now let us consider the Fourier transform of S with respect to the variable τ,  +∞ ˜ 0, K ) = 1 dτ e−iK τ S(e0 + τ, e0 ) S(e 2π −∞  A2M δ (K − K M ) . =

(5.8.20)

M

In the limit of large energies, e0 → ∞, both A M and K M are, as already mentioned, smooth functions of M. Thus no appreciable error results when the sum over M is replaced by an integral,  ˜ 0, K ) = S(e

d f M A2M δ (K − K M ) .

(5.8.21)

As a first step toward evaluating the M-integral, the length of the frequency vector in the amplitude A M must be reexpressed as   ˆ |∇m em |m = m M = f / m M · M

(5.8.22)

ˆ = M/|M|. Indeed, the energy surface relevant for the gradient in question where M is e = 1, and a nearby energy surface is characterized by the point m M (1 + ε), with a small real number ε, at which 1 + Δe = e(1+ε)m M = (1 + ε) f em M = (1 + ε) f ≈ 1 + f ε .

(5.8.23)

ˆ · εm such that the On the other hand, the distance between the two surfaces is M ˆ · m gives (5.8.22). Combining (5.8.15), (5.8.22), and (5.8.21) one quotient Δe/ε M gets the Fourier transformed density–density correlation function as ˜ 0, K ) = S(e  ˆ 2 δ(K − 2π M · m M )/ f e1−1/ f (m M · M) 0 dfM . ( f −1)/ f 2 f −1 2 ˆ f e0 |M| |det(∂ m · M/∂ξi ∂ξ j )|m = m M

(5.8.24)

A most remarkable simplification results now if one changes the integration vari1−1/ f M; the integral is independent of both e0 and K , ables: M → K e0 ˜ 0 , K ) = const . S(e

(5.8.25)

The fact that S˜ is independent of the energy e0 is not really a big surprise since the rescaling of the energy (5.8.3) was designed so as to make the spectrum statistically homogeneous. The lack of dependence of S˜ on K , however, implies merely the expected rapid falloff of S(τ ) with increasing |τ | :

5.8

Level Density Fluctuations for Integrable Systems

˜ S(τ ) = 2π S(0)δ(τ ).

163

(5.8.26)

˜ A little more geometry reveals that the prefactor 2π S(0) of the delta function is unity. ˆ with the The f integration variables M in (5.8.24) may be chosen as |M| and M unit vector determined in terms of f − 1 angles; the delta function allows one to perform the |M| integral, S˜ =

1 2π f



ˆ d f −1 M

ˆ · mM M . ˆ |det(∂ 2 m · M/∂ξ i ∂ξ j )|m = m M

(5.8.27)

ˆ of M determines a direction m ˆ of m = |m|m. ˆ Switching Now every direction M ˆ to the f − 1 integration variables m, S˜ =

1 2π f



ˆ d f −1 m

ˆ m| ˆ ˆ |m m ˆ · M| |∂ M/∂ . ˆ · (∂ 2 m m/∂ξ ˆ |det( M i ∂ξ j )|m = m M

(5.8.28)

As already mentioned in Sect. 5.7, the determinant in the denominator of (5.8.28) is related to the curvature of the energy surface. To establish an interpretation more useful in the present context, recall that the f −1 vectors t i = ∂ m/∂ξi are tangential ˆ is orthogonal to the energy surface, whereas M ˆ · ti = 0 . ˆ · ∂m = M M ∂ξi

(5.8.29)

This identity holds independently of ξ so that the increments of M and t i along an infinitesimal dξ are constrained to obey ˆ =0. ˆ ti + ti d M Md

(5.8.30)

A previous stipulation was that the f − 1 coordinates ξi be orthogonal to one another such that an arbitrary infinitesimal vector d m on the energy surface obeys dm =



t i dξi ,  2 2 t i dξi (d m) = 2

i

(5.8.31)

i

Now, it is convenient to require that the ξi are also locally Cartesian, d m2 =

 i

dξi2 ⇔ t i2 = 1 ,

(5.8.32)

164

5 Level Clustering

i.e., to make the t i unit vectors. Then, the identity (5.8.30) gives 2 ˆ ˆ ˆ ˆ ∂ t i = − ˆt i ∂ M = − ∂ M i ˆ ∂ m =M M ∂ξi ∂ξ j ∂ξ j ∂ξ j ∂ξ j

(5.8.33)

ˆ i denotes the ith Cartesian component of the vector d M. ˆ The determinant where d M in question,   ∂ M 2 ˆi ˆ  m ∂ M ∂  ˆ = det = det M  ,  ∂ξ  ∂ξi ∂ξ j ∂ξ j

(5.8.34)

is thus revealed as the Jacobian for a transformation from the f − 1 angular coordiˆ to the f − 1 Cartesian coordinates ξ . nates M The integral (5.8.28) now takes the form  ˆ   ˆ 1 ˆ m |∂ M/∂ m| ˆ m ˆ ·M d f −1 m ˆ 2π f m>0 |∂ M/∂ξ | ˆ        1 ˆ m  ∂ξ  , ˆ m ˆ ·M d f −1 m = ∂m ˆ 2π f

S˜ =

(5.8.35)

ˆ is which allows for the following geometric interpretation. The infinitesimal d f −1 m ˆ ξˆ /∂ m| is a Cartea surface element on the f -dimensional unit sphere and d f −1 m|∂ sian surface element on the energy surface e = 1 with the vectorial representation ˆ This vector has the component d f −1 m|∂ξ ˆ ·m ˆ ˆ M. ˆ ˆ M ˆ along m, ˆ /∂ m| /∂ m| d f −1 m|∂ξ which is simply the differential area on the energy surface “above” the element ˆ of the unit sphere. Written with the help of the polar representation of the d f −1 m ˆ ˆ f −1 , and the integral in (5.8.35) m)] energy surface, that latter element is d f −1 m[m( takes the form  ˜S = 1 ˆ [m(m)] ˆ f . d f −1 m (5.8.36) 2π f m>0 ˆ The physical meaning of (5.8.36) becomes apparent when the ( f − 1)-fold integral is blown up to the f -fold integral: 1 S˜ = 2π

 ˆ . d f m Θ [m − m(m)]

(5.8.37)

m>0

One recognizes the average level staircase σ¯ (e) evaluated at e = 1. Since σ¯ (e) = e, the final result reads 1 . S˜ = 2π

(5.8.38)

5.9

Exponential Spacing Distribution for Integrable Systems

165

In view of (5.8.26), the density–density correlation function becomes S(τ ) = δ(τ ) .

(5.8.39)

The local fluctuations of the level density at different energies bear no correlations, just as if the levels followed one another as the independent events of a Poisson process.

5.9 Exponential Spacing Distribution for Integrable Systems Now, I propose to show that the correlation function S(τ ) is related to the conditional probability density g(τ ) of finding a level in the interval [e+τ, e+τ +dτ ], given one at e. It was shown in Sect. 5.5 that this conditional probability density determines the spacing distribution P(σ ) according to P(σ ) = const g(σ )e−

$σ 0

dτ g(τ )

.

(5.9.1)

Denoting the local spectral average by the brackets  . . . , let us consider 1 − δ(τ ) + S(τ ) 8 = 1 − δ(τ ) + = −δ(τ ) + =







5 δ(e0 + τ − em ) − 1



59 δ(e0 − em ) − 1

m

m

δ(e0 + τ − em )δ(e0 − em )

m,m

δ(e0 + τ − em )δ(e0 − em ) .

(5.9.2)

m=m

The little calculation in (5.9.2) uses  = 1. By writing out the spectral average explicitly, one obtains 1 − δ(τ ) + S(τ ) 5   + Δσ   2 1 dσ δ(e0 + τ + σ − em ) δ (τ − em + em ) . = Δσ − Δσ2 m m (=m) (5.9.3) The sum over m gives the probability density of having a level at a distance τ away from em and the curly bracket averages over all levels em in the interval Δσ around e0 . The function considered in (5.9.3) is therefore the probability density for finding a level in [e + τ, e + τ + dτ ] given one at e, where e may lie anywhere in the spectral range Δσ around e0 , g(τ ) = 1 − δ(τ ) + S(τ ) .

(5.9.4)

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5 Level Clustering

Together with S(τ ) = δ(τ ), which is typical for integrable systems, one has g(τ ) = 1

(5.9.5)

and thus the exponential distribution of level spacings P(τ ) = e−τ .

(5.9.6)

5.10 Equivalence of Different Unfoldings The unfolding employed in Sect. 5.8 makes the rescaled levels homogeneous with degree f in the f quantum numbers m. That homogeneity was of critical importance at several stages of the argument. For one thing, all geometric considerations were referred to the single standard energy surface e = 1; moreover, the oscillatory dependence of the Mth density fluctuation  M on the energy could be made manifest. The question arises whether the exponential spacing distribution is really a general property of integrable systems with f ≥ 2 and not simply an artefact due to a peculiar unfolding. It was argued in Sect. 4.8 that the most natural unfolding involves the average level staircase σ¯ (E) as e = σ¯ (E) .

(5.10.1)

In practical applications, the average density is often used to rescale ¯ . e

= E (E)

(5.10.2)

This unfolding is only locally equivalent to (5.10.1), i.e., with respect to spectral regions within which the density ¯ is practically constant. The unfoldings (5.10.1) and (5.10.2) do not in general provide homogeneity of



and em . An exception arises only degree f to the rescaled semiclassical levels em for integrable Hamiltonians which themselves are homogeneous in the f actions I (see Problem 5.6). For all other integrable systems, it is not immediately obvious that (5.10.1) and (5.10.2) also yield exponentially distributed spacings. An important property of (5.10.1) is the monotonicity of e (E) since this ensures that ei > e j provided E i > E j . The density-based unfolding (5.10.2) need not strictly have that monotonicity but does in almost all cases of practical relevance. Not so for Berry and Tabor’s unfolding used in Sect. 5.8! It was in fact pointed out that the original levels E m may be reshuffled in their order by the rescaling (5.8.3) rather than just squeezed together or stretched apart to secure uniform mean spacing. Could the em owe their exponential spacing distribution to such reshuffling? I cannot formally prove that (5.10.1) and (5.10.2) yield exponentially distributed spacings whenever (5.8.3) does so. Nonetheless, a reasonably convincing argument

5.11

Problems

167

may be drawn from the following example. Consider a particle with f = 2 subjected to a harmonic binding in one coordinate and free in a finite interval with respect to the second coordinate. Then, the Hamiltonian takes the form H = α I1 + β I22

(5.10.3)

and the semiclassical levels read (again dropping the Maslov index for the oscillator part!) E m = αm 1 + β2 m 22 .

(5.10.4)

Clearly, intersections of levels become possible once Planck’s constant is replaced by a continuous variable h. Therefore, the Berry–Tabor levels em will not have precisely the same ordering as the E m . The possible reshuffling, however, is not due to any externally imposed randomness; rather, one is facing the fact that a single free parameter in H suffices to enforce a level crossing, i.e., a generic property of integrable systems. All effective randomness in the sequence of the em thus appears entirely due to the assumed integrability of the Hamiltonian H.

5.11 Problems 5.1 Give H (I) for the Kepler problem. 5.2 Show that the hydrogen spectrum is as pathological (nongeneric) as that of the harmonic oscillator. Give other examples of nongeneric spectra of integrable systems. 5.3 Show that the probability density for finding the k th neighbor of a level in the distance increment [S, S+dS], for a stationary Poissonian “process” is Pk (S) =

S k−1 −S e . (k − 1)!

For k = 2 the so-called semi-Poissonian distribution results which is usually written as P(S) = 4Se−2S , so as to secure S = 1; see [12]. 5.4 Show that the conditional probability g(S) defined in Sect. 5.5 is related to Dyson’s two-level cluster function (Sect. 4.10) Y (S) = 1 − g(S), provided that the spectrum is homogeneous. 5.5 Prove rigorously that L independent spectra with E μ (S) = (1/L 2 ) exp (−L S) superpose to yield a spectrum with exponentially spaced levels.

168

5 Level Clustering

5.6 Show that the Thomas–Fermi level staircase is homogeneous of degree f for integrable systems whose Hamiltonian is a homogeneous function of the f actions I. Give the explicit form of σ¯ (E). 5.7 In which sense may the frequencies ωi defined by (5.2.2) and (5.2.3) be taken to be positive? (see the remark on (5.7.14) following (5.7.19)).

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

I.C. Percival: Adv. Chem: Phys. 36, 1 (1977) M. Brack, R.K. Bhaduri: Semiclassical Physics (Addison-Wesley, Reading, MA, 1997) V.P. Maslov: Th´eorie des Perturbations et M´ethodes Asymptotiques (Dunod, Paris, 1972) V.P. Maslov, M.V. Fedoriuk: Semiclassical Approximation in Quantum Mechanics (Reidel, Boston, 1981) J.B. Delos: Adv. Chem. Phys. 65, 161 (1986) J.-P. Eckmann, R. S´en´eor: Arch. Rational Mech. Anal. 61, 153 (1976) R.G. Littlejohn, J.m. Robbins: Phys. Rev. A 36, 2953 (1987) N. Rosenzweig, C.E. Porter: Phys. Rev. 120, 1698 (1960) M.V. Berry, M. Robnik: J. Phys. A: Math. Gen. 17, 2413 (1984) M.V. Berry, M. Tabor: Proc. R. Soc. Lond. A349, 101 (1976) M.V. Berry, M. Tabor: Proc. R. Soc. Lond. A356, 375 (1977) E.B. Bogomolny, U. Gerland, C. Schmit: Phys. Rev. E59, R1315 (1999)

Chapter 6

Level Dynamics

6.1 Preliminaries As Pechukas discovered [1], the fate of the eigenvalues and eigenvectors of a Hamiltonian H = H0 + λV upon variation of λ can be described by a set of ordinary firstorder differential equations. That set was interpreted by Yukawa [2, 3] as Hamilton’s equations for fictitious classical particles moving in one dimension as the “time” λ elapses. The number N of these fictitious particles equals the number of levels of H. However, the phase space of the fictitious many-body systems has a dimension larger than 2N due to the fact that the coupling strengths for particle pairs become dynamic variables themselves. The nontrivial interactions notwithstanding, the fictitious-particle dynamics is integrable; this integrability follows from the equivalence between the dynamics in question and the quantum mechanical problem of diagonalizing the finite-dimensional matrix H = H0 + λV. The same strategy works for Floquet operators F = exp (−iλV ) exp (−iH0 ) of periodically kicked quantum systems [4–6] on which this chapter largely concentrates. Here, as the fictitious time λ elapses, the phase-space trajectory of the fictitious classical system winds around an N torus and tends to cover that torus ergodically [7]. The reformulation of level dynamics as the classical Hamiltonian flow of a fictitious one-dimensional gas is particularly useful for dynamic systems whose classical limit is globally chaotic. If there are two or more classical degrees of freedom and if H0 + λV is classically integrable, the quantum energy levels do not generically repel, at least not in any way reminiscent of repulsive interactions between particles. As we have seen in the previous chapter and shall check again here from a different perspective, levels from different multiplets cross freely. Intramultiplet encounters of levels, it shall be revealed here, typically take the form of just barely avoided crossings with the closest approach spacing often difficult to resolve, at least in the limit N → ∞. For chaotic dynamics, on the other hand, the limit N → ∞ allows to apply statistical mechanics to the fictitious particles and thus indirectly to the original quantum problem represented by the Hamiltonian H or the Floquet operator F. We shall in fact see that random-matrix theory for the original quantum problem arises as

F. Haake, Quantum Signatures of Chaos, Springer Series in Synergetics, 3rd ed., C Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-05428-0 6, 

169

170

6 Level Dynamics

equilibrium statistical mechanics for the fictitious particles: Spectral characteristics like the distribution of nearest-neighbor spacings or the two-point cluster function of the level density calculated with a uniform probability coverage of the torus come out as predicted by random-matrix theory to within corrections of relative weight N1 . Thus, an important step toward understanding the universality of spectral fluctuations according to the Bohigas–Giannoni–Schmit conjecture [8] can be made. Ergodic coverage of an N -torus means equality of time (here, “time” is the control parameter λ) and ensemble averages of spectral characteristics like the distribution of nearest neighbor spacings. Since the “time” interval needed for the average is of the order of a collision time of the fictitious gas and as such of the order N1ν ∝  f ν with some positive exponent ν, all averaged dynamical systems have the same classical limit. It is quite remarkable that the interpretation of spectral universality thus reached involves no semiclassical methods and thus stands in independence of the periodic-orbit based one to be presented in Chap. 10. Concepts and methods of nonequilibrium statistical mechanics also come into play. For instance, the transition from level clustering to level repulsion (which accompanies the classical transition from predominantly regular motion to global chaos) appears as a relaxation into equilibrium [9]. Another example is the transition from one universality class of level repulsion to another, when H or F changes its antiunitary symmetries as λ grows. Such relaxation processes will be illustrated by calculating the level spacing distribution P(S, λ) for some simple cases. In perhaps the most interesting application of irreversible statistical mechanics, I shall show that Dyson’s Brownian-motion model [10] is a rigorous consequence of level dynamics for autonomous systems. Throughout this chapter, Planck’s constant will be set equal to unity,  = 1, except where the disregard of Planck might hurt the feelings of even the toughest theorists.

6.2 Fictitious Particles (Pechukas-Yukawa Gas) We shall deal with time-dependent Hamiltonians of the form H (t) = H0 + λV

+∞ 

δ(t − n)

(6.2.1)

n = −∞

which entail the single-period Floquet operators F = e−iλV e−iH0 .

(6.2.2)

The Hilbert space will be assumed to have the finite dimension N . When the kick strength λ is varied, the eigenvalues and eigenvectors of F, defined by F|m = e−iφm |m, m = 1, 2, . . . N ,

(6.2.3)

6.2

Fictitious Particles (Pechukas-Yukawa Gas)

171

will change, and a description of that change is to be established. Our goal is similar to that of perturbation theory, but we demand more than just the first few terms of a power series in λ. We differentiate the identity Fmm = e−iφm with respect to the kick strength,   ˙ + m|m ˙ = Vmm , φ˙ m = Vmm + i m|m

(6.2.4)

assuming that the eigenstates |m are normalized to unity for all λ (a dot above a bra or ket denotes differentiation of that bra or ket). To find the variation of Vmm with λ, we differentiate again: ˙ V˙ mm = m|V |m + m|V |m ˙   ˙ ˙ m|nV = nm + Vmn n|m  .

(6.2.5)

n

˙ The quantities m|n and n|m ˙ are obtained by differentiating the eigenvalue equation (6.2.3) and taking the scalar product with |n(= |m), ∗ ˙ = n|m ˙ = m|n

−iVnm 1 − e−iφnm

for

m = n,

(6.2.6)

so that the rate of change of the diagonal element Vmm reads V˙ mm = i



 Vmn Vnm

n(=m)

1 1 − 1 − e−iφmn 1 − e+iφmn

 .

(6.2.7)

Note the shorthand φmn = φm − φn

(6.2.8)

for the difference between two quasi-energies. The off-diagonal elements Vmn are treated similarly, ˙ V˙ mn = m|V |n + m|V |n ˙ =



 ˙ ˙ m|lV ln + Vml l|n  .

(6.2.9)

l

Separating l = m, l = n from l = m, n and again using (6.2.6) we obtain   Vmn (Vmm − Vnn ) ˙ V˙ mn = m|m + n|n ˙ Vmn − i 1 − e−iφmn    1 1 +i Vml Vln − , m = n. 1 − e−iφml 1 − e−iφln l(=m,n) (6.2.10)

172

6 Level Dynamics

At this point it is well to realize that each eigenvector |m can be multiplied (“gauged”) with an arbitray phase factor, |m = e−iθm |m0 ,

(6.2.11)

and that each of the N real phases θm may arbitrarily depend on λ. No choice of these phases (“gauge”) can alter the eigenvalues φm (λ). Differentiation of (6.2.11) with respct to λ gives |m ˙ = −iθ˙m |m + e−iθm |m ˙0 , and the latter identity yields m|m ˙ = iθ˙m + 0 m|m ˙0 . Clearly, then, no generality is lost by setting 0 m|m ˙0 = 0 and thus m|m ˙ = −iθ˙m , while still retaining the arbitrariness of the phases θm . Similarly arguing for the “off-diagonal” products n|m ˙ we get n|m ˙ = ei(θn −θm ) 0 n|m ˙0 . For further reference, I put the upshot of the foregoing mini-excursion into gauge freedom on display as m|m ˙ = iθ˙m n|m ˙ = ei(θn −θm ) 0 n|m ˙0 , for i(θn −θm ) m|V |n = e 0 m|V |n0 ;

(6.2.12) m = n,

(6.2.13) (6.2.14)

the last of the foregoing identities follows immeadiately from the basic gauge freedom (6.2.11). As an immediate application we may introduce the arbitrary gauge phases in the evolution equation (6.2.10) for the off-diagonal matrix elements Vmn , Vmn (Vmm − Vnn ) V˙ mn = i(θ˙m − θ˙n )Vmn − i 1 − e−iφmn    1 1 +i Vml Vln − , m = n. 1 − e−iφml 1 − e−iφln l(=m,n)

(6.2.15)

The differential equations (6.2.4), (6.2.7) and (6.2.15) form a complete set. They suffice to determine the quasi-energies φm and all matrix elements of the perturbation V for arbitrary values of λ, once “initial” conditions are set at, say, λ = 0 and the gauge phases are fixed. It is well to realize that the eigenphases φm are gauge independent since the diagonal elements Vmm and the squared moduli |Vmn |2 are, due to the gauge identity (6.2.14). Perturbation theory could be extracted by solving in terms of a power series in λ, and for such an endeavor the gauge θm = 0 for m = 1, 2, . . . , N would be most convenient. For the goal of the present Section, however, it is advantageous to keep the gauge freedom manifest. The goal alluded to is a reformulation of the flow (6.2.4), (6.2.7) and (6.2.15) as a classical Hamiltonian flow, and to that I propose to turn now. The flow of the φm , Vmm , Vmn cannot be Hamiltonian in character since it is not free of sources,   ∂ φ˙ m m

  ˙  Vmm − Vnn ∂ Vmn ∂ V˙ mm + = −i = 0. + ∂φm ∂ Vmm ∂ Vmn 1 − e−iφmn m=n m=n

(6.2.16)

6.2

Fictitious Particles (Pechukas-Yukawa Gas)

173

The net divergence originates from the second term in (6.2.15). An intuitive next step therefore is to replace the off-diagonal elements Vmn by lmn = Vmn f mn

(6.2.17)

and to try to determine the factors f mn so as to cancel the second term in (6.2.15) in the evolution equation of lmn . That requirement yields the differential equation    ˙f mn iφ˙ mn iφ˙ mn φmn . = = (6.2.18) + ln sin f mn 1 − e−iφmn 2 2 which is easily integrated to f mn = const eiφmn /2 sin



φmn 2

 .

Choosing the integration constant as −2 I get   φmn ∗ lmn = −2Vmn eiφmn /2 sin . = −lnm 2

(6.2.19)

(6.2.20)

After a little algebra, the now source-free flow can be written as φ˙ m = pm p˙ m = −

 1 cos (φmn /2) lmn lnm 3 4 sin (φmn /2) n(=m)

l˙mn = i(θ˙m − θ˙n )lmn −

(6.2.21)

  lml lln  1 1 − 4 sin2 (φml /2) sin2 (φln /2) l(=m,n)

where pm = Vmm has been introduced as an intuitive shorthand. The first two equations in the foregoing set can clearly be interpreted as classical Hamiltonian equations ∂H = {H, φm } φ˙ m = ∂ pm ∂H p˙ m = − = {H, pm } ∂φm

(6.2.22)

with λ as the “time” and the Hamiltonian function H=

N 1  2  |lmn |2 pm + + ... ; 2 m =1 8 sin2 (φmn /2) m=n

(6.2.23)

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6 Level Dynamics

the dots stand for further terms independent of the φm , pm to be established presently. The familiar Poisson bracketsfor “position” and “momentum” variables have been employed here, { pm , φn } = δmn .

(6.2.24)

The dynamics arrived at suggests to speak of N classical particles of unit mass located on the unit circle, 0 ≤ φm < 2π, and interacting pairwise via a repulsive potential ∝ 1/ sin2 (φmn /2). The coupling strengths are not constants, however, but dynamical variables obeying the third equation in (6.2.21). The lmn cannot be looked upon as single-particle properties; each refers to a pair of particles. In order to elevate the source-free flow to a Hamiltonian one, all of the variables φm , pm , lmn must be assigned Poisson brackets. Pursuing that goal I first demand independence of the lmn from the pairs φm , pm in the sense 

   pm , li j = φm , li j = 0.

(6.2.25)

The Leibniz product rule { f, gh} = { f, g}h + g{ f, h},

(6.2.26)

as well as the Jacobi identity { f, {g, h}} + {g, {h, f }} + {h, { f, g}} = 0

(6.2.27)

must be respected with f, g, h functions of the φm , pm , lmn . The appropriate Poisson brackets for the lmn differ for the three Wigner-Dyson universality classes (orthogonal, unitary, and symplectic) of Floquet operators. The simplest case arises for the unitary class, with the brackets {lmn , li j } = δin lm j − δm j lin .

(6.2.28)

Before putting those brackets to use in our level dynamics it is well to note [11, 12] that they, together with the Jacobi identity (6.2.27), forbid the assumption of identically vanishing diagonal elements lmm , elements which have not yet popped up in the reasoning thus far. But indeed, with f = l pq , g = lik , h = lki the left hand side of the Jacobi identity erraneously yields the non-vanishing combination δ pq lqk + δiq l pi − δkq l pk − δ pi liq if one sets lii = lkk = 0 in intermediate steps; no such inconsistency arises if diagonal elements are admitted, as I shall henceforth do. The requirement l = −l † ,

(6.2.29)

already found above for the off-diagonal elements by demanding the evolution to be ∗ ; source-free in (6.2.20), restricts the diagonal elements to be imaginary, lmm = −lmm

6.2

Fictitious Particles (Pechukas-Yukawa Gas)

175

and that restriction will turn out unavoidable within the Hamiltonian reformulation under construction. The Hamiltonian H (6.2.23) still yields the evolution equations for φm and pm written out in (6.2.21). In order to also elevate the evolution equation for lmn to the Hamiltonian form l˙mn = {H, lmn } it is necessary to complement the Hamiltonian with a piece the gauge term i(θ˙m − θ˙n )lmn , and that complement is easily yielding N ˙ seen to be j=1 iθ j l j j . The full Hamiltonian thus reads H=

N N  1  2  |lmn |2 pm + iθ˙ j l j j + 2 2 m =1 8 sin (φ /2) mn m=n j=1

(6.2.30)

and is real valued precisely due to the diagonal elements l j j being imaginary. The above Poisson brackets now yield 



l˙mn = i(θ˙m − θ˙n )lmn −

lml lln

l(=m,n)

+ lmn (lmm − lnn )

1 4 sin2 (φml /2)

1 4 sin (φmn /2) 2

.

−

1



4 sin2 (φln /2) (6.2.31)

Somewhat surprisingly, the latter Hamiltonian evolution generalizes (6.2.21), by including the extra term lmn (lmm − lnn ) sin−2 (φmn /2) which involves the now admitted diagonal elements. Fortunately, there is no contradiction between the original level dynamics and the present Hamiltonian reformulation since the diagonal elements of the matrix l are immeadiately revealed as constants of the motion, {H, lmm } = 0;

(6.2.32)

therefore, if they vanish at some initial “moment” they will retain that property for all values of the “time” λ, lmm = 0 ;

(6.2.33)

the latter “constraint” will be adopted henceforth,1 and then the Hamiltonian equation (6.2.31) becomes identical in appearance with the original evolution equation in (6.2.21). It may be well to add once more that the N arbitrary functions θm (λ) of λ can be done away with by gauging as θ˙m = 0; in that latter “primitive gauge” the differential equations of level dynamics take their simplest form, which in fact was the only form known prior to [11, 12]. A final remark on the Poisson brackets (6.2.28) is appropriate. They may be read as those of generators of infinitesimal rotations in a complex N dimensional vector 1 One should keep in mind that the constraint l mm = 0 can be consistently imposed only after all Poisson brackets are evaluated.

176

6 Level Dynamics

space, i.e., the Hilbert space pertaining to Hamiltonians H and Floquet operators F from the unitary symmetry class. For that reason the lmn are often called “angular momenta”; see Problem 6.3. The latter remark is even helpful in passing to the orthogonal symmetry class where time reversal invariance allows to work with the quantum operators H or F in an N dimensional real Hilbert space, and therein infintesimal rotations are generated by angular momenta lmn obeying the Poisson brackets   lmn , li j =

1 2



 δm j lni + δni lm j − δn j lmi − δmi ln j .

(6.2.34)

These brackets reproduce the evolution equation (6.2.31) through l˙mn = {H, lmn }. Once again, the diagonal elements are conserved and the original level dynamics (6.2.21) is recovered by constraining as lmm = 0. If, in addition, the primitive gauge θ˙m = 0 is adopted the angular momentum matrix remains real antisymmetric, lmn = −lnm ,

(6.2.35)

if it is chosen so “initially”, at λ = 0. I shall not pause to construct the Poisson brackets pertaining to the symplectic symmetry class [13]; Problem 6.3 offers a different method for the derivation. The reader is invited in Problem 6.4 to establish the fictitious-particle dynamics corresponding to time-independent Hamiltonians H = H0 + λV, x˙ m =

∂H ∂ pm

p˙ m = −

∂H ∂ xm

(6.2.36)

l˙mn = (H, lmn ) H=

|lmn |2 1 2 1  , m pm + 2 m=n 2 (xm − xn )2

where the xm denote the eigenenergies of H . The constraint lmm = 0 as well as the primitive gauge θ˙m = 0 have been used in writing out the classical Hamiltonian flow (6.2.36). That flow was first established by Pechukas [1] and Yukawa [2, 3]; it can formally be obtained from (6.2.21) by linearizing via sin x → x, cos x → 1; thus the periodic potential proportional to sin−2 (φmn /2) is replaced by an inverse-square 2 . potential proportional to 1/xmn Prior to the discovery of their relevance to the diagonalization of quantum Hamiltonians and Floquet operators, the classical flows (6.2.36) and (6.2.21) had been known in the mathematical literature. The special cases of constant lmn are the Calogero–Moser and Sutherland–Moser dynamics [14–16]. Even the case of dynamic lmn had been investigated by Wojciechowski [17] as a “marriage of the

6.3

Conservation Laws

177

Euler equations with the Calogero–Moser system.” From the classical point of view, an interesting property of these systems is their integrability. In the present context, the integrability of (6.2.36) is not surprising since the fictitious-particle dynamics is equivalent to the diagonalization of an N × N matrix. A more powerful method of establishing level dynamics, as well as an extension to general matrices can be found in [18, 19].

6.3 Conservation Laws I should first recall the conservation of the N diagonal elements lmm among the angular momenta, l˙mm = 0 .

(6.3.1)

To find further conserved quantities for classical flow (6.2.21), it is convenient to rederive that flow slightly more abstractly. Let us imagine the matrices F = e−iλV F0 , V and F0 specified in some fixed representation independent of λ. In this section, lower case Latin letters will be reserved for matrices expressed in the eigenrepresentation of the Floquet operator F, defined by e−iφ = W † F W, φ = diag (φ1 , φ2 , . . . , φ N ) .

(6.3.2)

The diagonalizing unitary transformation W is not unique; given one, say W0 , multiplication from the right with an diagonal unitary matrix yields another one, W = W0 e−iθ ,

θ = diag(θ1 , . . . , θ N ).

(6.3.3)

The freedom of choosing the N phases θm is precisely the freedom to multiply an eigenvector |m0 with a phase factor e−iθm , termed gauge freedom in the previous section and highlighted in (6.2.11), (6.2.12), (6.2.13), and (6.2.14). The matrix v = W †V W

(6.3.4)

has the elements vmn = δmn pm + (1 − δmn )Vmn = δmn pm +

(1 − δmn )lmn . i(eiφmn − 1)

(6.3.5)

The “angular-momentum matrix”   l = i eiφ ve−iφ − v + l diag = −l †

(6.3.6)

has its off-diagonal elements related to those of v as in (6.2.20) while the diagonal elements lmm , assembled in l diag , are unrelated to v.

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6 Level Dynamics

We will also employ the anti-Hermitian matrix ˙ † W, a = −a † = W

(6.3.7)

where the dot again means differentiation with respect to λ. The aforementioned gauge freedom amounts to a † = iθ˙ + eiθ a0 e−iθ .

(6.3.8)

amm = iθ˙m + a0mm

(6.3.9)

The diagonal elements

are imaginary; due to the arbitrariness of the gauge phases θm no loss of generality is incurred by setting a0mm = 0, and I adopt that choice henceforth. I shall stay away, however, from the “primitive gauge” θ˙m = 0 which would annull the diagonal elements amm altogether. By differentiating e−iφ , v, and l with respect to λ, we obtain φ˙ = ia + v − ie−iφ a eiφ , v˙ = [a, v],

(6.3.10)

l˙ = [a, l]. The first of these equations yields φ˙ m = vmm = pm and, upon taking off-diagonal elements, one obtains the amn in terms of the Vmn and the quasi-energies φm as amn =

ivmn eiφmn lmn . =− iφ mn e −1 4 sin2 φ2mn

(6.3.11)

The equation of motion for v in (6.3.10) comprises (6.2.7) and (6.2.15) and thus especially the canonical equation for the momenta pm from (6.2.21), while the equation of motion of l is the canonical equation of motion (6.2.31) for the angular momenta. Most importantly, the commutator structure of the last two equations in the foregoing set, commonly called “Lax form” [20] , immediately yields an infinity of constants of the motion, Iμ1 ν1 μ2 ν2 ... = Tr {l μ1 v ν1 l μ2 v ν2 . . . } μ1 , ν1 , μ2 , ν2 , . . . = 0, 1, 2, 3, . . . .

(6.3.12)

Following Ku´s and coworkers [11, 12], I next propose to show that the infinite set of Iμ1 ν1 μ2 ν2 ... ’s contains precisely N (N − 1) independent constants of the motion. As a first step, the I ’s can be replaced by the set Cmn = Treiφ v m e−iφ v n ,

m, n = 0, 1, . . . .

(6.3.13)

6.3

Conservation Laws

179

Since eiφ v e−iφ = v − il and thus eiφ v m e−iφ = (v − il)m , the C’s are linear combinations of the I ’s. Functionally independent C’s arise as C0m = Cm0 = Trv m with m = 1, 2, . . . , N − 1 (which are N − 1 in number) and the Cmn with m, n = 1, 2, . . . , N − 1 (which are (N − 1)2 in number). Higher values of the exponents m, n yield only linear combinations, due to the Hamilton–Cayley theorem. Indeed, then, N (N − 1) is the number of independent constants of the motion of the type C (or, equivalently I ). It will be important to compare the number of independent dynamical variables with the number of “contraints” of our level dynamics. I shall do the counting for the unitary symmetry class, leaving to the reader the orthogonal and symplectic cases as simple exercises. There are N “coordinates” φm and as many momenta pm . The angular momenta comprise N imaginary diagonal ones and, due to the anti-Hermiticity of the matrix l, N (N − 1)/2 independent off-diagonal ones which are complex. Altogether, then, we confront 2N + N 2 real dynamical variables. Conserved, the other hand, are the N diagonal angular momenta lmm and the N (N − 1) independent C’s. Furthermore, the gauge freedom allows to fix N phases θm (λ) and thus N phases for the off-diagonal angular momenta lmn . We thus have a total of N + N 2 “constraints” for the flow of the 2N + N 2 independent variables φ, p, l. If no further restrictions exist the flow proceeds on a manifold of dimension N . An independent argument due to Ku´s [7] in fact reveals that level dynamics is confined to an N -torus. The λ dependence of the eigenvalues and eigenvectors of a Floquet operator is uniquely determined by the λ dependence of the matrix representative of F(λ) in any representation. Momentarily employing the eigenrepresentation of V , V |μ = ωμ |μ,

F(λ) eiH0 |μ = e−iλV |μ = e−iλωμ |μ,

(6.3.14)

we find the matrix F(λ) to depend on λ only through the N exponentials e−iλωμ which generically have non-dengenerate “frequencies” ωμ . We conclude that F(λ) moves through the space of unitary matrices along a “trajectory” lying on an N -torus. If we reparametrize matrix space by a change of representation, say, to the eigenrepresentation of F(λ), we get a transformed trajectory described by the Hamiltonian flow of level dynamics. However, the topology of the manifold to which the trajectory is confined cannot have changed and still is that of an N -torus. It is thus clear that the above reasoning has uncovered all independent conservation laws. We can also conclude that the N (N − 1) independent C’s and the N diagonal angular momenta lmm do not entail further independent constants of the motion as Poisson brackets {lkk , Cmn } or {Cmn , Cm n }; see Problem 6.5. It now appears natural to apply statistical mechanics to the N fictitious particles. The appropriate statistical ensemble to be used is a generalized microcanonical one represented by a product of delta functions, one for each of the independent constants of the motion. But before embarking onto statistical analyses, we need to clarify some more basic issues.

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6 Level Dynamics

6.4 Intermultiplet Crossings Even nonintegrable classical Hamiltonian flows or maps can have constants of the motion C, possibly symmetry based. For convenience, I formulate the following for a Hamiltonian flow and assume a single conserved quantity, but everything is easily transcribed to maps and any number of conservation laws. Such a C has a vanishing Poisson bracket with the Hamiltonian function H , {H, C} = 0. The two corresponding quantum operators, to be distinguished by a hat in the present section, ˆ , C] ˆ = 0. The quantum energy spectrum consists of multiplets, then commute, [ H ˆ each of which is labelled by an eigenvalue of C. ˆ (λ) = H ˆ 0 + λVˆ . If the Now let us consider a Hamiltonian of the structure H single conservation law we are allowing for holds for all values of λ, the conserved ˆ quantity will in general also depend on λ and be denoted by C(λ). Of course, Cˆ ˆ ˆ remains independent of λ if commuting with both H0 and V , and this is the natural ˆ 0 and Vˆ . I shall first discuss situation when one and the same symmetry reigns for H ˆ ˆ that simple case but still insist on [ H0 , V ] = 0. Levels from different multiplets have no reason not to cross when the conˆ 0 and Cˆ have common eigenvectors, trol parameter λ is varied. In fact, since H ˆ ˆ C|m, i; λ = Ci |m, i; λ, H |m, i; λ = E mi (λ)|m, i; λ, off-diagonal matrix elements of the perturbation Vˆ pertaining to different Ci , i.e., to different multiplets, vanish, m, i; λ|Vˆ |n, j; λ = 0. Therefore, the level dynamics (6.2.21) has vanishing intermultiplet couplings and splits into independent dynamics for the separate multiplets. Then, an intermultiplet crossing of levels has codimension 1 which means uninhibited crossings as λ varies. Loosely speaking, every multiplet has its own Pechukas–Yukawa gas. Intermultiplet crossings are still unhindered if the conserved quantity in question ˆ (λ) and depends on λ, provided only a pair of common eigenfunctions |i, λ of H ˆ C(λ) with close by energies, E 1 (λ) ≈ E 2 (λ), do not have the eigenvalues C1 (λ) and C2 (λ) in simultaneous close approach for, say λ ≈ λ0 . In that case, we may start ˆ (λ0 ) and C(λ ˆ 0 ) and calculate their successors from the the eigenfunctions |i, λ0  of H

ˆ ˆ 0 ). Since by |i, λ0 + δλ perturbatively, treating δλC (λ0 ) as a perturbation of C(λ assumption the eigenvalues C1 (λ0 ) and C2 (λ0 ) are not close to one another, the denominator C1 (λ0 ) − C2 (λ0 ) in the perturbation expansion is not small such that we can even calculate the shifted energies employing the zero-order eigenfunctions as E i (λ) = E i (λ0 )+δλi, λ0 |Vˆ |i, λ0  ≡ E i (λ0 )+δλVˆ ii . Thus, we expect the nearest crossing for δλ = −

E 1 (λ0 ) − E 2 (λ0 ) . V11 − V22

(6.4.1)

ˆ It is worth underscoring the responsibility of the conservation C(λ) for close encounters of levels from different multiplets to be associated with true crossings rather than anticrossings. Moreover, conserved quantities dependent on the strength of a perturbation is no more outlandish a phenomenon than hydrogen in an electric field [5].

6.5

Level Dynamics for Classically Integrable Dynamics

181

6.5 Level Dynamics for Classically Integrable Dynamics An integrable classical Hamiltonian flow with f freedoms has (at least) f conserved quantities which may be chosen as the actions of f action angle pairs of phase-space coordinates. If the f conservation laws remain rigorously intact quantum mechanically, the quantum energy spectrum consists of multiplets within each of which only the quantum number associated with a single conserved observable labels levels. What we have just learned about intermultiplet crossings remains true here as well. It remains to clarify whether intramultiplet crossings can generically occur or are avoided. That question can be addressed for f = 1, without loss of generality, and a satisfactory answer can be found with the help of a few examples. There are single-freedom systems without level crossings like the harmonic oscil1 2 p + 12 mω2 x 2 and V = −λx with λ a constant external force. lator with H0 = 2m The well-known levels E n (λ) = (n + 12 )ω − λ2 /2mω2 retain their spacing ω as λ varies. Another instructive example is the Hamiltonian   & λ ω Jz + √ Jx (6.5.1) H = ω Jz + λJx = ω2 + λ2 √ ω 2 + λ2 ω 2 + λ2 √ which generates uniform rotation rotationwith angular velocity ω2 + λ2 about  ω λ the axis defined by the unit vector eˆ = √ω2 +λ2 , 0, √ω2 +λ2 . In a Hilbert space with total angular √ momentum quantum number j, the levels of that Hamiltonian are E m (λ) = m ω2 + λ2 with m = − j, − j + 1, . . . , j. These never cross as λ grows. We could even speak of a single avoided crossing for each pair of nearest √ neighbor levels at λ = 0 since the spacing ω2 + λ2 does have a minimum there. However, the spectrum has no fluctuations at all inasmuch as we can find an explicit (λ-dependent) rescaling of the energy to get a rigid ladder of equidistant levels. √ Such rescalings are achieved by H = (ω Jz + λJx )/ ω2 + λ2 or, equivalently, H

= Jz cos λ + Jx sin λ. But let us not overhastily take complete rigidity of single multiplets as generic! Some Hamiltonians composed of angular momentum operators immediately tell us to beware of such a misconception. For instance,  4 2 1 Jz − 2 Jz − ( j + 12 ) sin λ (6.5.2) H (λ) = Jz cos λ + 2 j+1 has the 2 j + 1 levels E m (λ) = m cos λ +



4 m2 2 j+1

− 12 m − ( j + 12 ) sin λ

(6.5.3)

with m = − j, − j + 1, . . . , j, if the quantum number j is kept fixed. Figure 6.1 displays the level dynamics for the latter Hamiltonian as H (λ) is converted from 4 Jz2 − 12 Jz − ( j + 12 ). It is easy to see that the number H0 = Jz into V = 2 j+1 of crossings in the interval 0 ≤ λ ≤ π/2 is ∝ N 2 . These crossings arise since E m is a quadratic function of the quantum number m. Momentarily letting m range continuously, we encounter a minimum of E m at m min (λ) = ( 81 − 14 cot λ) ( j + 12 ). As

182

6 Level Dynamics

20

–4

E 10

–6

0

–8

–10

–10 0

π/4

λ

π/10

π/2

π/5

Fig. 6.1 Level crossings for the single-freedom Hamiltonian H (λ) = Jz cos λ +  Jz2 − j sin λ with j = 15. The right part is a blowup of the left

3π/10

1

J 2 z

+

4 2 j+1

long as that m min (λ) lies outside the interval [− j, j], the levels E m form a monotonic sequence and cannot cross as λ varies; but for λ exceeding the critical value λc determined by tan λ = 29 , the minimum of E m lies within the interval − j ≤ m ≤ j and then a further increase of λ leads to crossings. An even simpler example of a single-freedom Hamiltonian with level crossings is H (λ) = Jz cos λ− j12 Jz3 sin λ. I leave to the reader to show that within each multiplet of fixed j, each level crosses all others as λ grows from 0 to π/2. It is well to illustrate the absence of level repulsion for the quasi-energies of integrable quantum maps. Quasi-energies are defined modulo 2π , and taking the modulus is very effective in bringing about level crossings. The eigenphases of λ Jz2 )] display a never ending sequence of crossings as do F = exp[−i(Jz + 2 j+1 those of the even simpler Floquet operator F = exp(−iλJz ). See Fig. 6.2. The common feature of all of these examples is the absence of repeated avoided crossings between all neighboring levels, with closest approach spacings not much smaller than the mean spacing. If we still want to hold on to an analogy with

φ

τ

  λ Fig. 6.2 Level crossings for the integrable Floquet operator F(λ) = exp Jz + 2 j+1 Jz2 with j = 15

6.5

Level Dynamics for Classically Integrable Dynamics

183

a gas that PYG would have to be imagined as a rather ideal one. Frequent strongly avoided crossings seem to be the exclusive privilege of nonintegrable dynamics. For instance, if we destroy integrability by allowing for a kick such that the Floquet operator of the linear rotation, F0 = exp(−iλJz ), is accompanied by the torsion τ Jy2 ), one gets two multiplets, each of which displays avoided F1 (τ ) = exp(−i 2 j+1 crossings but no crossings; needless to say, as long as the torsion constant τ is sufficiently small, the avoided crossings are narrow and may be hard to resolve, but non-zero they definitely are; as soon as τ = O(1), we confront the predominance of chaos and universal level repulsion; Fig. 6.3 reveals the transition toward universal level repulsion for the Floquet operator F = F0 F1 (τ ) under discussion. We ought to dig deeper into the difference between integrable and nonintegrable dynamics manifest in avoided intramultiplet crossings. As becomes clear from Fig. 6.4, the single-freedom and thus integrable Hamiltonian   4 2 1  Jz − 2 Jz − ( j + 12 ) sin λ H (λ) = Jz +  Jx cos λ + 2 j+1

(6.5.4)

differs from (6.5.3) by the perturbation  Jx cos λ which does not commute with the remainder; for Fig. 6.4 that perturbation was chosen small,  = 0.4, so as to retain some similarity to the case  = 0 displayed in Fig. 6.2. Now, we encounter avoided crossings of a less trivial kind than those for (6.5.1). It will be noted, though, that the apparent crossings (which are not true crossings but unresolved avoided ones) outnumber the visibly avoided ones, and here lies the clue to the distinction in search. A good tool for the intended digging is the Einstein–Brillouin–Keller or WKB quantization presented in the previous chapter. For a comprehensive exposition of the techniques and ideas behind the more qualitative reasoning to be followed here, the reader is referred to P.A. Braun’s review [21].

λ Fig. 6.3 Level dynamics for the kicked top with the Floquet operator F = exp(−i 2 j+1 Jz2 ) exp(−i p Jy ), j = 25, and p = 1.7. Only eigenvalues pertaining to the positive eigenvalue of exp (iπ Jy ) are displayed. The classical transition from regular motion (λ = 0) to global chaos upward of λ ≈ 5 looks like an equilibration process whereas equilibrium appears to reign for λ > ∼5

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6 Level Dynamics

 Fig. 6.4 Some crossings are strongly avoided for the single-freedom Hamiltonian H (λ) = Jz +    2 4 J cos λ + 12 Jz + 2 j+1 Jz2 − j sin λ with j = 15. The right part is a blowup of the left 5 x

For a semiclassical treatment of the Hamiltonian (6.5.4), we must first degrade it to a classical Hamiltonian function. To that end, we introduce a pair of canonically conjugate phase-space coordinates p, ϕ through Jz /j → p,

Jx /j →

&

1 − p 2 cos ϕ,

Jy /j →

&

1 − p 2 sin ϕ;

(6.5.5)

this amounts to replacing the angular momentum J by the classical unit vector X = (X, Y, Z ) = lim j→∞ J/j and representing the latter by a polar angle θ and an azimuthal angle ϕ with p = cos θ . The canonical conjugacy of p and ϕ can be checked as follows: We start from the commutator i j[Jx /j, Jy /j] = −Jz /j. The usual replacement of commutators by Poisson bracketsyields the Poisson brackets 2 the latter in turn can be obtained {X, Y } = −Z of a classical&angular momentum;& 2 from { p, ϕ} = 1 and X = 1 − p cos ϕ, Y = 1 − p 2 sin ϕ, Z = p through the identity { f ( p), g(ϕ)} = f ( p)g (ϕ). The phase space thus turns out to be the rectangle −1 ≤ p ≤ 1, 0 ≤ ϕ < 2π in the plane spanned by p, ϕ; the rectangle may of course be seen as a projection of the unit sphere onto the said plane; therefore, the phase space is sometimes said to be spherical. It is well to keep the foregoing reasoning in mind in view of the ubiquity of angular-momentum dynamics (the kicked top, for instance) in this book. The classical Hamiltonian function H/( j + 12 ) → h( p, ϕ) reads &     h( p, ϕ, λ) = p +  1 − p 2 cos ϕ cos λ + 2 p 2 − 12 p − 1 sin λ (6.5.6) ≡ h 0 ( p, ϕ) cos λ + h 1 ( p) sin λ.

2 Note that we have set Planck’s constant equal to unity here such that its role is taken over by 1/j; ˆ B, ˆ ... in more conventional notation the transition from commutators of quantum observables A, ˆ B] ˆ → {A, B} = to Poisson brackets of the associated classical observables A, B, . . . reads i [ A, ∂A ∂B − ∂∂ Bp ∂∂qA . ∂ p ∂q

6.5

Level Dynamics for Classically Integrable Dynamics

185

The ensuing classical dynamics is open to the qualitative investigation of singlefreedom systems familiar from elementary textbooks, even though the Hamiltonian is not the sum of a kinetic and a potential term. Since | cos ϕ| ≤ 1, it is helpful to look into the plane spanned by the momentum p and the energy E and to draw the graphs of the two functions U+ ( p, λ) = h( p, 0, λ),

U− ( p, λ) = h( p, π, λ).

(6.5.7)

Figure 6.5 reveals that these two curves join smoothly with vertical slope at p = ±1. All points in the enclosed area correspond to classically allowed states. The shape of that area changes from a pancake to a banana as we increase the control parameter λ from zero toward π/2; at λ = π/2, the banana has shrunk in width to the parabolic segment h 1 ( p) in −1 ≤ p ≤ 1. The lower curve, U− ( p), has a single minimum irrespective of λ; the upper one, U+ ( p), has a single maximum for λ below a critical value λc but two maxima and an intervening minimum if λ > λc . It follows that p oscillates back and forth within only a single allowed interval if λ < λc whatever value the energy takes, whereas for λ > λc , there is an energy range with two separate accessible intervals for p. The classically forbidden region in between separate intervals of oscillation becomes penetrable by quantum tunneling. Consequently, energy levels within the energy range allowing for classically separate motions but well away from the “top of the barrier” (the minimum of U+ ( p) in the present example) can come in close

Fig. 6.5 The classically accessible region in the energy-momentum plane for the Hamiltonian (6.5.6) is surrounded by the curves U± ( p) given in (6.5.7). The top plot refers to a sub-critical value of λ for which only a single interval of oscillation exists while two separate such intervals arise for supercritical λ (middle and bottom); in the bottom case, the coalescence of the curves U± ( p) into the single line E m |m→ p according to (6.5.3) is imminent

186

6 Level Dynamics

$ pairs with the small splitting proportional to the factor exp{−| barrier dϕp|/}; in other words, tunneling through a high and thick barrier is exponentially suppressed and thus very slow. The three parts of Fig. 6.5 refer to three different values of λ, one of which is subcritical so as to allow for only one classically accessible p-interval; quantum mechanically. no level spacing is made small by tunneling if λ varies in the subcritical range, as is obvious from the level dynamics in Fig. 6.4. The middle part of Fig. 6.5 pertains to a slightly supercritical value in the neighborhood of which near crossings appear in the level dynamics which persist onto the third part which has its λ closer to π/2. We should clarify why some and which fraction of the avoided crossings are strongly avoided. These distinguished encounters happen near a line E(λ), not drawn into the level dynamics of Fig. 6.4, which gives the energy of the minimum of the upper boundary U+ ( p, λ) of the banana of Fig. 6.5, corresponding to the “top of the potential barrier.” Clearly, pairs of levels near that “top” cannot come very close to one another since here tunneling $ processes have but a small and thin barrier to penetrate; thus the factor exp{−| barrier dϕp|/} determining the splitting is not small. A substantial fraction (finite even when N → ∞) of all levels will get close to the curve E(λ) such that the number of strongly avoided crossings is proportional to N , quite small a minority indeed compared to the unresolved crossings whose number, we have seen, is of the order N 2 . It may be well to note that for a given value of λ the energy E(λ) defines a contour line in phase space which is the separatrix between periodic orbits encircling either one or the other of the two stable fixed points (the latter corresponding to the two minima of U+ ( p)) and those encircling both fixed stable points; the separatrix itself of course intersects itself at the hyperbolic fixed point. One final example of an integrable Hamiltonian is needed to convince us that strongly avoided crossings are exceptional events in the semiclassical limit N → ∞. In search of a Hamiltonian H (λ) whose N levels display roughly N 2 strongly avoided crossings and whose minimal spacings are a sizable fraction of the mean spacing, we might try a large-order polynomial in angular momentum components like a Chebyshev polynomial of the first kind, Tn (Jz /j) with n  1, in  H = Tn (Jz /j) + 2Jx /j cos λ + (Jz /j) sin λ.

(6.5.8)

It is easily checked that the ensuing level dynamics has the majority of crossings strongly avoided, if the dimension 2 j + 1 of the Hilbert space is comparable to the order n; but as soon as j  n, the majority of crossings become barely resolvable as avoided. If for the sake of illustration we fix n = 10 and count the percentage of closest approach spacings whose ratio with the mean spacing is in excess of e−4 , we find that percentage as 98, 44, and 31, respectively, for j = 10, 20, 100. To conclude, integrable dynamics have level dynamics corresponding to near ideal PYG’s. The weak nonideality lies in the rare strongly avoided crossings and the exponential smallness of most closest-approach spacings.

6.6

Two-Body Collisions

187

6.6 Two-Body Collisions As a final prelude to the promised statistical analyses, it is useful to consider what can be learned from the close encounter of two particles of the Pechukas–Yukawa gas corresponding to a single multiplet in a system with global classical chaos. We shall recover the insight already obtained by almost degenerate perturbation theory in Sect. 3.4. When two of the N particles, to be labeled 1 and 2, approach each other so closely that their mutual force by far exceeds the force exerted on either by any other particle, the collision effectively decouples from the rest of the dynamics. Restricting the consideration to the Floquet case, from (6.2.21), cos (φ12 /2) φ¨ 1 = −φ¨ 2 = 14 |l12 |2 3 sin (φ12 /2)

(6.6.1)

l˙12 = 0, the latter equation is due to the absence of small denominators sin2 (φ12 /2) in l˙12 given by the third of the Eq. (6.2.21). Obviously, the center of mass (φ1 + φ2 )/2 moves uniformly, while the relative coordinate φ = φ12 /2 obeys (setting l12 = l for short) 1 cos φ ∂V φ¨ = |l|2 3 = − 4 ∂φ sin φ

(6.6.2)

where the potential energy V (φ) =

1 |l|2 8 sin2 φ

(6.6.3)

is π-periodic in φ and confines the coordinate φ to [0, π ] if φ is in that interval initially. By using energy conservation, one easily constructs the solution   √ 2E(λ − λˆ . φ(λ) = arccos cos φˆ cos

(6.6.4)

Here λˆ denotes one of the times of closest approach of the two particles, 2φˆ is the corresponding angular distance, and E=

|l|2 8 sin2 φˆ

(6.6.5)

is the energy of the nonlinear oscillation. Consistent with the neglect of all other particles, we may assume that φˆ  1 and study the close encounter at times λ ≈ λˆ by expanding all cosines as cos x = 1 − x 2 /2. The result is

188

6 Level Dynamics

 ˆ 2 1/2 (λ − λ) φ = ± φˆ 2 + |l|2 . 4φˆ 2

(6.6.6)

The discriminant in (6.6.6) is a sum of two or three nonnegative terms depending on whether the angular momentum l is real or complex. Thus, The codimension of a level crossing is found once more as n = 2 and n = 3 for systems with, respectively, O(N ) and U (N ) as their canonical groups. As could have been expected on intuitive grounds, isolating a pair of colliding particles from the remaining N − 2 particles is equivalent to the nearly degenerate perturbation theory applied to an avoided level crossing in Sect. 3.2. The case in which Sp(N ) is the canonical group can be treated similarly, even though Kramers’ degeneracy looks slightly strange in the fictitious-particle picture: the particles have pairwise identical positions φ and an avoided crossing corresponds to a collision of two pairs of particles with φ1 = φ1¯ , φ2 = φ2¯ . Now, the equations of motion read (see Problem 6.6)  cos (φ12 /2) 1 φ¨ 1 = −φ¨ 2 = |l12 |2 + |l12¯ |2 4 sin3 (φ12 /2)

(6.6.7)

with the coupling strength |l12 |2 +|l12¯ |2 again effectively constant. The solution φ(λ) given in (6.6.6) for the nondegenerate cases holds here, too, but with the replacement |l12 |2 → |l12 |2 + |l12¯ |2 , which immediately yields the correct codimension of a level crossing, n = 5.

6.7 Ergodicity of Level Dynamics and Universality of Spectral Fluctuations 6.7.1 Ergodicity Now, we can take up one of the most intriguing questions of the field. Why do dynamical systems with global classical chaos display universal spectral fluctuations in fidelity to random-matrix theory, and for what reasons do exceptions occur? Our discussion will focus primarily on periodically driven systems and their quasienergy spectra. A motivating glance at Fig. 6.3 is recommended; there we see the λ Jz2 ) exp(−i p Jy )) with the quasi-energy spectrum of a kicked top (F = exp(−i 2 j+1 torsion strength λ varied from the integrable case λ = 0 to the range for which the classical motion is globally chaotic, λ > ∼ 5. Clearly, equilibrium reigns in the chaotic range while for λ up to, say, 5 we encounter an equilibration process. We had already seen in Sect. 6.2 that the level dynamics of an N × N Floquet matrix F = e−iλV e−iH0 proceeds on an N -torus. The pertinent frequencies are the eigenvalues ωμ of V for which we assume the generic case of no degeneracy. Moreover, if the eigenvalues ωμ are incommensurate, the motion on the N -torus will be ergodic. Ergodicity means that we can equate “time” averages, i.e., λ averages of certain observables with ensemble averages. The appropriate ensemble to use is the

6.7

Ergodicity of Level Dynamics and Universality of Spectral Fluctuations

189

generalized microcanonical ensemble nailing down the independent constants of the motion. In particular, it is not legitimate to work with the primitive microcanonical ensemble Z −1 δ(H − E) which fixes only the energy of the Pechukas– Yukawa gas, simply because the trajectory of level dynamics explores only an N torus rather than the full energy shell. As to the observables for which ensemble average equals “time” average (and ultimately even typical instantaneous values), the same thoughts as for other many-body systems apply. Quantities like spectrally averaged products of the level density or the spacing distribution can be expected to be self-averaging and thus to qualify for statistical description. Others, like the not spectrally averaged products of the level density, the (neither spectrally nor temporally averaged) form factor, or the localization length we shall meet in Chaps. 7, 10, and 11 allow for weaker statements only: Averages over the “time” λ still equal ensemble averages but there are strong ensemble fluctuations. Here, we are concerned mostly with self-averaging quantities. Time averages for the Pechukas–Yukawa gas are control parameter averages over a family of original dynamical systems: A λ average over, say, the spacing λ 2 distribution for F(λ) = e−i( 2 j+1 Jz +α Jz ) e−iβ Jx involves not a single kicked top but a whole one-parameter family of such. How can we draw conclusions for the actually observed universality of spectral fluctuations of individual dynamic systems like a single kicked top? The clue to the answer lies in the further question over how long an interval Δλ must we calculate time averages before the ensemble means are approached. One often stipulates that the interval Δλ is large in the sense Δλ → ∞. Such an extreme request allows us to accommodate and render weightless equilibration processes starting from initial conditions far from equilibrium. For level dynamics, such equilibration would arise if F(λ)|λ=0 = e−iH0 were the quantized version of an integrable classical dynamics and the switch-on of e−iλV paralleled the expansion of chaos to (almost) global coverage of phase space. On the other hand, if we bar such initial situations and assume that level dynamics displays avoided crossings right away, the interval need not be much larger than the mean control parameter distance between subsequent avoided crossings of a pair of neighboring levels. For the Pechukas–Yukawa gas, we could call that interval a collision time, i.e., the mean temporal distance between two subsequent collisions of a particle with its neighbors.

6.7.2 Collision Time The collision time just alluded to should scale with the number of particles (i.e., the number of levels N ) like a power [22], λcoll ∝ N −ν with ν > 0.

(6.7.1)

It is well to devote a little consideration to that power-law behavior and to indicate that the exponent ν cannot be expected to be universal, save for its positivity under conditions of chaos.

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6 Level Dynamics

First, let us consider a Hamiltonian of the structure H (λ) = H0 + λV and take H0 and V as independent random N × N matrices from, say, the GUE. Both should have zero mean and the same variance, H0i j = Vi j = 0,

|H0i j |2 = |Vi j |2 = 1/N .

(6.7.2)

According to Wigner’s semicircle law, see (4.7.2) and (4.7.3), the mean level spacing is Δ ≡ E i+1 − E i ∼ N1 . The level velocity vanishes in the ensemble mean and has the smean square pi2 = ψi |V |ψi 2 ∼

1 . N

(6.7.3)

Thus, a typical level velocity is p ∼ N−2 1

(6.7.4)

and the simple estimate λcoll ≈ Δ/ p ∼ N −1/2 yields the exponent ν = 1/2. The same value of the exponent results if we consider a Floquet operator F(λ) = e−iH0 e−iλV and again take H0 and V random as before. For the Floquet operator of the kicked top with V = Jz2 /(2 j + 1), the following argument will reveal ν = 32 . Fixing the number of levels as N = 2 j + 1 we get the  2 mean level velocity v ≡ N1 iN Vii = N1 TrV = N12 (1 + O( N1 )); but this describes a drift common to all levels and is irrelevant A typical velocity  N for collisions.  N relative V˜ ii2 , where again (Vii − v)2 ≡ N1 i=1 p results from the variance p 2 = N1 i=1 the matrix elements are meant in the eigenrepresentation of the Floquet operator F(λ); but in that representation the perturbation V˜ looks like a full matrix with  2  1  ˜2 1 N2 1 ˜ ˜ ˜2 i Vii ≈ 0 and i Vii ≈ N i j Vi j = N Tr V = 180 (1 + O( N )); we conclude

p ∝ N 2 . The mean level spacing 2π yields the collision time ∝ N − 2 and the N 3 exponent ν = 2 ; the latter value was confirmed by numerically following level dynamics for 10 < j < 160 [22]. At any rate, the asymptotic disappearance of the collision time under conditions of chaos is in sharp contrast to the parametric change of levels in integrable systems. No collision time could at all sensibly be established for the latter, as is clear from our discussion of Sect. 6.5. 1

3

6.7.3 Universality But inasmuch as Weyl’s law implies N ∝ (2π )− f , the minimal control parameter window under discussion can be specified as Δλ ∼  f ν and thus vanishingly small from a classical perspective. Thus, all of the different quantum systems involved in the control parameter alias “time” average can be said to have the same classical limit. Such classically vanishing control parameter intervals have recently been sug-

6.7

Ergodicity of Level Dynamics and Universality of Spectral Fluctuations

191

gested by Zirnbauer [23] in an attempt at demonstrating the universality of spectral fluctuations by the superanalytic method to be expounded in Chap. 11. The assumption of a close-to-equilibrium initial condition for the Pechukas– Yukawa gas means roughly, as already indicated above, that the original dynamical system should have global chaos to begin with. That assumption is crucial since an equilibration of the gas corresponding to the classical transition from regular to dominantly chaotic behavior takes a time of classical character independent of N ∝ − f and thus much larger than λcoll . Back to the assumption of incommensurate eigenvalues ωμ of V which entails ergodicity on the N -torus. We can and should partially relax that assumption; we should since we often encounter commensurate eigenvalues, most notably perhaps λ 2 for the kicked top with V ∝ Jz2 . The unitary operator e−i 2 j+1 Jz is periodic in λ with period 2π (2 j + 1) if 2 j + 1 is odd and 8π (2 j + 1) if 2 j + 1 is even. We may be permissive since, inasmuch as that period is O(N ) while the collision time is O(N −1 ), the distinction of strict and approximate ergodicity on the torus is quite irrelevant. In such cases we can still equate time averages with ensemble averages for the Pechukas–Yukawa gas, choosing Δλ of the order of λcoll and the generalized microcanonical ensemble to describe equilibrium on the torus. To understand finally why spectral fluctuations for systems with global classical chaos are universal and faithful to random-matrix theory, we now have to take the most difficult step and show that equilibrium statistical mechanics for the Pechukas– Yukawa gas entails random-matrix type behavior for the relevant spectral characteristics. This will be the object of the next section. Before precipitating ourselves into that rather serious adventure, it is well to mention exceptional classes of dynamical systems that, although globally chaotic in their classical behavior, do not display level repulsion and therefore do not fall into the universality classes we are principally dealing with here. Prototypical for one exceptional class is the kicked rotor which we shall treat in some detail in Chap. 7. Classical chaos notwithstanding, the eigenfunctions of the Floquet operator are exponentially localized in the H0 basis. It follows that matrix elements of V with respect to different quasi-energy eigenstates are exponentially small if the states involved do not overlap; should the corresponding levels engage in an avoided crossing, the distance of closest approach would be exponentially small compared to the mean spacing. But since almost all pairs of states have that property in a Hilbert space whose dimension N is large compared to the (dimensionless) localization length l, almost all avoided crossings become visible only after a blow-up of the eigenphase scale by a factor of the order el  1; on the scale of a mean spacing such narrow anticrossings look like crossings and this is why quantum localization comes with Poissonian spectral fluctuations. The qualitative reasoning just presented about localization may be formalized in the fashion of our above treatment of two-body collisions. By appeal to conservation of the energy 12 ϕ˙ 2 + U (ϕ) according to (6.6.2) and (6.6.3), we can express the distance of closest approach 2ϕˆ in terms of an initial distance 2ϕ0 taken to equal the v2 ϕ0 2 mean distance 2π /N and an initial velocity −ϕ˙ 0 = v0 as ( sin ) = 1 + |V120 |2 where sin ϕˆ

192

6 Level Dynamics

the matrix element V12 refers to the two Floquet eigenstates of the colliding levels at the initial “time”; that matrix element is exponentially small for nonoverlapping states and makes the avoided crossing look like a crossing. A second class of exceptions is constituted by some billiards on surfaces of constant negative curvature that display what is sometimes called arithmetic chaos. These have certain symmetries of the so-called Hecke type that produce effectively independent multiplets of quantum energy levels; though their classical effect is not to provide integrals of the motion which would make for integrability; they rather provide degenerate periodic orbits whose degeneracy increases exponentially with the period [24]. Quantum symmetries without classical counterparts have also been found for cat maps [25] and shown to cause spectral statistics not following the usual association of classical symmetries with quantum universality classes.

6.8 Equilibrium Statistics In the limit of a large number of levels N , the fictitious N -particle system calls for a statistical description [26, 27]. As already mentioned, the phase-space trajectory ergodically fills an N -torus and the appropriate equilibrium phase-space density is therefore the generalized microcanonical one .-

-

,   δ Cmn (φ, p, l) − C¯ mn (φ, p, l) ∼ mn

N ,

. δ(lmm )

.

(6.8.1)

m=1

Accounted for in the distribution function (6.8.1) are the N (N − 1) independent constants of the motion Cmn (φ, p, l) identified in Sect. 6.3 as well as the N conserved diagonal angular momenta lmm . Inasmuch as only the reduced distribution of the eigenphases,  P(φ) =

dp dl(φ, p, l),

(6.8.2)

will be of interest in the remainder of the present chapter we do not have to worry about the gauge phases; whether or not we fix N phases for the off-diagonal angular momenta to sharp values, P(φ) will not be affected. In order not to carry along unnecessary burden I propose to imagine the primitive gauge, θ˙m = 0, such that neither the Hamiltonian H nor the Hamiltonian equations of motion for the φ, p, l contain the gauge phases any longer. The φ dependence of P(φ) arises through the off-diagonal elements of the matrix v given in (6.3.5). As is obvious from that definition, for a fixed finite value of the angular momentum lmn , the associated vmn tends to infinity when the two coordinates φm and φn become equal; the phase-space functions Cμ (φ, p, l) involving v diverge then, too, and the coordinate distribution thus vanishes. To find out how

6.8

Equilibrium Statistics

193

P(φ) approaches zero as two particles suffer a close encounter, we may change the integration variables in (6.8.2) according to lmn → vmn =

lmn , m = n. i(eiφmn − 1)

(6.8.3)

The Jacobian of this transformation,    ,   ∂l  e−iφm − e−iφn  , J (φ) = det =  ∂v (m,n)

(6.8.4)

is a function of the coordinates φ deserving special attention. The product in (6.8.4) is over all distinct pairs of particles, each pair counted with a multiplicity characteristic of the group of canonical transformations of the underlying Floquet operator. In the orthogonal case, the lmn = −lnm are real, and N (N − 1)/2 in number and the multiplicity in question is one. When U (N ) applies, however, the multiplicity is two ∗ are complex and the original l integral is over N (N − 1)/2 since the lmn = −lnm complex planes. The symplectic case, finally, produces the multiplicity four, as can be seen from the following symmetry argument. The time reversal covariance T F T −1 = F † yields, on differentiation w.r.t. λ, T V T −1 = F † V F.

(6.8.5)

¯ We assume that the eigenbasis of F is organized such that |m and T |m ≡ |m are the two eigenvectors pertaining to the quasi-energy φm . Taking matrix elements in (6.8.5) between the states pertaining to a pair of quasi-energies, one obtains Vmn = e−iφmn Vn¯ m¯ ,

Vm n¯ = −e−iφmn Vn m¯ .

(6.8.6)

Now, it is obvious that, of the four matrix elements of V associated with the pair of levels φm , φn with m < n, Vmn , Vmn ¯ , Vm n¯ , Vm¯ n¯ , only two are independent. Thus, the integral in (6.8.2) is over the complex plane for both lmn and lm n¯ , and the Jacobian (6.8.4) acquires four factors |e−iφm − e−iφn | for each pair m < n. To summarize, the Jacobian in question can be written as  n−1 ,  ,  φi j  −iφi −iφ j n−1   e −e = J (φ) = 2 sin 2  i< j i< j

(6.8.7)

where n is the codimension of a level crossing, ⎧ ⎪ ⎨2 O(N ) n = 3 U (N ) ⎪ ⎩ 5 Sp(N ).

(6.8.8)

194

6 Level Dynamics

It is quite noteworthy that J (φ) is, to within a normalization constant, the joint probability density (4.11.2) for the eigenphases of unitary random matrices from Dyson’s circular ensembles. Moreover, the reader will appreciate that the factor J (φ) arose as a Jacobian for Dyson’s ensembles as well, in that case from the transformation from matrix elements to eigenvalues and the parameters in the diagonalizing transformation. (Actually, in Chap. 4 the argument was given only for the Gaussian ensembles, but it is easy to carry over to the circular ensembles.) When the substitution (6.8.3) is made in (6.8.2), the generalized microcanonical distribution  depends on the coordinates φ only through the lmn = vmn i (exp(iφmn ) − 1) in the conserved phase-space functions Cmn . Now, a crossing φmn → 0 (at fixed vmn ) no longer implies a divergence of any of these C. When the reduced distribution P(φ) is written as  ˜ ˜ P(φ) = J (φ) P(φ), P(φ) ∼ dp dv , (6.8.9) ˜ the integral P(φ) does not in general vanish at a crossing. Therefore, the behavior of P(φ) near crossings is dominated by the Jacobian J (φ), i.e., it is the same as that postulated by random-matrix theory. As an immediate consequence, one recovers a level spacing distribution obeying the repulsion law P(S) ∼ S n−1 ∼ S β for S → 0.

(6.8.10)

To fully establish random-matrix theory as an implication of equilibrium statistical mechanics of the fictitious N -particle system, one would have to show that ˜ P(φ) is effectively a constant, at least for those particle configurations to which J (φ) assigns appreciable weight. It is amusing to note that one would strictly obtain P(φ)/J (φ) = const if one restricted the generalized microcanonical ensemble (6.8.1) so as to admit only constants of the flow (6.3.10) of the form Tr {v μ }; evidently, the φ dependence of P(φ) in that case would be exclusively due to the Jacobian J (φ). Even more amusingly, the Hamiltonian function H is of precisely that form: H =

|lmn |2 1 2 1 1 pm + Tr {v 2 } = . 2 2 2 m 8 m=n sin (φmn /2)

(6.8.11)

Thus, the usual microcanonical ensemble is among the ensembles equivalent to random-matrix theory with respect to the statistics of eigenvalues, as first remarked by Yukawa in the analogous case of autonomous quantum systems [2]. Needless to say, this observation does not constitute a proof that P(φ)/J (φ) ≈ const for the ergodic motion on the N -torus. Rather, the functionally independent constants of the flow determining the N -torus can definitely not all be of the form Tr {v m }, since, of the latter quantities, only N − 1 are linearly independent. Arguments supporting the effective constancy of P(φ)/J (φ) will be given in the next section. We shall

6.9

Random-Matrix Theory as Equilibrium Statistical Mechanics

195

see that “effective” means that any φ dependence of P(φ)/J (φ) affects the spacing distribution and low-order cluster functions of the level density by no more than corrections of relative order 1/N . With that result established, our goal to establish the universality of spectral fluctuations on the basis of level dynamics will be completed.

6.9 Random-Matrix Theory as Equilibrium Statistical Mechanics 6.9.1 General Strategy This section is devoted to showing that equilibrium statistical mechanics for the fictitious-particle dynamics of Sect. 6.2 becomes equivalent to random-matrix theory as the number of levels N goes to infinity. For technical convenience, we consider Floquet operators F(λ) = e−iλV e−iH0 without antiunitary symmetries and shall aim to calculate their distribution of nearest neighbor spacings; for a more complete treatment, the reader is referred to [26, 27]. Starting from the generalized microcanonical ensemble I immediately deal with the constraints for the diagonal angular momenta by setting lmm = 0 in the N (N −1) independent constants of the motion Cmn (φ, p, l). Next, since the microcanonical ensemble (6.8.1) is hard to work with, we employ a generalized canonical ensemble,   αC(φ, p, l) , (φ, p, l) = Z −1 exp −

(6.9.1)

which fixes the constants of the motion C not sharply but only in the ensemble mean. The coefficients α in (6.9.1) are Lagrange multipliers determined by the mean values of the corresponding C, C = −

∂ ln Z . ∂α

(6.9.2)

The spacing distribution P(S) = E

(S) will be characterized by its second integral, the gap probability E(S). To demonstrate that the generalized canonical ensemble yields the same E(S) as Dyson’s circular unitary ensemble (CUE),  E(S) =



π −π S/N

dφ −π +π S/N

 = E CUE (S) 1 + O

 dl 

dp (φ, p, l)  1 , N

(6.9.3)

196

6 Level Dynamics

we shall proceed in close analogy with the calculation of E CUE (S) in Sect. 4.12. The latter calculation amounted to performing the N -fold integral in  π −π S/N d N φ PCUE (φ) (6.9.4) E CUE (S) = −π +π S/N

over the joint distribution of all eigenvalues PCUE (φ) =

1, ... N  −iφ  1 e i − e−iφ j 2 . N N !(2π) i< j

(6.9.5)

As a warm-up for treating the more numerous integrals in (6.9.3), a look back at Sect. 4.12 may be advisable. If afraid of technicalities, the reader might even prefer to jump ahead to the concluding remarks in Sect. 6.9.5. As a first step toward evaluating E(S), we change integration variables as in (6.8.3), thereby incurring the Jacobian J (φ) ∼ PCUE (φ),  π −π S/N ˜ d N φ PCUE (φ) P(φ), (6.9.6) E(S) = −π +π S/N

˜ P(φ) = N −1

 d (N

2

−N )

vd N pe−



αC

.

(6.9.7)

Needless to say, for the N 2 real integrals in (6.9.7) to exist, the N (N −1) constants of the motion C must  be selected from the set of all Cmn such that the highest powers αC have even exponents and positive coefficients. A further of |vi j | and p j in precaution will be necessary: The coordinate integrals in (6.9.6) become manage able only after formally extracting all φ-dependent terms in exp (− αC) from the exponent by a Taylor expansion in powers of (some of) the Lagrange multipliers α. It must be ensured that this series converges uniformly since otherwise term-by-term integration over the v, p, and φ would not be permissible. Therefore, a suitable part  of αC, to be identified presently, must remain unexpanded. None of the N − 1 independent constants of the motion of the type C0n = Tr {v n }, n = 1, 2, . . . , N − 1

(6.9.8)

depends on φ and hence these may be kept in the exponent. The remaining C’s do depend on the coordinates through lmn = ivmn (eiφmn − 1). Let the highest power of vmn in any of these C be |vmn | M with M even and necessarily M ≥ N . Now, consider the coefficient of |vmn | M , and let its maximum value, attained for some coordinate configuration(s), be denoted by N  2

T =

i = N +1

αi Ti ,

(6.9.9)

6.9

Random-Matrix Theory as Equilibrium Statistical Mechanics

197

where αi Ti is the contribution from Ci ; of course Ti = 0 for all Ci not containing |vmn | M ; the summation index i summarily denumbers the (N − 1)2 constants of the motion Cmn with m, n = 1, 2, . . . , N − 1. Then, by separating as N 

N 

2

2

αi Ci = T Tr {v } + M

i = N +1

  αi Ci − Ti Tr {v M }

(6.9.10)

i = N +1

and expanding exp (−αi Ci ) in powers of αi , a uniformly (for all values of v and φ) converging series is obtained. The following remark is an aside, meant to explain the separation (6.9.10) in terms of an elementary example in which only a single integral over a real variable v arises with an integrand depending on a parameter φ,  I =

+∞ −∞

dv e−T (φ)v . 2

(6.9.11)

For this “toy” integral to exist, the function T (φ) must be positive for all admissible values of φ. Let T be the maximum value attained by T (φ), such that 0 < T (φ) ≤ T. The latter bounds for T (φ) imply |T (φ) − T | 0 so that λ → ∞ corresponds to τ → ∞, whereupon Γ can be absorbed in the scale of the ”time” τ. Within the latter freedom for the direction and the unit of τ , the solution of (6.13.6, 6.13.9) is unique and reads λ(τ )2 = e2τ − 1, f (τ ) = e−2τ , f =

1 . 1 + λ2

(6.13.10)

The resulting conditional probability density (6.13.7) turns out to be the Green’s function of the N 2 -dimensional Ornstein–Uhlenbeck process [53]. It obeys the Fokker–Planck equation    ∂2 1 ∂ ∂ Hμ + Dμ P(H, τ ) = P(H, τ ), ∂τ ∂ Hμ 2 ∂ Hμ2 μ

(6.13.11)

where the shorthand Hμ denotes the N 2 independent variables, i.e., Hii and, for i < j, Re {Hi j } and Im {Hi j }. The diffusion constants Dμ are the same as for the free diffusion encountered before the rescaling, i.e., Dμ = 1/2 for the diagonal elements Hi , and Dμ = 1/4 for the off-diagonal elements Re {Hi j } and Im {Hi j }. The Fokker–Planck equation (6.13.11) is invariant under unitary transformations of H ; thus the representation of H may be chosen at will. The distribution P(H, τ ) interpolating between PGOE (H ) = P(H, 0) and PGUE (H ) = P(H, ∞) can also be obtained explicitly by doing the N (N + 1)/2 Gaussian integrals in (6.13.5), P(H, τ ) =

, exp [−H 2 /(1 + e−2τ )] & ii π(1 + e−2τ ) i  , exp [−(Re {Hi j })2 2/(1 + e−2τ )] & × (π/2)(1 + e−2τ ) i< j  exp [−(Im {Hi j })2 2/(1 − e−2τ )] & × . (π/2)(1 − e−2τ )

(6.13.12)

228

6 Level Dynamics

It is interesting to note the invariance of P(H, τ ) under orthogonal transformations for 0 ≤ τ < ∞ while in the limit τ → ∞, the full symmetry of the GUE is established. The Ornstein–Uhlenbeck process for the matrix elements of a Hamiltonian is known in random-matrix theory as Dyson’s Brownian-motion model [10]. It was suggested by Dyson as the simplest random process displaying relaxation into a stationary regime described by one of the three Gaussian random-matrix ensembles. The foregoing arguments elevate the Brownian-motion model to a rigorous description of the transition of a Hamiltonian from a typical member of the GOE to a typical member of the GUE when an additive term λV breaking time-reversal invariance is switched on. Evidently, the considerations outlined above for the transition GOE → GUE can be extended immediately to the other transitions between pairs of universality classes. The significance of this result lies in its apparent universality. Indeed, inasmuch as the time-reversal invariant Hamiltonian H0 of a dynamic system generically displays GOE fluctuations in its spectrum, while, similarly, a perturbation V without antiunitary symmetries has √ f (H0 + λV ) should be faithful GUE fluctuations, the perturbed Hamiltonian to (6.13.12). A further comment about the rescaling (6.13.10) is in order. According to Wigner’s semicircle law [see (4.7.3) and the remark following that equation] both √ H0 and V have a mean level spacing proportional to 1/ N in the middle of their spectra. The family of Hamiltonians described by (6.13.12) should share that property since the widths of the distributions of the matrix elements of H interpolate monotonically between the two limits at τ = 0 and τ → ∞. Indeed, by using the fact that Hi j H ji  = 1 independent of τ for the distribution (6.13.12), it is easy to see that the arguments of Sect. 4.7 yield the same semicircle law as for the GOE and the GUE (Problem 6.7). The Ornstein–Uhlenbeck process (6.13.11) contains a subset of N stochastic variables, namely, the eigenvalues E i of the matrix H, which itself undergoes a separate Markovian process. The joint probability density of the E i obeys a Fokker– Planck equation of the form ⎤ ⎡ 1 ... N 1 ... N 2 ∂ ∂ ∂ 1 Di (E) + Di j (E)⎦ (E, τ ). (E, τ ) = ⎣− ∂τ ∂ E 2 ∂ E ∂ E i i j i ij (6.13.13) Different paths may be followed to prove the validity of (6.13.13) and to determine the drift coefficients Di (E) and the diffusion matrix Di j (E). In the fashion of Sect. 4.3, a transformation of variables may be performed from the N 2 matrix elements Hi j to the N eigenvalues E i and the N 2 − N angles that specify the unitary matrix diagonalizing H. When these angles are integrated out of the Fokker– Planck equation (6.13.11), the reduced Fokker–Planck equation (6.13.13) results. A more convenient route is opened by the observation that the Green’s function

6.13

Dyson’s Brownian-Motion Model

229

(E + δ E, τ + δτ |E, τ ) of (6.13.13), for an infinitesimal time increment δτ, has the mean and mean-squared eigenvalue increments [53] δ E i  = E i (τ + δτ ) − E i = Di (E)δτ + O(δτ 2 )

(6.13.14)

δ E i δ E j  = Di j (E)δτ + O(δτ 2 ).

These identities can in fact be obtained immediately from (6.13.13). On the other hand, they may be derived from the short-time version of the Green function (6.13.7) of the Ornstein–Uhlenbeck process,  P (H + δ H, τ + δτ |H, τ ) =

1 2δτ

N 2 /2

 PGUE

δ H + H δτ √ 2δτ

 ,

(6.13.15)

and the perturbative relationship δ E i = δ Hii +

 δ Hi j δ H ji + ... Ei − E j j(=i)

(6.13.16)

between the matrix increment δ H and the eigenvalue increment δ E i . Obviously, the relation (6.13.16) refers to the eigenrepresentation of H, which, according to the “unitary” invariance of the Fokker–Planck equation (6.13.11) of the Ornstein– Uhlenbeck process, we are free to adopt at time τ. If the Green’s function (6.13.15) is also taken in that representation, it has the moments δ Hii  = −E i δτ 

(δ Hii )2 = δτ  

(Re {δ Hi j })2 = (Im {δ Hi j })2 = δτ/2, i < j,

(6.13.17)

and these, together with (6.13.16), imply   δ E i  = −E i + j(=i)

1 E i −E j



δτ

δ E i δ E j  = δi j δτ.

(6.13.18)

In all increment moments (6.13.14), (6.13.17), and (6.13.18) powers of δτ higher than the first have been neglected. To that accuracy, higher-than-second orders in δ H could be neglected in the perturbation expansion (6.13.16). Comparison of the moments in (6.13.18) and (6.13.14) now yields the drift and diffusion coefficients for the Brownian motion of the eigenvalues Di (E) = −E i +

 j(=i)

1 , Di j = δi j . Ei − E j

(6.13.19)

230

6 Level Dynamics

Like the Ornstein–Uhlenbeck process of the Hi j the Brownian motion of the E i has a harmonic binding force in the drift. The additional two-body interaction term in Di (E) can be interpreted as a repulsive coulombic force between charged particles in two spatial dimensions; the interaction energy for a pair of particles varies logarithmically with the distance. In view of this interaction the process is often referred to as to Dyson’s stochastic Coulomb gas model [10]. While the repulsive character reflects level repulsion, the harmonic binding follows from the energy rescaling f specified in (6.13.5), (6.13.10): the purpose of that rescaling is to prevent the “particles” from flying apart indefinitely for λ → ∞. Interestingly, the (logarithmic) Coulomb gas interaction differs from the (inverse-square) two-body interaction in Pechukas’ level dynamics (6.2.36). See Problem 6.10. Since the Fokker–Planck equation (6.13.13) and (6.13.19) of the Coulomb gas obeys the condition of detailed balance [53], the stationary solution (E, ∞) can be given immediately in closed form. As it must, (E, ∞) comes out as the joint probability density of the eigenvalues for the GUE [see (4.3.15) with n = 2 or (6.13.22) below]. The time-dependent solution (E, τ ) originating from the joint eigenvalue density of the GOE at τ = 0 is also known. It was determined by Pandey and Mehta by integrating P(H, τ ) as given in (6.13.12) over the N 2 − N “angles” alluded to above, that “integration over the unitary group” yields [54, 55], for even N , ⎤   , Ei − E j ⎦ (E, τ ) = N (τ )−1 ⎣ (E i − E j )Pf erf √ e2τ − e−2τ i< j .  E i2 × exp − , 1 + e−2τ i ⎡

⎡ N (τ )−1 = 23N /2 e−(N

2

−N )τ/2

,

⎣

j

(6.13.20)

⎤   j ⎦ Γ 1+ . 2

& Here, we encounter the Pfaffian Pf (ai j ) = det (ai j ) of an antisymmetric N × N matrix ai j with even N introduced in (4.13.13). For the case of odd N , the reader may consult [54, 55]. To check on the limiting form of (E, τ ) for τ → ∞, both the Pfaffian and the normalization factor N (τ ) must be inspected for their time dependence. The argument ai j of the Pfaffian simplifies as ai j → erf [e−τ (E i − E j )] and may be expanded in powers of e−τ (E i − E j ). The exponential blowup of N (τ ) can be compensated for only when the N /2 factors ai j in each additive term of the Pfaffian build up (N 2 − N )/2 factors e−τ (E i − E j ). Such contributions arise when all the ai j join the (N − 1)th-order term of their Taylor expansions. At any rate, since the Pfaffian is totally antisymmetric in the variables E i , it must have the structure

6.13

Dyson’s Brownian-Motion Model

231

   Pf erf e−τ (E i − E j ) ⎤ ⎡ ,   2 = e−(N −N )τ/2 ⎣ (E i − E j )⎦ Fsym {e−τ E j }

(6.13.21)

i< j

where Fsym is totally symmetric in its arguments. The stationary form of (E, τ ) follows as ⎡

,

(E, ∞) = 2−3N /2 ⎣ ⎡ ×⎣

j

,

 Γ

⎤  −1 j ⎦ 1+ Fsym ({0}) 2 ⎤

(E i − E j )2 ⎦ e−

 i

E i2

i< j

= GUE (E).

(6.13.22)

The constant factor Fsym ({0}) may either be constructed from the Pfaffian or simply determined by normalization. Verifying the correct initial behavior of the solution (6.13.20) is also instructive, starting from ai j (τ ) → (E i − E j )/|E i − E j |. Then, the Pfaffian has the value (−1)Q where Q is the permutation i 1 i 2 . . . i N of 1 2 . . . N corresponding to E i1 > E i2 > . . . > E i N . But + the same permutation Q determines the sign of the Vandermonde determinant i< j (E i − E j ) as (−1)Q . Thus, the Pfaffian may be replaced by the modulus operation on the Vandermonde determinant, whereupon (E, 0) = GOE (E) is established. An important conclusion may be drawn from the solution (6.13.20) for the time scale of the transition from GOE to GUE behavior. Clearly, the Gaussian factors in (E, τ ) are irrelevant for the change in the degree of level√repulsion; their √ principle role is to keep the spectrum confined, roughly, to − N < ∼E< ∼+ N according to the semicircle law. It is actually the product of the Pfaffian and the Vandermonde √ determinant that signals the transition. As long as the arguments (E i − E j )/ e2τ − e−2τ of the error functions in the Pfaffian, for a typical spectrum {E i }, are all large, GOE behavior still prevails; the transition to the GUE is more or less complete once these √ arguments have all become small. Again taking the typical spacing E i − E j ≈ 1/ N from the semicircle law, the time scale τˆ of the transition may be estimated as N sinh 2τˆ ≈ 1, i.e., τˆ ≈

1 1 ⇔ λˆ ≈ √ . N N

(6.13.23)

ˆ the degree of level repulsion still appears to be unity on For τ  τˆ or λ  λ, a scale given by the mean level spacing. It follows that in the limit of large N a rather small symmetry-breaking addition λV to the time-reversal invariant H0 brings about quadratic level repulsion. It also follows that the crossover in the inverse

232

6 Level Dynamics

Fig. 6.8 Progress of breaking of time-reversal invariance in the two-level correlation function [normalized density of two eigenvalues P(E 1 , E 2 )/P(E 1 )P(E 2 ) as implied by (6.9.19). The full curves pertain to the limit N → ∞ which becomes accessible after referring the spacing E 1 − E 2 to the mean spacing and the control parameter to the scale (6.9.23)

& & 2 (V + H /λ) ≡ ˜f (V + λH ˜ 0 ) takes place transition, GUE → GOE, in H = f λ 0 √ ˜ at λ ≈ N ; a comparatively large time-reversal-invariant addition is needed to overwhelm a Hamiltonian V without antiunitary symmetries. Figure 6.8 illustrates the transition. Incidentally, the “time” scale (6.13.23) for the transition GOE → GUE can also be obtained from the matrix density (6.13.12). To this end, we consider the imaginary part of Hi j as a small perturbation which shifts the ith level by δ Ei =

 (Im {Hi j })2 . Ei − E j j(=i)

(6.13.24)

In the average over the ensemble (6.13.12),

 (Im {Hi j })2 = 14 (1 − e−2τ ) ≈ 12 τ.

(6.13.25)

The N dependence of the sum √ on the r.h.s. in (6.13.24) can be estimated from the mean level spacing, ΔE ∼ 1/ N . Finally, the transition will be more or less completed once the mean shift δ E i  has grown to the order of magnitude of the

6.13

Dyson’s Brownian-Motion Model

233

mean spacing. Putting all of these estimates into (6.13.24), one indeed recovers the result (6.13.23) for the transition time τˆ . While the foregoing discussion was confined to transitions between the GOE and the GUE, the other transitions between universality classes, GOE ↔ GSE and GUE ↔ GSE, can be treated similarly, at least as far as the interpolating matrix densities P(H, τ ) are concerned. Even though the interpolating reduced densities of the eigenvalues are not known in closed form for the latter transitions, the transition times can The result again is √ be determined using the above perturbative argument.√ λˆ ∼ 1/ N for the easy direction (symmetry breaking) and λˆ ∼ N for the inverse transition (symmetry restoring). It is worth emphasizing once more that a given transition (with fixed sense) may be achieved either through breaking or restoring a symmetry. For instance, GSE → GUE takes place when (1) a time-reversal invariance of H0 with T 2 = −1 is broken in the absence of any geometric symmetry or (2) both H0 and V are invariant under T with T 2 = −1 while V introduces a parity not shared by H0 (Sect. 2.7); clearly, the first transition is easy, and the second one hard. A particularly interesting crossover arises when H0 is classically integrable but H0 + λV with λ > 0 nonintegrable, such that the classical phase space becomes dominated by chaos once λ is sufficiently large. As a quantum parallel to this spreading of classical chaos, one expects that the level clustering typically present for λ = 0 eventually gives way to level repulsion. A convenient random-matrix model for such a case assumes a diagonal N × N matrix H0 whose eigenvalues are independent random numbers with a Gaussian density of zero mean and variance (H0ii )2 √= N ; the latter normalization implies an average eigenvalue spacing of the order 1/ N , in coincidence with that assumed for V. For the sake of concreteness, taking V as an N × N matrix without antiunitary symmetry, one is led to consider, in analogy with (6.13.5), the matrix ensemble √    & −N 2  H − f H0 d H0 PP (H0 )PGUE P(H, τ ) = λ f , √ λ f

(6.13.26)

where the integral is over the N diagonal elements (i.e. eigenvalues) of H0 . The density (6.13.26) interpolates between the initial “Poissonian” ensemble (PE) PP (H0 ) =

N ,

(2π N )−1/2 e−(H0ii ) /2N 2

(6.13.27)

i =1

and the final GUE. The N -fold integral in (6.13.26) is once more Gaussian and thus assigns to the interpolating density a Gaussian form as well; all matrix elements are distributed independently of one another; they have zero means and the variances  1 1 + (2N − 1)e−2τ , (Hii )2 = 2 @  2 A @ 2 A 1  1 − e−2τ , i = j. Im {Hi j } = Re {Hi j } 4

(6.13.28)

234

6 Level Dynamics

One would like to extract from the Gaussian matrix density defined by (6.13.28) an explicit expression for, e.g., the spacing distribution P(S, λ) which interpolates between P(S, 0) = e−S and P(S, ∞) = PGUE (S). Unfortunately, the integrals involved have thus far resisted all attempts at evaluation. See, however, Problem 6.16 and [61, 62]. The discussions of this section may be looked upon from a different perspective. Once the functions g(τ ) and λ(τ ) are specified, the Hamiltonian g(H0 + λV ) moves deterministically along a line in the space of complex Hermitian N × N matrices. The family of Hamiltonians described by Green’s function (6.13.7) and (6.13.10) corresponds to a bundle of such trajectories, all of which originate from a single point at τ = 0. At each moment τ > 0, the bundle of trajectories defines a cloud of points. The bundle is made unique by the cloud reached at τ → ∞, and that final cloud represents the GUE. Thus, the matrix-valued Ornstein–Uhlenbeck process may be thought of as determined by a teleological average over a bundle of deterministic trajectories.

6.14 Local and Global Equilibrium in Spectra As explained in Sect. 4.8, the notion of an average density of levels (E) = ΔN /ΔE for a given Hamiltonian would not make sense if (E) were not smooth, and in fact constant, on the energy scale of a mean level spacing. The scale on which (E) is allowed to vary in the limit of large N is a “global” one on which fluctuations in the level sequence tend not to be noticeable. For instance, if the Hamiltonian H is a typical member of any of the Gaussian ensembles of random matrices, (E) obeys the semicircle law and thus has the radius of the semicircle as its natural scale of energy. To obtain a well-defined limiting form of the density of levels for N → ∞, one must therefore refer the energy variable to a global unit (such that, e.g., the radius of the semicircle law becomes independent of N ). The energy scale on which fluctuations in the sequence of levels become manifest is smaller by a factor of the order N , however. Indeed, the natural energy unit for quantities describing such fluctuations is the local mean spacing 1/(E). In discussing fluctuations of the density of levels in Sect. 4.10, I used the unfolded energy e = E(E) and reported Dyson’s results (4.10.15) for the two-level cluster functions of the Gaussian ensembles of random matrices; these results indeed imply that density fluctuations at different energies become statistically independent when the energies differ by many mean spacings; more quantitatively speaking, the two-point correlation function approaches the product of two mean densities as 1/(e − e )2 . Similar results hold for the matrix ensemble (6.13.12) and (6.13.20) which smoothly interpolates between the GOE and the GUE, as explicitly verified by Pandey and Mehta [54, 55] who calculated all n-point correlation functions by integrating the joint density (6.13.20) of all N eigenvalues over N − n of its arguments. Intimately related to the energy scale separation just described is a separation of “time” or control parameter scales for “processes” in which a spectrum

6.14

Local and Global Equilibrium in Spectra

235

changes its global and local properties as a control parameter λ in the Hamiltonian H = H0 + λV or the Floquet operator F = e−iλV e−iH0 is varied: The level density (E, λ) changes much more slowly with λ than any of the quantities describing local fluctuations. For instance, the transitions between the Gaussian ensembles of random matrices√discussed in Sect. 6.13 have a level density independent of the “time” τ = ln 1 + λ2 while the spacing distribution and low-order correlation functions of density fluctuations have transition “time” scales of the order 1/N [see (6.13.23)]. The transition from the Poissonian ensemble to the GUE briefly treated at the end of the previous section does not display a strictly vanishing time scale ratio for the density fluctuations and the mean density but still one of order 1/N . Indeed, the characteristic time for the density of levels in that case can be revealed to be of order unity by looking at the mean-squared matrix elements (6.13.28) and recalling the derivation of the semicircle law in Sect. 4.7. Initially, at τ = 0, the Gaussian level density is governed by the diagonal elements Hii ; at τ → ∞, as is clear from Sect. 4.7, the validity of the  law hinges on the negligibility of the  semicircle diagonal elements in the sense i Hii2 / i= j |Hi j |2  = O(1/N ). Since the latter ratio has the order N during the early stages during which τ = O(1/N ), the transition from the Poissonian to the final behavior may roughly be located at times of order unity, when the ratio in consideration itself has the intermediate order unity. Note that according to the mean-squared matrix elements (6.13.28), the ratio falls to O(1/N ) at τ = O(ln N ). However, local equilibrium within clusters of levels extending over a few mean spacings should be attained more rapidly on a time scale ∼ 1/N , just as for the transition, say, from the GOE to the GUE. Unfortunately, the latter expected behavior cannot be considered obvious from the interpolating matrix density defined by (6.13.28) since the corresponding level spacing distribution and low-order correlation functions have not yet been evaluated. Lacking such explicit rigorous results, perturbative arguments have been invoked in the literature [56] to indicate the proportionality of the characteristic time of local fluctuations to 1/N . The separations of time and energy scales just described is certainly expected by analogy with ordinary many-particle systems. There, too, the approach to global equilibrium is characterized by long “hydrodynamic” times while local equilibrium across microscopically correlated clusters is established within a few collision times. The remainder of this section is devoted to a more systematic discussion of the scaling properties of the correlation functions  1 d E n+1 . . . d E N  N (E 1 . . . E N , τ ), (6.14.1) n (E 1 . . . E n , τ ) = (N − n)! which are normalized as  d E 1 . . . d E n n (E 1 . . . E n , τ ) =

N! . (N − n)!

(6.14.2)

Note that the correlation function  N (E 1 . . . E N , τ ) of all N levels differs from the joint probability density P(E 1 . . . E N , τ ) of all N levels by normalization,

236

6 Level Dynamics

 N (E 1 . . . E N , τ ) = N !P(E 1 . . . E N , τ ). These correlation functions arise naturally as the following ensemble averages, >1 ... N 

1 (E 1 , τ ) =

?

δ(E − E i ) 8 1 ... N n (E 1 . . . E n , τ ) = i

i 1 =i 2 = ... =i n

9

(6.14.3)

δ(E 1 − E i1 ) . . . δ(E n − E in ) ,

the averages are defined with P(E 1 . . . E N , τ ) as the weight. Of course, 1 (E) ≡ (E) is the average density of levels referred to above. The reader is invited to establish the relationship of 2 (E 1 , E 2 ) to the “structure factor” S(E 1 , E 2 ) employed in Sect. 4.10. By integrating the Fokker–Planck equation (6.13.13) and (6.13.19) over N − n energy variables, one obtains the following set [57] of integrodifferential equations for the correlation functions n : ∂ n (E 1 . . . E n , τ ) ∂τ ⎛ ⎞ 1 ... n 1 ... n ∂ ⎝ 1 1 ∂ ⎠ Ei − n (E 1 . . . E n , τ ) + = ∂ Ei E − Ej 2 ∂ Ei i j(=i) i −

1 ... n i

∂ P ∂ Ei

 d E n+1

1 n+1 ((E 1 . . . E n+1 , τ ), E i − E n+1

(6.14.4)

which in the context of ordinary many-particle theory is called the BBGKY hierarchy. Now, I propose to show that this hierarchy is consistent with the separation of time and energy scales discussed above. Let us first consider (6.14.4) for n = 1 in the form   

∂ 1 ∂ ∂

1 (E , τ ) + 1 (E, τ ) = E − P dE 1 (E, τ ) ∂τ ∂E E − E

2 ∂E   1 ∂ 1 (E, τ )1 (E , τ ) − 2 (E, E , τ ) . P d E

+ ∂E E − E

(6.14.5) Note that I have added and subtracted a piece bilinear in 1 so that Dyson’s two-level cluster function 1 (E, τ )1 (E , τ ) − 2 (E, E , τ ) appears in the last integral; the latter function was already met in Sect. 4.10 (4.10.14), albeit expressed in different units. Now, I shall argue that the last integral in (6.14.5) vanishes. To that end, I assume, subject to later proof of consistency with (6.14.4) for n > 1, that the two-level cluster function has the scaling properties announced before: it approaches an adiabatic equilibrium, contingent on the local and current value of the density 1 (E, τ ),

6.14

Local and Global Equilibrium in Spectra

237

roughly N times faster than 1 relaxes to its final semicircular form. The assumed time-scale separation entails, as will become clear presently, that the two-level cluster function makes negligible contributions to the last integral in (6.14.5) once E and E are further apart than ΔE = N ε with −1/2 < ε < +1/2. To appreciate the required N dependence of the interval ΔE, the√reader must recall that in the units chosen the final√semicircle law has a radius ∼ N while the mean level spacing is of the order 1/ N . Due to the assumed adiabaticity of the temporal variation of 1 (E, τ ) with respect to the two-level cluster function, the latter may, under the last integral in (6.14.5), be taken to be in adiabatic equilibrium during the whole relaxation of 1 (E, τ ) toward the semicircle law, 1 (E, τ )1 (E , τ ) − 2 (E, E , τ ) = 1 (E, τ )1 (E , τ )YGUE (e − e )

(6.14.6)

with e − e = E1 (E, τ ) − E 1 (E , τ ) and with YGUE (e) given in (4.10.15). Then, consider the contribution to the last term in (6.14.5) from the integration range outside the interval ΔE ∼ N ε around E, ∂ 1 (E, τ ) ∂E



E−ΔE/2

−∞





∞

+

E+ΔE/2

d E

1 (E , τ ) YGUE (e − e ). (6.14.7) E − E

By order of magnitude, we may put 1 (E , τ ) ≈ that the term (6.14.7) turns out to have the weight

√

N and YGUE (e) ≈ 1/e2 such

√ √ 4 ∂ ∂ ≈ 1 (E, τ ) N 1 (E, τ ) N 4N −1−2ε , 2 ∂E N (ΔE) ∂E

(6.14.8)

i.e., smaller than the weight of the linear drift term ∂∂E E1 (E, τ ) in (6.14.5) by a factor N −1−2ε and thus indeed negligible in the limit N → ∞ for −1/2 < ε. It remains to discuss the contribution to the last term in (6.14.5) from the integration interval ΔE = N ε around E. Instead of (6.14.7), we are facing ∂ 1 (E, τ ) ∂E



E+ΔE/2 E−ΔE/2

d E

1 (E , τ ) YGUE (e − e ). E − E

(6.14.9)

Assuming √ once more that the variation in energy of 1 (E , τ ) is characterized by the scale N given by the final semicircle law and now requiring ε < 1/2 we conclude that 1 (E , τ ) cannot vary noticeably across the interval ΔE; thus the integral in (6.14.9) is seen to vanish since Y (e) is an even function of e. With the whole last term in (6.14.5) now having disappeared, the integrodifferential equation for the level density is decoupled from the rest of the hierarchy (6.14.4)

238

6 Level Dynamics

with 1 < n ≤ N . One further simplification actually arises in the Eq. (6.14.5) √ for 1 (E, τ ). Inasmuch as 1 (E, τ ) indeed varies on a global energy scale ∼ N , a well-defined limiting form of the density for N → ∞ becomes accessible only through the change of units 1 (E, τ ) =

√

˜ τ ) ; E˜ = √E . N ˜ 1 ( E, N

(6.14.10)

The radius of the final semicircle is thereby assigned a value independent of N , while the mean spacing of neighboring levels appears to be of the order 1/N , and $ ˜ τ ) = 1. When (6.14.5) the new density ˜ 1 is normalized as a probability, d E˜ ˜ 1 ( E, is expressed in these new units, the diffusion term takes the form (1/2N )∂ 2 /∂ E˜ 2 ; ˜ τ ) in the limit N → ∞. clearly, such a term cannot affect the E˜ dependence of ˜ 1 ( E, Thus, the final form of the evolution equation, derived by Dyson [58] and independently by Pastur [59], reads ˜ τ) ∂ ∂ ˜ 1 ( E, = ∂τ ∂ E˜



 E˜ − P

d E

 ˜ 1 (E , τ ) ˜ τ ). ˜ 1 ( E, E˜ − E

(6.14.11)

& ˜ ∞) = (1/π ) 2 − E˜ 2 . It is easy to check that this has the stationary solution ˜ 1 ( E, When turning to the hierarchy (6.14.4) with n > 1, the most interesting situation to discuss arises when all n energy arguments in n (E 1 . . . E n , τ ) lie in a single √ clusters cluster of extension ΔE ∼ 1/ N ; otherwise, if the {E i } form several such√ and the intercluster distances are noticeable on the global energy scale N , n would factor into a product in which each local subcluster would be represented by its own correlation function. Assuming that n describes a single local cluster, it is helpful to start the discussion of (6.14.4) by noting the order of magnitude of the various terms. We have O(n ) = O(1n ) = N n/2 . In contrast to the situation discussed above, the√energy scale for all n with n > 1 is of the order of a mean spacing, i.e., ∼ 1/ N , such that formally O((∂/∂ E i )n ) = N (n+1)/2 ; however, a slight complication is of importance: the n energy arguments of n can be reorganized into one “center-of-energy” variable √ and n − 1 relative energies; only the latter can be expected to have a range ∼ 1/ N . The dependence of n on the centerof-energy variable, on the other hand, must certainly be weaker, i.e., must have the √ global scale N ; otherwise, a conflict would arise with the previous assumption √ that constant across a mean spacing 1/1 ∼ 1/ N and the density of levels 1 (E) is √ $varies only on the global scale N . For instance, the definition (6.14.1) implies that d√E 2 2 (E 1 , E 2 ) = (N − 1)1 (E 1 ) and therefore 1 (E 1 ) would vary on the scale 1/ N if 2 (E 1 , E 2 ) were allowed to do so independently in both of its arguments. It n (∂/∂ E i )n ) = N (n−1)/2 , a point to be returned to several times follows that O( i=1 below. The foregoing order-of-magnitude estimates imply that, of the three terms involving n on the right-hand side in (6.14.4), the Coulomb drift and the diffusion overwhelm the linear drift (∂/∂ E i )E i n by a factor N . Indeed, the linear restoring

6.14

Local and Global Equilibrium in Spectra

239

force, essential for the global behavior of the density of levels 1 , is entirely negligible on the smaller energy scale of relevance for local fluctuations. Conversely, the diffusion was ineffective globally but is now as important as the drift caused by the Coulombic “interparticle” force. Incidentally, there are two types of Coulomb terms in (6.14.4) for n > 1, both of equal weight in the sense of the estimates in consideration: The first such term comprises all intracluster interactions while the second refers to pairs of “particles” with one member particle within the cluster and the other outside; the second term involves the (n + 1)-level correlation function n+1 . Having commented on all terms on the right-hand side of (6.14.4) for n > 1, we should also look at the left-hand member, (∂/∂τ )n . Its formal weight is O(N n/2 /τ ), and the equation therefore requires that O(τ ) = 1/N . It is at this point that we get back, self-consistently, the time-scale separation for local fluctuations and global density equilibration. A convenient implementation of the scaling with N just discussed employs new units, n (E 1 , . . . E n , τ ) → ˜ n (e1 , . . . en , τ˜ ), such that the latter function has a welldefined limiting form for N → ∞. Following Pandey [57], I choose 

Ei

ei =

d E 1 (E , τ ) ≈ (E i − E)1 (E, τ ),

E

τ˜ = τ 1 (E, τ )2 ∼ τ N ,

(6.14.12)

˜ n (E 1 , . . . E n , τ ) = 1 (E, τ )n ˜ n (e1 , . . . en , τ˜ ). The introduction of the ei is equivalent to the familiar unfolding of the spectrum to unit mean spacing (see Sects. 4.8, 4.10). It is important to realize that the ei are effectively independent of τ on the time scale of relevance to local fluctuations. The reference energy E specifies the location of the cluster. The new time τ˜ is effectively linear in τ and independent of E throughout the cluster. The reader should note the difference between the scalings (6.14.10) for n = 1 and (6.14.12) for n > 1. In the limit N → ∞, the new n-point correlation function must be restricted so as to become independent of the center of energy. To see this, consider a displacement of the original energies Ei → Ei +

ε ε ei =E+ + 1 1 1

(6.14.13)

and the corresponding correlation function n . Displaying only variables of relevance for the argument, I have the identity 

n

Ei +

ε 1



  ε n ˜ n ({ei }) = 1 E + 1 = 1 (E)n ˜ n ({ei + ε}) .

(6.14.14)

240

6 Level Dynamics

To first order in ε this yields  ∂ 1 n ∂ ˜ n ({ei }) = 2 1 . ˜ n ({ei }) j = 1 ∂e j 1 ∂ E

(6.14.15)

As long as n remains finite for N → ∞, the right-hand side in (6.14.15) is of order 1/N , and thus ⎞ n  1 ∂ ˜ n ⎠ = → 0 for N → ∞. O⎝ ∂e N j j =1 ⎛

(6.14.16)

Of course, this restriction on n and ˜ n can be imposed only at some initial time, e.g., at τ = 0. We shall see below, however, that the dynamics will preserve local homogeneity at later times. When inserting the rescaling (6.14.12) into (6.14.4) it is once more important to use the effective constancy of the density 1 (E i , τ ) across the cluster, ∂ ∂ ∂ = 1 (E i , τ ) ≈ 1 (E, τ ) ∂ Ei ∂ei ∂ei 2 ∂2 2 ∂ ≈  (E, τ ) , 1 ∂ E i2 ∂ei2

(6.14.17)

thereby neglecting corrections vanishing in the limit N → ∞. Similarly, in the denominator of the intracluster Coulomb drift, we may put  Ei d E 1 (E , τ ) ≈ (E i − E j )1 (E, τ ). (6.14.18) ei − e j = Ej

Dropping the linear-drift term, I obtain the rescaled hierarchy as ⎛ ⎞ 1 ... n 1 ... n ∂ ∂ ⎝  1 1 ∂ ⎠ + ˜ n (e1 , . . . en , τ˜ ) = − ˜ n (e1 , . . . , en , τ˜ ) ∂ τ˜ ∂ei e − ej 2 ∂ei i j(=i) i −1 (E, τ )−(n+2)

1 ... n i

∂ ∂ Ei

 d E n+1

1 n+1 (E 1 , . . . , E n+1 , τ ). E i − E n+1 (6.14.19)

It remains to discuss the last term in (6.14.19) which describes the Coulomb drift of levels within the cluster caused by levels outside. As was done in treating the analogous term for n = 1, I split the integral into an integral over an interval ΔE ∼ N ε with −1/2 < ε < 1/2 around the center of the cluster and a remainder. Note that ΔE is chosen larger than the extension of the cluster such that the contribution to the integral from outside ΔE, the (n + 1)-level correlation function, factors: n+1 → n 1 , is

6.14

Local and Global Equilibrium in Spectra

1−(n+2)

1 ... n i

241

 ∂ 1 n (E 1 , . . . E n , τ ) outside d E n+1 1 (E n+1 , τ ). ∂ Ei E i − E n+1 ΔE (6.14.20)

This “remainder” is easily seen to vanish asymptotically: since ε > −1/2 the intracluster positions E i in the Coulomb-force denominator can be replaced by the center energy E of the cluster; after inserting the rescaling (6.14.12) and (6.14.13) for the n-level correlation function, the local homogeneity (6.14.15) and (6.14.16) can be invoked. Finally, the integral over the interval ΔE ∼ N ε can, using ε > −1/2 once more, be extended over the whole energy axis such that the hierarchy takes the form ⎛ ⎞ 1 ... n 1 ... n ∂ ⎝  1 1 ∂ ⎠ ∂ + ˜ n = − ˜ n ∂ τ˜ ∂ei e − ej 2 ∂ei i j(=i) i −

1 ... n i

∂ P ∂ei

 den+1

˜ n+1 , n > 1. ei − en+1

(6.14.21)

Now, the correlation function ˜ n+1 under the integral represents a local cluster of n + 1 levels. Remarkably enough, apart from the absence of the linear drift term from (6.14.21), the original hierarchy (6.14.4) and the rescaled one have identical structures. Having invoked local homogeneity for the correlation functions ˜ n of local clusters in showing the “locality” of the integral term in (6.14.21), it is imperative to demonstrate the consistency of the homogeneity property (6.14.16) with the hierarchy (6.14.21). Indeed, by differentiating (6.14.21) and integrating by parts in the last term, it is easy to check that the quantities Dn = i1 ... n (∂/∂ei )˜ n obey the same hierarchy of integrodifferential equations (6.14.21) as the correlation functions ˜ n . Consequently, if all Dn are of order 1/N initially, none of them can grow enough to exceed that order of magnitude. To appreciate this statement, the reader should realize that both the diffusion and the Coulomb repulsion tend to make ˜ n smooth and thus Dn small. The separation of energy and time scales for local fluctuations and global relaxation of the density must hold for any reasonable initial condition. A particularly interesting example is the transition from the GOE to the GUE treated in the preceding section. The transition from the Poissonian ensemble to the GUE, also briefly mentioned in Sect. 6.13, must abide by that separation as well. It follows that the latter transition cannot be considered the quantum parallel of classical crossovers from predominantly regular motion to global chaos, except when that classical transition is abrupt. In fact, I know of no classical system with a Hamiltonian H0 +λV or a Floquet operator e−iλV e−iH0 which, while integrable for λ = 0, is globally chaotic for all non-zero values of λ. Such abrupt classical crossovers can happen for billiards upon changes of the boundaries [60]. In that case, however, level dynamics does

242

6 Level Dynamics

not apply: Even a small deformation of the shape of a billiard boundary amounts to strong perturbations for sufficiently high degrees of excitation. Neither the Gaussian ensembles of Hermitian random matrices nor their Brownianmotion dynamizations could yield good models of Hamiltonians of quantum systems with globally chaotic classical limits, were it not for the disparity of scales under discussion. In fact, concrete dynamic systems do not in general obey the semicircle law for their level densities; it is only the local fluctuations in the spectra which tend to be faithful to random-matrix theory. Were there not a complete decoupling of local fluctuations and global variations of the density, provided by the scale separation, the predictions of random-matrix theory would be as unreliable locally as they in fact are globally [58].

6.15 Problems 6.1 Determine the number of independent dynamic variables of the fictitious N particle system when O(N ), U (N ), and Sp(N ) are the groups of canonical transformations. 6.2 Show that the angular momentum matrix l fulfills l˜ = −l, l † = −l, and l˜ Z = −Zl when the Floquet operator has O(N ), U (N ), and Sp(N ), respectively, as its canonical group. 6.3 Having done this problem you will understand why the variables lmn are often called angular momenta and why their Poisson brackets define the Lie algebras o(N ), u(N ), and sp(N ) in the orthogonal, unitary, and symplectic cases, respectively. Imagine an N -dimensional real configuration space with Cartesian coordinates x1 . . . x N . A 2N -dimensional phase space arises by associating N conjugate momenta p1 . . . p N . Rotations in the configuration space are generated by the angular momenta lmn = 12 (xm pn − xn pm ). The Poisson brackets (6.2.34) then follow from { pm , xn } = δmn . Think about how to generalize to complex x, p to get the Poisson brackets (6.2.28) for the unitary case where the lmn generate rotations in an N -dimensional complex vector space. In what vector space do the lmn generate rotations for the symplectic case? 6.4 Verify x˙ m = pm p˙ m = − l˙mn =





2lml llm 3 (x m − xl ) l(=m)

 lml lln (xm − xl )−2 − (xl − xn )−2

l(=m,n)

as the level dynamics for time-independent Hamiltonians H = H0 + λV.

6.15

Problems

243

6.5 Due to the Jacobi identity (6.2.27) the Poisson bracket of two conserved quantities is conserved as well. As a check on the reasoning of Sect. 6.3 it might be interesting to verify by explicit calculation that the Poisson bracket of any two of the independent constants of the motion either vanishes identically or is a linear combination of the independent ones. Please do not try to do this for the general case {Cmn , Cm n }. However, an instructive and easy exercise is to check {lkk , Cmn } = 0 and {C0m , C0n } = 0. 6.6 Prove (6.6.7) by showing that l11¯ = l22¯ = 0 and |l12 |2 + |l12¯ |2 = const. Use the T -invariance of F with T 2 = −1 and the ensuing identity for T V T −1 . 6.7 Using the arguments of Sect. 4.7 show that the ensemble (6.13.12) implies Wigner’s semicircle law for the mean level density in the limit N → ∞. 6.8 Show that the free diffusion described by (6.13.2) with λ2 as a “time” implies a Brownian motion with Coulomb gas interaction for the eigenvalues of H. Discuss how this differs from (6.13.13) and (6.13.19). 6.9 Study a two-body collision in the Coulomb gas model (6.13.13) and (6.13.19). 6.10 Write the Fokker–Planck equation (6.13.13) and (6.13.19) in the compact  √ −1 ∂  GUE , ˜ and show form ˙ = L, L = i i GUE ∂i GUE . Transform as  = that the transformed generator L˜ is a Hermitian differential operator. Furthermore, show that L˜ contains pair interactions inversely proportional to the squared distance between the “particles”, as does the Hamiltonian level dynamics (6.2.36). Appreciate the conceptual difference between the latter and the motion generated ˜ by L. 6.11 Derive the Hamiltonian flows generated by the Hamiltonians (6.14.13) and (6.14.15) by inserting the transformation E m (τ ) = g(τ ) · xm (λ(τ )) into Pechukas’ dynamics (6.2.36). 6.12 Proceeding as in Sect. 6.3, find the Lax form of the Pechukas equations (6.2.36). Hint: The Pechukas equations follow from (6.2.21) by linearizing as cos x → 1, sin x → x. 6.13 Find the Lax form of the modified Pechukas equations pertaining to the Hamiltonian function (6.14.15). Hint: Proceed as in Sect. 6.3 but use (6.14.6). 6.14 Establish the most general canonical ensemble for the modified Pechukas–Yukawa gas (with the confining potential) which rigorously gives the joint distribution of eigenvalues (4.3.15) of the Gaussian ensembles of random matrices. Use the results of Problem 6.13, and argue as in Sect. 6.5. Where does the Gaussian factor come from?

244

6 Level Dynamics

6.15 Find the spacing distribution P(S, λ) interpolating between the Gaussian orthogonal and unitary ensemble of 2 × 2 matrices. Use the method of Sect. 4.5, but start from the matrix density (6.13.12), taking the latter for N = 2. The result is [61] & 2 2 P(S, λ) = 1 + λ2 /2Sφ(λ)2 e−S φ(λ) /2 erf (Sφ(λ)/λ) where φ(λ) =

&

&

-

2 1 + λ2 /2 π/2 1 − π



√ . λ 2 λ arctan √ − 2 2 + λ2

and λ2 = e2τ − 1. 6.16 Repeat Problem 6.15 but for the transition of Poisson to Wigner. Start from the Gaussian defined by (6.13.28) to find 2 −S 2 ψ(λ)2 /4



P(S, λ) = Sλψ(λ) e

∞

dξ e−ξ

2

−2ξ λ

I0 [Sψ(λ)] .

0

By requiring S¯ = 1, the scale factor ψ(λ) can be determined and is related to the Kummer function [61, 62].

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

P. Pechukas: Phys. Rev. Lett. 51, 943 (1983) T. Yukawa: Phys. Rev. Lett. 54, 1883 (1985) T. Yukawa: Phys. Lett. 116A, 227 (1986) F. Haake, M. Ku´s, R. Scharf: Z. Phys. B65, 381 (1987) K. Nakamura, H.J. Mikeska: Phys. Rev. A35, 5294 (1987) H. Frahm, H.J. Mikeska: Z. Phys. B65, 249 (1986) M. Ku´s: Europhys. Lett. 5, 1 (1988) O. Bohigas, M.J. Giannoni, C. Schmit: Phys. Rev. Lett. 52, 1 (1984) F. Haake, M. Ku´s: Europhys. Lett. 6, 579 (1988) F. Dyson: J. Math. Phys. 3, 140 (1962); see also M.L. Mehta: Random Matrices (Academic, New York 1967; 2nd edition 1991; 3rd edition Elsevier 2004) K. Mnich: Phys. Lett. A176, 189 (193) M. Hardej, M. Ku´s, C. Gonera, P. Kosi´nski: J. Phys. A40, 423 (2007); for a more extende version see arXiv:nlin.cd/0608026 v1 11August2006 P. Gaspard, S.A. Rice, H.J. Mikeska, K. Nakamura: Phys. Rev. A42, 4015 (1990) F. Calogero, C. Marchioro: J. Math. Phys. 15, 1425 (1974) B. Sutherland: Phys. Rev. A5, 1372 (1972) J. Moser: Adv. Math. 16, 1 (1975) S. Wojciechowski: Phys. Lett. 111A, 101 (1985) M. Ku´s, F. Haake, D. Zaitsev, A. Huckleberry: J. Phys. A 30, 8635 (1997)

References 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62.

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D. Zaitsev, M. Ku´s, A. Huckleberry, F. Haake: J. Geometry Phys., to appear P.D. Lax: Comm. Pure Appl. Math. 21, 467 (1968) P.A. Braun: Rev. Mod. Phys. 65, 115 (1993) ˙ P.A. Braun, S. Gnutzmann, F. Haake, M. Ku´s, K. Zyczkowski: Found. Phys. 31, 614 (2001) M.R. Zirnbauer: In I.V. Lerner, J.P. Keating, D.E. Khmelnitskii (eds.) Supersymmetry and Trace Formulae; Chaos and Disorder (Kluwer Academic/Plenum, New York, 1999) E.B. Bogomolny, B. Georgeot, M.-J. Giannoni, C. Schmit: Phys. Rep. 291, 219 (1997) J.P. Keating, F. Mezzadri, Nonlinearity 13, 747 (2000) B. Dietz: Dissertation, Essen (1991) B. Dietz, F. Haake, Europhys. Lett. 9, 1 (1989) F. Haake, G. Lenz: Europhys. Lett. 13, 577 (1990) D.J. Thouless: Phys. Rep. 13C, 93 (1974) P. Gaspard, S.A. Rice, H.J. Mikeska, K. Nakamura: Phys. Rev. A42, 4015 (1990) D. Saher, F. Haake, P. Gaspard: Phys. Rev. A44, 7841 (1991) B.D. Simons, A. Hashimoto, M. Courtney, D. Kleppner, B.L. Altshuler: Phys. Rev. Lett. 71, 2899 (1993) M. Kollmann, J. Stein, U. Stoffregen, H.-J. St¨ockmann, B. Eckhardt: Phys. Rev. E49, R1 (1994) D. Braun, E. Hofstetter, A. MacKinnon, G. Montambaux: Phys. Rev. B55, 7557 (1997) D. Braun, G. Montambaux: Phys. Rev. B50, 7776 (1994) J. Zakrzewski, D. Delande: Phys. Rev. E47, 1650 (1993); ibid. 1665 (1993); D. Delande, J. Zakrzewski: J. Phys. Soc. Japan 63, Suppl. A, 101 (1994) F. von Oppen: Phys. Rev. Lett. 73,798 (1994); Phys. Rev. E51, 2647 (1995) Y.V. Fyodorov, H.-J. Sommers: Phys. Rev. E51, R2719 (1995); Z. Physik B 99, 123 (1995) S. Iida, H.-J. Sommers: Phys. Rev. E49, 2513 (1994) E. Akkermans, G. Montambaux: Phys. Rev. Lett. 68, 642 (1992) Y.V. Fyodorov, A.D. Mirlin: Phys. Rev. B51, 13403 (1995) M. Sieber, H. Primack, U. Smilanski, I. Ussishkin, H. Schanz: J. Phys. A28, 5041 (1995) H.-J. St¨ockmann: Quantum Chaos, An Introduction (Cambridge University Press, Cambridge, 1999) M. Barth, U. Kuhl, H.-J. St¨ockmann: Phys. Rev. Lett. 82, 2026 (1999) M.V. Berry: J. Phys. A10, 2083 (1977) V.N. Prigodin, : Phys. Rev. Lett. 75, 2392 (1995) X. Yang, J. Burgd¨orfer: Phys. Rev. A46, 2295 (1992) B.D. Simons, B.L. Altshuler: Phys. Rev. Lett. 70, 4063 (1993); Phys. Rev. B48, 5422 (1993) J. Zakrzewski: Z. Physik B98, 273 (1995) ˙ I. Guarneri, K. Zyczkowski, J. Zakrzewski, L. Molinari, G. Casati: Phys. Rev. E52, 2220 (1995) G. Lenz, F. Haake: Phys. Rev. Lett. 65, 2325 (1990) G. Lenz: Dissertation (Essen, 1992); H. Risken: The Fokker Planck Equation, Springer Ser. Synergetics, Vol. 18 (Springer, Berlin, Heidelberg 1984) A. Pandey, M.L. Mehta: Comm. Math. Phys. 87, 449 (1983) M.L. Mehta, A. Pandey: J. Phys. A16, 2655 (1983) J.B. French, U.K.B. Kota, A. Pandey, S. Tomsovic: Ann. Phys. (N.Y.) 181, 198 (1988) A. Pandey: In T.H. Seligman, H. Nishioka (eds.) Quantum Chaos and Statistical Nuclear Physics (Springer, Berlin, Heidelberg, 1986) F.J. Dyson: J. Math. Phys. 13, 90 (1972) L.A. Pastur: Th. Math. Phys. 10, 67 (1972) L.A. Bunimovich: Funct. Anal. Appl. 8, 73 (1974) G. Lenz, F. Haake: Phys. Rev. Lett. 67, 1 (1991) E. Caurier, B. Grammaticos, A. Ramani: J. Phys. A23, 4903 (1990)

Chapter 7

Quantum Localization

7.1 Preliminaries This chapter will focus mainly on the kicked rotator that displays global classical chaos in its cylindrical phase space for sufficiently strong kicking. The chaotic behavior takes the form of “rapid” quasi-random jumps of the phase variable around the cylinder and “slow” diffusion of the conjugate angular momentum p along the cylinder. A quantum parallel of this time-scale separation of the motions along the two principal directions of the classical phase space is the phenomenon of quantum localization: the quasi-energy eigenfunctions turn out localized in the angular momentum representation [1–3]. (We shall be concerned with the generic case of irrational values of a certain dimensionless version of Planck’s constant – for rational values extended eigenfunctions arise [4].) The difference between classical and quantum predictions for the rotator is nothing like a small quantum correction, even if states with large angular-momentum quantum numbers are involved. Classical behavior would be temporally unlimited diffusive growth of the kinetic energy,  p 2  ∝ t. Quantum localization must set an end to any such growth starting from a finite range of initially populated angularmomentum states, since only those Floquet states can be excited that overlap the initial angular momentum interval. Given localization of all quasi-energy eigenfunctions, it is clear that classical chaos cannot be accompanied by repulsion of all quasi-energies: Indeed, eigenfunctions without overlap in the angular momentum basis have no reason to keep their quasi-energies apart. On the other hand, if a level spacing distribution is established by including only eigenfunctions with overlapping supports one expects, and does indeed find, a tendency to level repulsion again [5]. From the point of view of random-matrix theory, the Floquet operator F of the kicked rotator appears as nongeneric in other respects as well. In the angular momentum representation, F takes the form of a narrowly banded matrix. In appreciating this statement, the reader should realize that the angular momentum representation is the eigenrepresentation of a natural observable of the kicked rotator, not merely a representation arrived at through an incomplete diagonalization. Banded random matrices will be considered separately in Chap. 11.

F. Haake, Quantum Signatures of Chaos, Springer Series in Synergetics, 3rd ed., C Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-05428-0 7, 

247

248

7 Quantum Localization

In the following I shall discuss some peculiarities of the dynamics of the kicked rotator related to localization. Particular emphasis will be placed on the analogy of the kicked rotator to Anderson’s hopping model in one dimension. Moreover, I shall contrast the kicked rotator with the kicked top. The latter system does not display localization except in a very particular limit in which the top actually reduces to the rotator. The absence of quantum localization from the generic top, it will be seen, is paralleled by the absence of any time-scale separation for its two classical phasespace coordinates. Correspondingly, there is no slow, nearly conserved variable. No justice can be done to the huge literature on localization, not even as regards spectral statistics of systems with localization. The interested reader can pick up threads not followed here in [6–8] and in Efetov’s and Imry’s recent books [9, 10]. Before diving into our theoretical discussions, it is appropriate to mention that quantum localization has been observed in rather different types of experiments. Microwave ionization of Rydberg states was investigated by Bayfield, Koch, and co-workers for atomic hydrogen [11, 12] and by Walther and co-workers for rubidium, as reviewed in [13]. The degree of atomic excitation in these experiments was so high that one might expect and in some respects does find effectively classical behavior of the valence electron. Quasi-classical behavior would, roughly speaking, amount to diffusive growth of the energy toward the ionization threshold. What is actually found is a certain resistance to ionization relative to such classical expectation. Quantitative analysis reveals an analogy to the localization-imposed break of diffusion for the kicked rotator. For reviews on this topic, the reader is referred to [14, 15]. A second class of experiments realizing kicked-rotator behavior was suggested by Graham, Schlautmann and Zoller [16] and conducted by Raizen and co-workers, on cold (some micro-Kelvins) sodium atoms exposed to a standing-wave light field whose time dependence is monochromatic, apart from a periodic amplitude modulation [17–19]. The laser frequency is chosen sufficiently close to resonance with a single pair of atomic levels so that all other levels are irrelevant. The selected pair gives rise to a dipole interaction with the electric field of the standing laser wave. Thus, the translational motion of the atom takes place in a light-induced potential U (x, t) = −K cos(x −λ sin ωt) where ω is the modulation frequency, λ is a measure of the modulation amplitude, and K is proportional to the squared dipole matrix element. The periodically driven motion in that potential is again characterized by localization-limited diffusion of the momentum p. An account of the physics involved in the quantum optical experiments on localization is provided by St¨ockmann’s recent book [20]. Experiments and the theory of localization in disordered electronic systems are reviewed in [21].

7.2 Localization in Anderson’s Hopping Model In preparation for treating the quantum mechanics of the periodically kicked rotator, we now discuss Anderson’s model of a particle whose possible locations are the equidistant sites of a one-dimensional chain. At each site, a random potential Tm

7.2

Localization in Anderson’s Hopping Model

249

acts and hopping of the particle from one site to its r th neighbor is described by a hopping amplitude Wr . The probability amplitude u m for finding the particle on the mth site obeys the Schr¨odinger equation [22–24] Tm u m +



Wr u m+r = Eu m .

(7.2.1)

r

If the potential Tm were periodic along the chain with a finite period q such that Tm+q = Tm , the solutions of (7.2.1) would, as is well known, be Bloch functions, and the energy eigenvalues would form a sequence of continuous bands. However, of prime interest with respect to the kicked rotator is the case in which the Tm are random numbers, uncorrelated from site to site and distributed with a density (Tm ). The hopping amplitudes Wr , in contrast, will be taken to be non-random and will decrease fast for hops of increasing length r. In the situation just described, the eigenstates of the Schr¨odinger equation (7.2.1) are exponentially localized. They may be labelled by their center ν such that u νm ∼ e−|ν−m|/l for |ν − m| → ∞ .

(7.2.2)

The so-called localization length l is a function of the energy E and may also depend on other parameters of the model such as the hopping range and the density (Tm ). For the special case of Lloyd’s model, defined by a Lorentzian density (Tm ) and hoppings restricted to nearest-neighbor sites, an exact expression for l will be derived in Sect. 7.4. Typical numerical results for the eigenfunctions u νm are depicted in Fig. 7.1. Numerical results for the energy eigenvalues E ν indicate that in the limit of a long chain of sites (N  1), a smooth average level density arises. Moreover, the level spacings turn out to obey an exponential distribution just as if the levels were statistically independent. It is easy to see, in fact, that the model must display level clustering rather than level repulsion due to the exponential localization of the eigenfunctions: two eigenfunctions with their centers separated by more than a localization

Fig. 7.1 Exponential localization of quasi-energy eigenfunctions of kicked rotator From [1]

250

7 Quantum Localization

length l have an exponentially small matrix element of the Hamiltonian and thus no reason to keep their energies apart. A further aspect of localization is worthy of mention. The Hamiltonian matrix Hmn implicitly defined by (7.2.1) is a random matrix with independent random diagonal elements Hmm = Tm and with (noticeably) non-zero off-diagonal elements Hm,m+r = Wr only in a band around the diagonal; the “bandwidth” is small compared to the dimension N of the matrix. Upon numerically diagonalizing banded random matrices, one typically finds localized eigenvectors and eigenvalues with Poissonian statistics. Both the small bandwidth and the randomness of H are essential for that statement to hold true: As we have seen in Chap. 4, full random matrices whose elements are independent to within the constraints of symmetry typically display level repulsion. On the other hand, any Hermitian matrix with repelling eigenvalues can be unitarily transformed so as to take a banded and even diagonal form, but then the matrix elements must bear correlations. We shall return to banded random matrices in Chap. 11 and show there that localization takes place if the bandwidth b is small in the sense b2 /N < 1. An interesting argument for Anderson localization can be based on a theorem due to Furstenberg [25, 26]. This theorem deals with unimodular random matrices Mi (i.e., random matrices with determinants of unit modulus) and states that under rather general conditions   1 ln Tr M Q M Q−1 . . . M2 M1 ≡ γ > 0. Q

lim

Q →∞

(7.2.3)

To apply (7.2.3) in the present context, we restrict the discussion to nearest neighbor hops (W±1 = W, Wr = 0 for |r | > 1) and rewrite the Schr¨odinger equation (7.2.1) in the form of a map for a two-component vector, 

u m+1 um







(E − Tm )/W −1 = 1 0

um u m−1



 ≡ Mm

 um . u m−1

(7.2.4)

The 2 × 2 matrix in this map is indeed unimodular, det Mn = 1, and random due to the randomness of the potential Tm . Starting from the values of the wave function on two neighboring sites, say 0 and 1, one finds the wave function further to the left or to the right by appropriately iterating the map (7.2.4), 

u m+1 um

 = Mm Mm−1

  u1 . . . M2 M1 . u0

(7.2.5)

Far away from the starting sites, i.e., for |m| → ∞, Furstenberg’s theorem applies and means that the unimodular matrix Mm Mm−1 . . . M2 M1 has the two eigenvalues exp (±mγ ). It follows that almost all “initial” vectors u 1 , u 0 will give rise to wave functions growing exponentially both to the left and to the right. Thus, the parameter γ is appropriately called the Lyapunov exponent of the random map under consideration. By special choices of u 0 and u 1 , it is possible to generate wave

7.3

The Kicked Rotator as a Variant of Anderson’s Model

251

functions growing only in one direction and decaying in the other. However, to enforce decay in both directions, special values of the energy E must be chosen. In other words, the eigenenergies of the one-dimensional Anderson model cannot form a continuum and in fact typically constitute a discrete spectrum. The corresponding eigenfunctions are exponentially localized and the Lyapunov exponent turns out to be the inverse localization length (we disregard the untypical exception of so-called singular continuous spectra). An interesting consequence arises for the temporal behavior of wave packets. An initially localized packet is described by an effectively finite number of the localized eigenstates. As a sum of an effectively finite number of harmonically oscillating terms, the wave packet must be quasi-periodic in time. At any site m, the occupation amplitude u m (T ) will get close to its initial value again and again as time elapses. A particle initially placed at, say, m = 0 such that u m (0) = δm,0 may, while hopping to the left and right in the random potential, display diffusive behavior (Δm)2  ∼ t. As soon as a range of the order of the localization length has been explored, however, the diffusive growth of the spread (Δm)2  must give way to manifestly quasi-periodic behavior.

7.3 The Kicked Rotator as a Variant of Anderson’s Model The periodically kicked rotator, one of the simplest and best investigated models capable of displaying classical chaos, has a single pair of phase-space variables, an angle Θ and an angular momentum p. As quantum operators, these variables obey the canonical commutation rule [ p, Θ] =

 . i

(7.3.1)

Their dynamics is generated by the Hamiltonian [27] H (t) =

+∞  p2 I δ(t − nτ ), + λ V (Θ) 2I τ n = −∞

(7.3.2)

where I is the moment of inertia, τ the kicking period, and λ a dimensionless kicking strength. The dimensionless potential V (Θ) must be 2π -periodic in Θ. We shall deal for the most part with the special case V (Θ) = cos Θ.

(7.3.3)

Due to the periodic kicking, a stroboscopic description is appropriate with the Floquet operator F = e−iλ(I /τ )V e−i p

2

τ/2I 

(7.3.4)

252

7 Quantum Localization

which accounts for the evolution from immediately after one kick to immediately after the next. The wave vector at successive instants of this kind obeys the stroboscopic map Ψ (t + 1) = FΨ (t).

(7.3.5)

Equivalently, the Heisenberg-picture operators obey X t+1 = F † X t F .

(7.3.6)

In (7.3.5) and (7.3.6) and throughout the remainder of this chapter, the time variable t is a dimensionless integer counting the number of periods executed. Special cases of (7.3.6) are the discrete-time Heisenberg equations pt+1 = pt − λ τI V (Θt+1 ) , Θt+1 = Θt + τI pt ,

(7.3.7)

which follow from the commutator relation (7.3.1) (Problem 7.1). Due to the absence of products of p and Θ, the recursion relations (7.3.7) also hold classically as discrete-time Hamiltonian equations. A word about parameters and units is now in order. The parameter I /τ has the dimension of an angular momentum. Upon introducing a dimensionless angular momentum P = pτ/I , the stroboscopic equations of motion (7.3.7) simplify to reveal the dimensionless kicking strength λ as the only control parameter of classical dynamics. On the other hand, in the commutator relation [P, Θ] = τ/I i and also in the Floquet operator F, the dimensionless version of Planck’s constant τ/I appears as a second parameter controlling the effective strength of quantum fluctuations. With these remarks in mind, we may simplify the notation and henceforth set τ and I equal to unity. It is appropriate first to comment on the classical behavior of the rotator. The period-to-period increment of angle Θ becomes greater, the larger the chosen kick strength λ. For sufficiently large λ, the angle may in fact jump by several 2π, without Θt+1 (mod 2π) showing any obvious preference for any part of the interval [0, 2π ]. It follows that the momentum (which reacts only to Θt+1 (mod 2π )) tends to undergo a random motion for strong kicking. In a rough idealization, we may assume that a series of successive values of Θ uniformly fills the interval [0, 2π ] and keeps no kick-to-kick memory; thus, the force −V (Θ) in (7.3.7) appears as noise with no kick-to-kick correlation. Then, the map (7.3.7) can be approximated by pt = p 0 −

t  ν =1

λV (Θν )

(7.3.8)

7.3

The Kicked Rotator as a Variant of Anderson’s Model

253

and yields a constant mean momentum, pt = p0 −

t 

λV (Θν ) = p0 ,

(7.3.9)

ν =1

since the mean force 1 2π

V (Θ) =



2π

dΘ V (Θ) = 0

(7.3.10)

0

vanishes for a periodic potential. Similarly, the mean-squared momentum, pt2 = p02 +

t 

λ2 V (Θμ )V (Θν )

μ,ν = 1



=

p02

1 + λ 2π



2π

2



dΘ V (Θ)

 2

t,

(7.3.11)

0

grows linearly with time inasmuch as the force displays no kick-to-kick correlations, V (Θμ )V (Θν ) = δμν V (Θ)2 .

(7.3.12)

Evidently, the crude idealization of the dynamics (7.3.7) for λ  1 by a random process suggests diffusive behavior of the (angular) momentum with the diffusion constant  λ2 2π dΘ V (Θ)2 . (7.3.13) D= 2π 0 In particular, for V (Θ) = cos Θ, D=

λ2 . 2

(7.3.14)

In fact, by numerically iterating Chirikov’s standard map [27] pt+1 = pt + λ sin Θt+1 Θt+1 = (Θt + pt ) mod (2π )

(7.3.15)

for an ensemble of initial angles Θ0 and vanishing initial momentum, one does find the expected diffusive behavior if λ is chosen sufficiently large, λ > ∼ 1.5. Figure 7.2 depicts this numerical evidence. The effectively diffusive behavior of the momentum is tantamount to chaos, as follows from the linearized version of the standard map that describes the fate of infinitesimally close phase-space points under iteration,

254

7 Quantum Localization

Fig. 7.2 Classical (full curve) and quantum (dotted) mean kinetic energy of the periodically kicked rotator, determined by numerical iteration of the respective maps. The quantum mean follows the classical diffusion for times up to some “break” time and then begins to display quasi-periodic fluctuations. After 500 kicks, the direction of time was reversed; while the quantum mean accurately retraces its history, the classical mean reverts to diffusive growth thus revealing the extreme sensitivity of chaotic systems to tiny perturbations (here round-off errors). Courtesy of Dittrich and Graham [28]



δpt+1 δΘt+1



     δpt δpt 1 + λ cos Θt+1 λ cos Θt+1 = ≡ Mt . (7.3.16) 1 1 δΘt δΘt

Inasmuch as Θt is effectively random, Mt is a unimodular random matrix. Furstenberg’s theorem again applies (now in a classical context, in contrast to the application in Sect. 7.2) and secures a positive Lyapunov exponent. Exponential separation of classical trajectories, i.e., chaos, is manifest. Now, we turn to the quantum version of the kicked rotator. Slightly changing notation, we rewrite the quantum map (7.3.5) as |Ψ + (t + 1) = F|Ψ + (t) , t = 0, 1, 2, . . . .

(7.3.17)

Instead of looking at the wave vector |Ψ + (t) immediately after the tth kick, one may study the wave vector just before that kick |Ψ − (t) = e−iH0 / |Ψ + (t − 1) , H0 =

p2 . 2

(7.3.18)

To solve Schr¨odinger’s equation for either |Ψ + (t) or |Ψ − (t), a representation must now be chosen. The free motion in between kicks is most conveniently described in the basis constituted by the eigenvectors of p,

7.3

The Kicked Rotator as a Variant of Anderson’s Model

p|n = n|n , n = 0, ±1, ±2, . . . ,

255

(7.3.19)

since the Hamiltonian is diagonal in that representation: 2 2 n |n. 2

H0 |n =

(7.3.20)

By expanding as |Ψ ± (t) =



Ψn± (t)|n ,

(7.3.21)

n

one finds that the free motion is described by Ψn− (t + 1) = e−in

2

/2

Ψn+ (t) .

(7.3.22)

An individual kick, on the other hand, is most easily dealt with in the Θ representation, |Ψ ± (t) =



2π

dΘ Ψ ± (Θ, t)|Θ ,

(7.3.23)

Ψ + (Θ, t) = e−iλV (Θ)/ Ψ − (Θ, t) .

(7.3.24)

0

since

Combining the two steps in the p representation, one encounters the map +∞  2 Jm−n e−in /2 Ψn+ (t) , Ψm+ (t + 1) = n = −∞ $ 2π 1 i(m−n)Θ −iλV (Θ) e / . Jm−n = 2π 0 dΘ e

(7.3.25)

For the special case V = cos Θ, the matrix element of the unitary kick operator becomes the Bessel function  π 1 dΘ eiz cos Θ cos nΘ . (7.3.26) Jn (z) = n πi 0 To establish a relation between the kicked rotator and Anderson’s model [1], we must consider the eigenvalue problem for the Floquet operator F, F|u +  = e−iφ |u +  ,

(7.3.27)

256

7 Quantum Localization

which, in the momentum representation, reads 

Jm−n e−in

2

/2 + un

= e−iφ u + m .

(7.3.28)

n

The corresponding problem with the order of kick and free motion reversed will also play a role; the corresponding eigenvectors |u −  are most easily related to the |u +  in the Θ representation, u − (Θ) = eiλV (Θ)/ u + (Θ) ,

(7.3.29)

where u − (Θ) and u + (Θ) pertain to the same eigenphase φ of e−iH0 / e−iλV / and F = e−iλV / e−iH0 / , respectively. With the help of (7.3.27), the relation (7.3.29) can be rewritten as u − (Θ) = ei(φ−H0 /) u + (Θ)

(7.3.30)

and thus in the p representation i(φ−m u− m =e

2

/2) + um

.

(7.3.31)

Now, we represent the unitary kick operator in terms of a Hermitian operator W , e−iλV / =

1 + iW λV , W = −tan , 1 − iW 2

(7.3.32)

and define the vector |u =

1 2



 |u +  + |u −  .

(7.3.33)

From (7.3.29), this vector is obtained as u(Θ) =

u + (Θ) u − (Θ) = 1 + iW (Θ) 1 − iW (Θ)

(7.3.34)

and together with (7.3.31) yields an integral equation for the function u(Θ), [1 − iW (Θ)] u(Θ) = ei(φ−H0 /) [1 + iW (Θ)] u(Θ) .

(7.3.35)

This equation assumes an algebraic appearance, however, when expressed in the p representation, Tm u m +

 r(=0)

Wr u m+r = Eu m ,

(7.3.36)

7.3

The Kicked Rotator as a Variant of Anderson’s Model

257

where we have introduced   2 1 − ei(φ−m /2) φ − m 2 /2 = tan , Tm = i 2 1 + ei(φ−m 2 /2)

(7.3.37)

E = −W0 . Now, the desired relationship between Anderson’s hopping model and the kicked rotator is established: the algebraic equation (7.3.36) for u m = m|u does indeed have the form of the Schr¨odinger equation (7.2.1) for a particle on a lattice, where Tm is a single-site potential and Wr is a hopping amplitude. Interestingly, the quasienergy φ of the rotator has become a parameter in the potential Tm while the zeroth Fourier component W0 of the function W (Θ) has formally taken on the role of the energy of the hopping particle. A subtle difference between the kicked rotator and the Anderson model deserves discussion. The amplitudes Tm defined in (7.3.37) are not strictly random but only pseudorandom numbers. In fact, since the quantities (φ − m 2 /2) enter Tm as the argument of a tangent, they become effective modulo π. According to a theorem of Weyl’s [29], the sequence (φ − m 2 /2) mod π is ergodic in the interval [0, π ] and covers that interval with uniform density as ±m = 0, 1, 2, . . . . It follows that the Tm have a density W (T )dT = dφ/π. With dT /dφ = 1 + T 2 , one finds W (Tm ) =

1 , π(1 + Tm2 )

(7.3.38)

i.e., a Lorentzian distribution. Nevertheless, the Tm are certainly not strictly independent from one value of m to the next. Therefore, it may be appropriate to speak of the kicked rotator in terms of a pseudo-Anderson model or, actually, a pseudoLloyd model since the Anderson model with truly random Tm distributed according to a Lorentzian is known as Lloyd’s model [30]. There is another difference between the simplest version of the Anderson model and the simplest rotator. Rather than allowing for hops to nearest neighbor sites, the kicking potential V (Θ) = cos Θ gives rise to   λ cos Θ (7.3.39) W (Θ) = −tan 2 with Fourier components 

2π

Wr = − 0

  dΘ irΘ λ cos Θ e tan . 2π 2

(7.3.40)

These hopping amplitudes can be calculated in closed form for λ < π and, it may be shown, fall off exponentially with increasing r. Even though nobody has proved localization for the kicked rotator, the pseudorandomness of the potential Tm and the finite range of the hopping amplitude Wr

258

7 Quantum Localization

strongly suggest such behavior. Moreover, all numerical evidence available favors an effective equivalence of the Anderson model and the kicked rotator.

7.4 Lloyd’s Model It may be well to digress a little and add some hard facts to the localization lore. Anderson’s Schr¨odinger equation allows an exact evaluation of the localization length l provided that the hopping amplitude is non-zero only for nearest neighbor hops and that the potential Tm is independent from site to site and distributed according to a Lorentzian [30], Tm u m +

κ (u m−1 + u m+1 ) = Eu m , 2

(7.4.1)

1 . π(1 + Tm2 )

(7.4.2)

(Tm ) =

Now, we shall prove1 the following well-known result [24, 31] for the ensembleaveraged localization constant γ ≡ 1/l, cosh γ =

& 1 & (E − κ)2 + 1 + (E + κ)2 + 1 2κ

(7.4.3)

which implies γ > 0 for all values of the energy E. The starting point is the tridiagonal Hamiltonian matrix Hmn = Tm δmn +

 κ δm,n+1 + δm,n−1 2

inserted in the Green function   detnm (E − H ) 1 1 1 G mn = = (−1)m−n . N E − H mn N det (E − H )

(7.4.4)

(7.4.5)

The determinant detmn (E − H ) is obtained from det (E − H ) by cancelling the nth row and the mth column. Assuming a finite number N of sites, the element G 1N , i.e., the one referring to the beginning and end of the chain of sites, takes an especially simple form since  κ N −1 . det1N (E − H ) = − 2

1

I am indebted to H.J. Sommers for showing this proof to me.

(7.4.6)

7.4

Lloyd’s Model

259

It is an obvious result of the tridiagonality of Hmn that the element G 1N reads simply G 1N =

N 1  κ N −1 B , (E − E ν ). + N 2 ν =1

(7.4.7)

On the other hand, by invoking the spectral representation of Green’s function, G mn =

1  u νm u νn , N ν E − Eν

(7.4.8)

and comparing the residues of the pole at E = E ν in (7.4.7) and (7.4.8), one obtains  κ N −1 B , (E ν − E μ ) . u ν1 u νN = + 2 μ(=ν)

(7.4.9)

This latter expression lends itself to a determination of the localization length for the νth eigenvector. If that eigenvector were extended like a Bloch state, one would have u ν1 u νN ∼ 1/N , i.e., |u ν1 u νN |1/N ∼ (1/N )1/N ∼ exp (−(ln N )/N ) → 1. For an exponentially localized state, however, u ν1 u νN ∼ Ae−γν N and thus |u ν1 u νN |1/N ∼ | A|1/N e−γν → e−γν < 1. Therefore, the localization length 1/γν can be obtained as  κ 1   ln E ν − E μ  − ln . N μ(=ν) 2

γν =

(7.4.10)

The limit N → ∞ is implicitly understood, so we may replace the sum in (7.4.10) by an integral with the level density (E) as a weight  γ (E) =

d x (x) ln |E − x| − ln

κ , 2

(7.4.11)

with the normalization of (x) chosen as that of a probability density, 

+∞ −∞

d x (x) = 1 .

(7.4.12)

More easily accessible than γ (E) itself is its derivative γ (E) = P

 dx

(x) , E−x

(7.4.13)

since the principal-value integral occurring here is easily related to Green’s function. Indeed, the spectral representation (7.4.8) entails

260

7 Quantum Localization

Tr {G(E)} =

 1 1  (x) → dx . N ν E − Eν E−x

(7.4.14)

The continuum approximation endowed Green’s function with a cut along the real axis in the complex energy plane, so different limiting values of G arise immediately above and below the real energy axis2   Tr G(E ± i0+ ) = P

 dx

(x) ∓ iπ (E) . E−x

(7.4.15)

Usually, one is interested in the imaginary part of this equation which yields a strategy for calculating the level density (Sect. 4.7). Here, however, we compare the real part with (7.4.13) and conclude that    γ (E) = Re Tr G(E − i0+ ) .

(7.4.16)

The latter identity holds for any configuration of the potential Tm . Averaging over all configurations {Tm } with the Lorentzian density (7.4.2) gives [31]    γ (E) = Re Tr G(E ± i0+ ) ,

(7.4.17)

a result often referred to as Thouless’ formula. Due to the assumed site-to-site independence,  (...) =

, m

+∞

−∞

 1 dTm (...) . π(1 + Tm2 )

(7.4.18)

To carry out the average of G(E), it is convenient to represent Green’s function by a multiple Gaussian integral, 

N , n  ,

+∞

i = 1 α = 1 −∞

 d Siα

 S 1p Sq1

exp −i



+

(E − H − i0

)i j Siα S αj



i jα

 n/2 πN N + . = G pq (E − i0 ) 2 det (E − H − i0+ )

(7.4.19)

An especially useful identity arises from (7.4.19) when the parameter n, after performing the integral, is elevated from integer to continuous real and then sent to zero,

2

Here and in the following 0+ denotes an arbitrarily small positive number.

7.4

Lloyd’s Model

261

 +

N G pq (E − i0 ) = lim 2 n→0



N , n  ,

 d Siα

S 1p Sq1

i =1 α=1

× exp −i



 (E − H − i0+ )i j Siα S αj .

(7.4.20)

i jα

This representation of the inverse of the matrix (E − H − i0+ ) pq by a multiple Gaussian integral with the subsequent manipulation of the parameter n is known in the theory of disordered spin systems as the “replica trick” [32]. Its virtue is that the random numbers Tm now appear in a product (of exponentials), each factor involving a single Tm . Therefore, the average (7.4.18) can be taken site by site according to N     , exp [iT j S αj S αj ] dT j exp i T j S αj S αj = π (1 + T j2 ) j j =1    α α = exp − Sj Sj .

(7.4.21)

j

The average Green function then takes the simple form  N G pq (E − i0+ ) = lim

n→0

2

,,



 d Siα

S 1p Sq1

α

i

× exp −i



˜ − i0+ )i j Siα S αj (E − H

 (7.4.22)

i jα

with the matrix   ˜ pq = iδ pq + κ δ p,q+1 + δ p,q−1 . H 2

(7.4.23)

˜ pq differs from the original Hamiltonian only by the It is remarkable that H replacement T p → i. At any rate, one can now read the multiple integral (7.4.21) ˜ − i0+ )−1 , i.e., backwards as a representation of (E − H 

N −1 G

−1

 pq

= (E − i)δ pq −

 κ δ p,q+1 + δ p,q−1 . 2

(7.4.24)

With the average over the random potential done, it remains to invert the tridiagonal matrix (7.4.23). To that end, we again employ (7.4.5) and write Tr {N G} =

 detmm (E − H ˜) ∂ ˜) = ln det(E − H ˜) ∂E det (E − H m

(7.4.25)

262

7 Quantum Localization

where in the last step we have exploited the fact that the energy E enters only the ˜ ). Indeed, simplifying the notation for a moment in diagonal elements of det (E − H ˜ ) along the first row, an obvious manner and expanding det (E − H ∂ det = det11 + (E − i)∂ det11 −

κ ∂ det12 = det11 + ∂1 det 2

(7.4.26)

where the differentiation ∂1 has to leave the first diagonal element untouched; similarly expanding along the second row gives ∂1 det = det22 + ∂12 det where ∂12 now has to leave the first two diagonal elements of the determinant undifferentiated; after N such steps, one arrives at ∂ det =

N 

det pp .

p=1

Finally, we must evaluate the N × N determinant ˜). D N ≡ det (E − H

(7.4.27)

That goal is most easily achieved recursively since D1 = E − i D2 = (E − i)2 −

κ2 4

(7.4.28)

κ2 Dn = (E − i)Dn−1 − Dn−2 . 4 These relations obviously allow the extension D0 = 1. Since the recursion relation in (7.4.28) has coefficients independent of n, it can be solved by the ansatz Dn ∼ x n where x is determined by the quadratic equation x 2 − (E − i)x +

κ2 =0. 4

(7.4.29)

The two solutions x± =

1 2

3

E −i±

&

(E − i)2 − κ 2

4 (7.4.30)

together with the “initial” conditions D0 = 1, D1 = (E − i) give N +1 N +1 ˜ ) = x+ − x− . det (E − H x+ − x−

Combining (7.4.16, 7.4.25, 7.4.31),

(7.4.31)

7.5

The Classical Diffusion Constant as the Quantum Localization Length

γ (E)

∂ = Re ∂E



1 ln N



x+N +1 − x−N +1 x+ − x−

263

5 .

(7.4.32)

This result simplifies considerably in the limit N → ∞. Since |x+ | > |x− |, ∂ ln |x+ | . ∂E

γ (E) =

(7.4.33)

Integrating and fixing the integration constant by comparing with (7.4.10) for E → ∞ gives   6   2  E −i  E − i  γ (E) = ln  − 1 . + κ  κ 

(7.4.34)

A little elementary algebra finally brings the localization constant to the form (7.4.3).

7.5 The Classical Diffusion Constant as the Quantum Localization Length It was argued in Sect. 7.3 that the classical map pt+1 = pt + λV (Θt+1 ) Θt+1 = (Θt + pt ) mod (2π )

(7.5.1)

may, for sufficiently strong kicking λ, be simplified to pt+1 = pt + noise ,

(7.5.2)

where the noise imparted to the momentum is uncorrelated from period to period. The ensuing diffusive behavior of the momentum,  pt2  −  p02  = Dt ,  pt  =  p0  ,

(7.5.3)

with the diffusion constant [27] D=

λ2 2π



2π

dΘ V (Θ)2

(7.5.4)

0

is in fact in good agreement with numerical data obtained by following the bundle of solutions of (7.5.1) originating from a cloud of many initial points; see Fig. 7.2.

264

7 Quantum Localization

For the quantum rotator, however, diffusive growth of the squared momentum can prevail only until the spread of the wave packet along the momentum axis has reached the localization length l. Afterward,  pt2  must display quasi-periodicity in time. The transition from diffusive to manifestly quasi-periodic behavior, depicted in Fig. 7.2, will take place at a “break time” t ∗ , whose order of magnitude may be estimated by  (Δpt )2 = Dt ∗ = l 2 2 .

(7.5.5)

An independent estimate for t ∗ or l is needed to complement (7.5.4) and (7.5.5) for a complete set of equations for D, t ∗ , and l. Chirikov et al. [33] have found a third relation that will be explained now. A quasi-periodic wave packet is spanned by about l basis states, for example, eigenstates of either the momentum p or the Floquet operator F. The corresponding quasi-energies, roughly l in number, all lie in the interval [0, 2π] and thus have a mean spacing of the order 2π/l. The inverse of this spacing is the minimum time needed for the discreteness of the spectrum to manifest itself in the time dependence of the wave packet. Clearly, the time in question may be identified with the break time t ∗ , whereupon one has the important order-of-magnitude result t ∗ ≈ l ≈ −2 D ≈ −2

λ2 2π



2π

dΘ V (Θ)2 .

(7.5.6)

0

Shepelyansky [34] has numerically verified this result for the kicked rotator with λV (Θ) = λ cos Θ, and the kicking strength ranging over the interval 1.5 ≤ λ ≤ 29. While the above estimate of t ∗ certainly implies that  pt2  must display quasiperiodicity for t > t ∗ , it does not explicitly suggest that  pt2  precisely follows ∗ classical diffusion for 0 ≤ t < ∼ t . Clearly, for classical behavior to prevail at early times the, dimensionless version of  must be sufficiently small.

7.6 Absence of Localization for the Kicked Top It should be pointed out that not all periodically kicked systems display localization analogous to that of Anderson’s model. An interesting case is provided by kicked tops [35–38] which have already been alluded to several times and will be discussed now somewhat more systematically. Kicked tops have the three components of an angular momentum vector J as their only dynamic variables. In their Floquet operators F = e−iλV e−iH0 both H0 and V are polynomials in J such that the squared angular momentum is a constant of the motion,3 While τ/I for the kicked rotator naturally arises as a dimensionless measure of Planck’s constant, that role will be played by 1/j for the kicked top. Therefore, it is convenient to set  = 1 in this section.

3

7.6

Absence of Localization for the Kicked Top

J 2 = j( j + 1) , j = 12 , 1, 32 , . . . .

265

(7.6.1)

The classical limit is attained when the quantum number becomes large, j → ∞. The simplest model capable of chaotic motion in the classical limit is given by F = e−iλJz /2 j e−iα Jx . 2

(7.6.2)

The second factor in this F clearly describes a linear precession of J around the x-direction by the angle α. Similarly, the first factor may be said to correspond to a nonuniform “rotation” around the z axis; instead of being a constant c number the rotational angle is itself proportional to Jz . The dimensionless coupling constant λ might now be called a torsion strength. The quantum number j appears in V = Jz2 /2 j to provide λV and H0 = α Jz with the same asymptotic scaling with j when j → ∞ for finite constant λ and α. By using the angular momentum commutators [Ji , J j ] = iεi jk Jk ,

(7.6.3)

one finds the stroboscopic Heisenberg equations of motion J t+1 = F † J t F in the form

    λ ˜ 1 ˜ Jx,t+1 = Jx , cos Jz − − J˜ y , sin . . . j 2 i i cos . . . , J˜ y + sin . . . , J˜ x , 2 2

    λ ˜ 1 ˜ Jz − = Jx , sin + J˜ y , cos . . . j 2 +

Jy,t+1

+

i i sin . . . , J˜ y − cos . . . , J˜ x , 2 2

Jz,t+1 = J˜ z , J˜ x = Jx,t ,

(7.6.4)

J˜ y = Jy,t cos α − Jz,t sin α , J˜ z = Jy,t sin α + Jz,t cos α . ˜ a linear precession by α around These clearly display the sequence J t → J, the x-axis and J˜ → J t+1 , a nonlinear precession around the z-direction. Since the latter rotation is by an angle involving the z component of the intermediate vector ˜ symmetrized products as well as commutators of noncommuting operators occur J, in the corresponding equations in (7.6.4), { A, B} = ( AB + B A)/2. In the classical

266

7 Quantum Localization

limit, j → ∞, the first two equations in (7.6.4) simplify inasmuch as the contribution λ/2 j to the rotational angle is negligible and the symmetrized operator products become ordinary products of the corresponding c number observables while the commutators disappear. Formally, the classical limit can be achieved by first rescaling the operators J as X=

J j

(7.6.5)

whereupon the commutators (7.6.3) take the form [X, Y ] =

i Z. j

(7.6.6)

In the limit j → ∞, the vector X tends to a unit vector with commuting components, and the stroboscopic equations of motion take the form already described above, X t+1 = X˜ cos λ Z˜ − Y˜ sin λ Z˜ Yt+1 = X˜ sin λ Z˜ + Y˜ cos λ Z˜ Z t+1 = Z˜ X˜ = X t

(7.6.7)

Y˜ = Yt cos α − Z t sin α Z˜ = Yt sin α + Z t cos α . Due to the conservation law X 2 = 1, the classical map (7.6.7) is two dimensional, expressible as two recursion relations for two angles defining the orientation of X, for example, X = sin Θ cos φ , Y = sin Θ sin φ , Z = cos Θ .

(7.6.8)

Actually, the surface of the unit sphere X 2 = 1 is the phase space of the classical top with Z = cos Θ and φ as canonical variables. This can be seen most easily by replacing the quantum commutators (7.6.6) with the classical Poisson brackets {X, Y } = −Z and checking that the latter are equivalent to the canonical Poisson brackets {Z , φ} = 1. Indeed, from {Z , φ} = 1, one finds { f (Z ), g(φ)} = f (Z )g (φ), and this immediately gives {X, Y } = −Z when the spherical representation (7.6.8) is used for X and Y. The classical trajectories generated by the map (7.6.7) depend in their character on the values of the torsional strength λ and the precessional angle α. As shown in Fig. 7.3, the sphere X 2 = 1 is dominated by regular motion for α = π/2, λ< ∼ 2.5, whereas chaos prevails, at fixed α = π/2, for λ > ∼ 3. It is important to

7.6

Absence of Localization for the Kicked Top

267

Fig. 7.3 Classical phase space portraits for the kicked top (7.6.7) with α = π/2 and λ = 2 (a); λ = 2.5 (b); λ = 3 (c); and λ = 6 (d). Periodic orbits are marked by numerical labels

realize that as soon as chaos has become global with increasing torsional strength, the typical trajectory traverses the spherical phase space rapidly in time: a single kick suffices for the phase-space point to hop all around the sphere in any direction. Such stormy exploration is in blatant contrast to the behavior of the kicked rotator in its cylindrical phase space: While the cylinder may be surrounded in the direction of the angular coordinate once or even several times between two subsequent kicks, the momentum is capable of only slow quasi-random motion along the cylinder; the distance covered in the p-direction in a finite time is only a vanishing fraction of the infinite length of the cylinder.

268

7 Quantum Localization

Similarly striking is the difference in the quantum mechanical behavior of the top and of the rotator. The rotator displays localization along the one-dimensional angular momentum lattice, but the top, as we shall now proceed to show, does not. Rather, under conditions of classical chaos, wave packets pertaining to the top are in general limited in their spreads only by the finite size of the Hilbert space. To illustrate the typical behavior of the top, it is convenient to consider wave packets originating from coherent initial states. Coherent states of an angular momentum [39, 40] with the quantum number j assign a direction to the observables J that can be characterized by angles Θ and φ as Θφ|Jz |Θφ = j cos Θ 

Θφ|Jx ± iJy |Θφ = je±iφ sin Θ .

(7.6.9)

Clearly, the geometric meaning of these angles is the same as that of the angles used in (7.6.8) to specify the direction of the classical vector X. However, due to the noncommutativity of the components Ji , it is with a finite precision only that the coherent state |Θφ defines an orientation. To find that precision, one may observe that two of the states |Θφ have Θ = 0 and Θ = π, i.e., Jz  = ± j and therefore can be identified with the joint eigenstates | jm of J 2 and Jz pertaining to Jz = m = ± j and J 2 = j( j + 1). The relative variance of J with respect to these special coherent states is ( J 2  −  J2 )/j 2 = 1/j. All other states | jm with m = ± j have larger variances of J and so do all of their linear combinations (for fixed j) except the other coherent states |Θφ. In fact, all coherent states |Θφ can be generated from the “polar” state |Θ = 0, φ = | j, j by a rotation,4 |Θφ = R(Θ, φ)| j, j ,

(7.6.10)

   R(Θ, φ) = exp iΘ Jx sin φ − Jy cos φ ∗

= eγ J− e−Jz ln (1+γ γ ) e−γ J+ , where γ = eiφ tan (Θ/2) and J± = Jx ± iJy . The variance of J remains unchanged under the rotation in question, 

B 2  1 j = , Θφ| J 2 |Θφ − Θφ| J|Θφ2 j

(7.6.11)

and indeed constitutes the minimum uncertainty of the orientation of J permitted by the angular-momentum commutators. A solid angle ΔΩ = 1/j may be associated with the relative variance (7.6.11), and with respect to the classical unit sphere X 2 = 4

The reader will pardon the sloppiness of denoting the coherent state by |Θφ rather than | jΘφ.

7.6

Absence of Localization for the Kicked Top

269

1, a coherent state |Θφ may be represented by a spot of size ΔΩ located at the point (Θ, φ). Such spots of the minimal size allowed quantum mechanically are often called Planck cells. In the classical limit j → ∞, the spot in question shrinks to the classical phase-space point (Θ, φ). Some further properties of the states |Θφ are worth mentioning for later reference. First, we should note the expansion in terms of the states | jm. It follows from (7.6.10) that |γ  ≡ |Θφ = (1 + γ γ ∗ )− j eγ J− | j, j

(7.6.12)

and thus ∗ −j

6

 jm|Θφ = (1 + γ γ ) γ

j−m

2j j −m

 .

(7.6.13)

Therefore, the probability of finding Jz = m in the coherent state |Θφ is given by the binomial distribution | jm|Θφ|2 = (1 + γ γ ∗ )−2 j (γ γ ∗ ) j−m



2j j −m

 ,

(7.6.14)

which we shall use presently. The expression (7.6.12) for the coherent state allows us to easily calculate expectation values like (7.6.9). For instance, obviously, γ |J− |γ  = (1 + γ ∗ γ )−2 j

∂ 2 jγ ∗ (1 + γ ∗ γ )2 j = . ∂γ 1 + γ ∗γ

(7.6.15)

Next, by employing the easily checked identities e−γ J− Jz eγ J− = Jz − γ J− ,

e−γ J− J+ eγ J− = J+ + 2γ Jz − γ 2 J−

(7.6.16)

we get, similarly, γ |J− |γ  =

2 jγ ∗ , 1 + γ ∗γ

γ |Jz |γ  = j

 2 γ |J+ J− |γ  = γ |J− |γ  +

1 − γ ∗γ , 1 + γ ∗γ

2j . (1 + γ ∗ γ )2

(7.6.17)

A little trigonometry may finally be exercised to express these expectation values in terms of the angles Θ and φ recover (7.6.9).

270

7 Quantum Localization

1.0 Var(J/j) 0.5

0

0

50

j

100

Fig. 7.4 Time-dependent variance (Δ J/j)  for the kicked top (7.6.2) (α = π/2, λ = 3, j = 100) for a coherent initial state localized well outside classical islands of regular motion in Fig. 7.3c. The dotted line refers to the classical variance based on a bundle of 1000 classical trajectories 2

A coherent state |Θφ does not in general remain coherent under the time evolution generated by the Floquet operator F. Figure 7.4 depicts the variance of J with respect to a state F t |Θφ as the time grows. The behavior shown corresponds to the classical phase-space portrait of Fig. 7.3c, i.e., to α = π/2 and λ = 3 and to initial angles well outside the islands of regular motion. The angular-momentum quantum number was chosen as j = 100 to make the spot size ΔΩ = 1/j smaller than the solid angle range of the islands of regular motion. Several features of the full curve in Fig. 7.4 are worth noting. Figure 7.4 also displays the variance of the classical vector X t for a bundle of phase-space trajectories originating from a cloud of 1 000 initial points. The cloud was chosen to have uniform density and circular shape, and to be equal in location and size to the spot corresponding to the coherent initial state of the quantum top. Due to classical chaos, one would expect the classical and the quantum variance to become markedly different for times of the order ln j (a typical quantum uncertainty ∼ 1/j is amplified to order unity within such a time by exponentially separating chaotic trajectories); this expectation is consistent with the behavior displayed in Fig. 7.4. A further qualitative difference between the two curves in Fig. 7.4 becomes manifest for times exceeding the inverse mean level spacing (2 j + 1)/2π : While the classical curve becomes smooth, the quantum curve displays quasi-periodicity. Incidentally, the recurrent events are quite erratic in their sequence. Most importantly, there appears to be no limit to the spread of either the quantum wave packet or the classical cloud of phase-space points other than the finite size, respectively, of the Hilbert space and the phase space. Further light may be shed on the problem of localization if coherent states |Θφ are represented as vectors with respect to the eigenstate of the Floquet operator F, 

2 j+1

|Θφ =

μ=1

Cμ (Θ, φ)|μ .

(7.6.18)

7.6

Absence of Localization for the Kicked Top

271

Taking the basis vectors |μ as normalized, 

|Cμ |2 = 1 .

(7.6.19)

μ

Useful information can be gained by ordering the components Cμ according to decreasing modulus and truncating the representation (7.6.18) so as to include only the minimum number Nmin of basis vectors necessary to exhaust the normalization of |Θφ to within, say, 1 %. Numerical work [36] using α = π/2, λ = 3 indicates that Nmin scales quite differently with j, depending on whether the initial state lies in a region of classical chaos or in an island of classically regular motion, Nmin ∝ j x  1 chaotic x= 1 regular 2

(7.6.20)

The proportionality between Nmin and j for coherent states located in the classically chaotic region supports the conclusion that the Floquet eigenstates related to classical chaos are not confined in their angular spread to a region of solid angle smaller than the range of classical chaos on the unit sphere depicted in Fig. 7.3c. Moreover, Nmin ∼ j is precisely the result that random-matrix theory would suggest [41]. Conversely, the value 1/2 for the exponent x implies that Floquet eigenstates tend to be localized if they correspond to classically regular motion within islands around √ periodic orbits. Indeed, in the limit j → ∞, a vanishingly small fraction to build a coherent state located in such ∼ 1/ j of the full set of eigenstates suffices √ an island. The proportionality Nmin ∼ j is most easily understood when the island in question contains a fixed point of the classical map. A regular orbit surrounding such a fixed point must closely resemble linear precession of the vector X around the direction defined by the fixed point. Similarly, the eigenstates of the quantum dynamics must be well approximated by the states | jm provided the z-axis is oriented toward the classical fixed point in question. Then, the weight | jm|Θφ|2 of the approximate eigenstate | jm in the coherent state |Θφ is given by the binomial distribution (7.6.14) and that distribution is easily seen to have a width of the order √ 2 j when j is large. Numerical evidence for the scaling (7.6.20) is presented in Fig. 7.5, which again corresponds to the classical situation of Fig. 7.3c. The three curves pertain to j = 200, 400, and 500. The drop of Nmin as the coherent state enters the island of classically regular motion is so pronounced that Nmin might even be taken as an approximate quantum measure of the classical Lyapunov exponent. The plots are consistent with Nmin ∝ j 1/2 in the classically regular region and support Nmin ∝ j quite convincingly whereas the coherent state ranges in the domain of classical chaos.

272

7 Quantum Localization

j

j/2

0

0

1

Θy

2

3

Fig. 7.5 Minimum number of eigenmodes of the kicked top (7.6.2) necessary to exhaust the normalization of coherent states to within 1% versus the polar angle Θ y (with respect to positive y-axis) at which the coherent state is located; the azimuthal angle is fixed at φ y = π/4. Coupling < constants as in Fig. 7.3c. Classically, there is regular motion for 0.6 < ∼ Θ y ∼ 1.4 (the regular island in the lower right part of the y > 0 hemisphere in Fig. 7.3c). The curves follow the classical Lyapunov exponent better, the larger the j value (chosen here as 200, 400, and 500)

The “regular localization” and “chaotic delocalization” of the Floquet eigenvectors for the top5 have interesting consequences for the time evolution of expectation values. In both cases quasi-periodicity must become manifest after a time of the order j, i.e., the inverse of the average quasi-energy spacing 2π/(2 j + 1). However, as illustrated in Fig. 7.6, quasi-periodicity arises in two rather differentvarieties:

Fig. 7.6 Quasi-periodic behavior in time of Jx  for the kicked top with the Floquet operator (7.6.2) for α = π/2, λ = 3, j = 100. Initial states are coherent, see (7.6.10), localized within a classically regular island around a classical fixed point (see Fig. 7.3c) for curve (a) and in a classically chaotic region for curve (b). Note the near-periodic alternation of collapse and revival in the “regular” case (a) and the erratic variety of quasi-periodicity in the “irregular” case (b)

5 The top does not, in general, display the phenomenon of quantum localization under conditions of classical chaos; from this fact it may be understood that the localization length is larger than 2 j + 1; see, however, the subsequent section.

7.6

Absence of Localization for the Kicked Top

273

a nearly periodic sequence of alternate collapses and revivals modulates oscillations of Jx ; those oscillations correspond to orbiting of the wave packet around the fixed point in one of the islands of Fig. 7.3c; this nearly periodic kind of quasi-periodicity accompanies Nmin ∝ j 1/2 and may be interpreted as a quantum beat phenomenon dominated by a very small number of excited eigenmodes. When Nmin ∝ j, on the other hand, the wave packet has neither a fixed point to orbit around nor are there any regular modulations; instead, recurrent events form a seemingly erratic sequence; this type of behavior might be expected for broadband excitation of eigenmodes. Yet another quantum distinction of regular and chaotic motion follows from the different localization properties of the Floquet eigenvectors just discussed [42]. The “few” eigenvectors localized in an island of regular motion around a fixed point (as shown in Fig. 7.3c) should not vary appreciably when a control parameter, say λ, is altered a little, as long as the classical fixed point and the surrounding island of regular motion are not noticeably changed. On the other hand, an eigenvector that is spread out in a large classically chaotic region might be expected to react sensitively to a small change of λ, since it must respect orthogonality to the “many” other eigenfunctions spread out over the same part of the classical phase space. To check on the expected different degree of sensitivity to small changes of the dynamics, one may consider the time-dependent overlap of two wave vectors, both of which originate from one and the same coherent initial state but evolve according to slightly different values of λ, for example,    Θ, φ|F(λ , α)†t F(λ, α)t |Θφ 2 for t = 0, 1, 2, . . . .

(7.6.21)

Figure 7.7 displays that overlap for α = π/2, λ = 3, λ = 3(1 + 10−4 ). The classical phase-space portraits for these two sets of control parameters are hardly distinguishable and appear as in Fig. 7.3c. For the upper curve in Fig. 7.7, the initial state was chosen that it lies within one of the regular islands of Fig. 7.3c; in accordance with the above expectation, the overlap (7.6.21) remains close to unity for all times in this “regular” case. The lower curve in Fig. 7.7 pertains to an initial state lying well within the chaotic part of the classical phase space shown in Fig. 7.3c; the overlap (7.6.21), it is seen, falls exponentially from its initial value of unity down to a level of order 1/j. This highly interesting quantum criterion to distinguish regular and irregular dynamics, based on the overlap (7.6.21), was introduced by Peres [42]. Incidentally, the “regular” near-periodicity and “irregular” quasi-periodicity seen in Fig. 7.6 is again met in Fig. 7.7 on time scales of the order j. To summarize, the top and the rotator display quite different localization behavior. Under conditions of classical chaos, the eigenvectors of the rotator dynamics localize while those for the top do not. Somewhat different in character is the localization described above for eigenvectors confined to islands of regular motion; the latter behavior, well investigated for the top only, must certainly be expected for the rotator and other simple quantum systems as well.

274

7 Quantum Localization

Fig. 7.7 Overlap (7.6.18) of two wave vectors of the kicked top with the Floquet operator (7.6.2) for α = π/2, λ = 3, λ = 3(1 + 10−4 ), j = 1, 600. Initial coherent state as in Fig. 7.6a for upper curve (regular case) and as in Fig. 7.6b for lower curve. Note the extreme sensitivity to tiny changes of the dynamics in the irregular case. Courtesy of A. Peres [42]

7.7 The Rotator as a Limiting Case of the Top Now, I proceed to show that the top can be turned into the rotator by subjecting the torsional strength λ and the precessional angle α to a special limit [43]. From a classical point of view, that limit must confine the phase-space trajectory of the top

7.7

The Rotator as a Limiting Case of the Top

275

to a certain equatorial “waistband” of the spherical phase space to render the latter effectively indistinguishable from a cylinder. The corresponding quantum mechanical restriction makes part of the (2 j + 1)-dimensional Hilbert space inaccessible to the wave vector of the top: with the axis of quantization suitably chosen, the orientational quantum number m must be barred from the neighborhoods of the extremal values ± j. To establish the limit in question, one may look at the Floquet operators of the top FT = e−iλT Jz /(2 j+1) e−iα Jx / 2

2

(7.7.1)

and the rotator FR = e−i p

2

τ/2I  −iλR (I /τ ) cos Θ

e

.

(7.7.2)

To avoid possible confusion, all factors , τ, and I are displayed here. The reader should note that the sequence of the unitary factors for free rotation and kick has been reversed with respect to the previous discussion of the rotator; the operator FT defined in (7.7.2) describes the time evolution of the rotator from immediately before one kick to immediately before the next. This shift of reference is a matter of convenience for the present purpose. By their very appearance, the two Floquet operators suggest the correspondence λT =

I 1 τ j , α = λR . I τ j

(7.7.3)

For j → ∞ with τ, I, and λR fixed, this clearly amounts to suppressing large excursions of J away from the “equatorial” region Jz /j ≈ 0. To reveal the rotatorlike behavior of the classical top in the limit (7.7.3), we call upon the classical recursion relations (7.6.7) of the top setting Jz = p, Jx =  j cos Θ, and Jy =  j sin Θ and obtain p = p + λR τI sin Θ Θ = Θ + τI p .

(7.7.4)

These are indeed the equations of motion for the rotator. In fact, to bring (7.7.4) into the form (7.3.7), we must shift back the reference of time to let the free precession precede the nonlinear kick, Θ = Θt+1 , p = pt . The equivalence of the quantum mechanical Floquet operators (7.7.1) and (7.7.2) in the limit (7.7.3) becomes obvious when their matrix representations are considered with respect to eigenstates of Jz and p, respectively, Jz |m = m|m , − j ≤ m ≤ + j p|m = m|m , m = 0, ±1, ±2, . . . .

(7.7.5)

276

7 Quantum Localization

Then, the left-hand factors in FT and FR coincide once λT is replaced by  jτ/I according to (7.7.3). The exponents in the right-hand factors of FT and FR read @ α  A 1 3&   ( j − m)( j + m + 1)δm ,m+1 m  Jx  m = α  2 4 & + ( j + m)( j − m + 1)δm ,m−1 ,

(7.7.6)

 ? >     I I  1  m λR cos Θ  m = λR δm ,m+1 + δm ,m−1 . τ 2 τ These matrices become identical when the correspondence (7.7.3) is inserted and the limit j → ∞ with m/j → 0 is taken, the latter condition constituting the quantum mechanical analogue of a narrow equatorial waist-band of the classical sphere. It is important to realize that quantum localization under conditions of classical chaos, impossible when α and λT remain finite for j → ∞, arises in the limit (7.7.3). According to (7.5.6) the localization length can be estimated as l ≈ (λI /τ )2 /2 and for localization to take place this length must be small compared to j, l  j. The limit (7.7.3) clearly allows that condition to be met.

7.8 Problems 7.1 Derive the discrete-time Heisenberg equations for the kicked rotator. 7.2 The random-phase approximation of Sect. 7.3 does not assign Gaussian behavior to the force ξ = λV (Θ). Verify this statement by calculating ξ n  for V = cos Θ and uniformly distributed phases Θ. 7.3 The result of Problem 7.3 notwithstanding, the moments  pt2n  tend to display Gaussian behavior for t → ∞. Show this with the help of the random-phase approximation. 7.4 Rewrite the quantum map (7.3.17) in the Θ-representation. 7.5 Show that V (Θ) = −2 arctan (κ cos Θ − E) for the kicked rotator corresponds to nearestneighbor hops in the equivalent pseudo-Anderson model. ˜ )−1 with 7.6 Discuss the analyticity properties of Green’s function G(E) = (E − H ˜ given by (7.3.23). H

References

277

7.7 Generalize the simple Gaussian integrals 

:

+∞

−αs 2

ds e −∞



+∞

=

∂ =− ∂α

2 −αs 2

ds s e −∞

:

π α

π 1 = α 2α

:

π α

to the multiple integrals 

+∞



−

N

d se

i, j

: αi j Si S j

−∞



+∞

−∞

N

−

d S S1 S2 e

 i, j

αi j Si S j

= 

=

πN det α

1 2α

 1,2

:

πν det α

for a positive symmetric matrix α. Use the eigenvectors and eigenvalues of α. 7.8 Diagonalize the matrix Vi j = κ(δi, j+1 + δi, j−1 )/2. 7.9 Use (7.4.29) and (7.4.30) to determine the eigenvalues of the matrix Vi j = κ(δi, j+1 + δi, j−1 )/2. Locate the poles of Green’s function given in (7.4.24). What happens to these poles in the limit N → ∞? 7.10 Verify the stroboscopic Heisenberg equation (7.6.4) for the kicked top with the Floquet operator (7.6.2). 7.11 Determine the amplitude of the temporal fluctuations of var ( J)/j 2 from the point of view of random-matrix theory.

References 1. S. Fishman, D.R. Grempel. R.E. Prange: Phys. Rev. Lett. 49, 509 (1982); Phys. Rev. A29, 1639 (1984) 2. D.L. Shepelyansky: Phys. Rev. Lett. 56, 677 (1986) 3. G. Casati, J. Ford, I. Guarneri, F. Vivaldi: Phys. Rev. A34, 1413 (1986) 4. S.-J. Chang, K.-J. Shi: Phys. Rev. A34, 7 (1986) 5. M. Feingold, S. Fishman, D.R. Grempel, R.E. Prange: Phys. Rev. B31, 6852 (1985) 6. M.V. Berry, S. Klein: Eur. J. Phys. 18, 222 (1997) 7. T. Dittrich, U. Smilansky: Nonlinearity 4, 59 (1991);ibid. 4, 85 (1991) 8. N. Argaman, Y. Imry, U. Smilansky: Phys. Rev. B47, 4440 (1993) 9. K.B. Efetov: Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997)

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10. 11. 12. 13.

Y. Imry: Introduction to Mesoscopic Physics (Oxford University Press, Oxford, 1997) J.E. Bayfield, P.M. Koch: Phys. Rev. 33, 258 (1974) M.R.W. Bellermann, P.M. Koch, D.R. Mariani, D. Richards: Phys. Rev. Lett. 76, 892 (1996) O. Benson, G. Raithel, H. Walther: In G. Casati, B. Chirikov: Quantum Chaos (Cambridge University Press, Cambridge, 1995) R. Bluemel, W.P. Reinhardt: Chaos in Atomic Physics (Cambridge University Press, Cambridge, 1997) G. Casati, B. Chirikov: Quantum Chaos (Cambridge University Press, Cambridge, 1995) R. Graham, M. Schlautmann, P. Zoller: Phys. Rev. A 45, R19 (1992) F.L. Moore, J.C. Robinson, C. Bharucha, P.E. Williams, M.G. Raizen: Phys. Rev. Lett. 73, 3974 (1994) J.C. Robinson, C. Bharucha, F.L. Moore, R. Jahnke, G.A. Georgakis, Q. Niu,, M.G. Raizen, B Sundaram: Phys. Rev. Lett. 74, 3963 (1995) F.L. Moore, J.C. Robinson, C. Bharucha, B Sundaram, M.G. Raizen: Phys. Rev. Lett. 75, 4598 (1995) H.-J. St¨ockmann: Quantum Chaos, An Introduction (Cambridge University Press, Cambridge, 1999) B. Kramer, A. MacKinnon: Rep. Prog. Phys. 56, 1469 (1993) P.W. Anderson: Phys. Rev. 109, 1492 (1958); Rev. Mod. Phys. 50, 191 (1978) D.J. Thouless: In Session XXXI, 1979, Ill-Condensed Matter R. Balian, R. Maynard, G. Thoulouse (eds.) Les Houches, (North-Holland, Amsterdam, 1979) K. Ishii: Prog. Th. Phys. Suppl. 53, 77 (1973) H. Furstenberg: Trans. Ann. Math. Soc. 108, 377 (1963) A. Crisanti, G. Paladin, A. Vulpiani: Products of Random Matrices (Springer, Berlin, 1993) B.V. Chirikov: preprint no. 367, Inst. Nucl. Physics Novosibirsk (1969); Phys. Rep. 52, 263 (1979) T. Dittrich, R. Graham: Ann. Phys. 200, 363 (1990) H. Weyl: Math. Ann. 77, 313 (1916) P. Lloyd: J. Phys. C2, 1717 (1969) D.J. Thouless: J. Phys. C5, 77 (1972) S.F. Edwards, P.W. Anderson: J. Phys. F5, 965 (1975) B.V. Chirikov, F.M. Izrailev, D.L. Shepelyansky: Sov. Sci. Rev. 2C, 209 (1981) D.L. Shepelyansky: Phys. Rev. Lett. 56, 677 (1986) F. Haake, M. Ku´s, J. Mostowski, R. Scharf: In F. Haake, L.M. Narducci, D.F. Walls (eds.): Coherence, Cooperation, and Fluctuations (Cambridge University Press, Cambridge, 1986) F. Haake, M. Ku´s, R. Scharf: Z. Phys. B65, 381 (1987) M. Ku´s, R. Scharf, F. Haake: Z. Phys. B66, 129 (1987) R. Scharf, B. Dietz, M. Ku´s, F. Haake, M.V. Berry: Europhys. Lett. 5, 383 (1988) F.T. Arecchi, E. Courtens, G. Gilmore, H. Thomas: Phys. Rev. A6, 2211 (1972) R. Glauber, F. Haake: Phys. Rev. A13, 357 (1976) K. Zyczkowski: J. Phys. A23, 4427 (1990) A. Peres: Quantum Theory: Concepts and Methods (Kluwer Academic, New York, 1995) F. Haake, D.L. Shepelyansky: Europhys. Lett. 5, 671 (1988)

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43.

Chapter 8

Dissipative Systems

8.1 Preliminaries Regular classical trajectories of dissipative systems eventually end up on limit cycles or settle on fixed points. Chaotic trajectories, on the other hand, approach so-called strange attractors whose geometry is determined by Cantor sets and their fractal dimension. In analogy with the Hamiltonian case, the two classical possibilities of simple and strange attractors are washed out by quantum fluctuations. Nevertheless, genuinely quantum mechanical distinctions between regular and irregular motion can be identified. The main goal of this chapter is the development of one such distinction, based on the generalization of energy levels to complex quantities whose imaginary parts are related to damping. An important time scale separation arising for dissipative quantum systems will also be expounded: Coherences between macroscopically distinct states tend to decay much more rapidly than quantities with well-defined classical limits. An example relevant in the present context is the dissipative destruction of quantum localization for the kicked rotator. For background information on dissipation the reader may consult the comprehensive treatise of U. Weiss [1]. More special issues related to quantum chaos are dealt with by D. Braun [2].

8.2 Hamiltonian Embeddings Any dissipative system S may be looked upon as part of a larger Hamiltonian system. In many cases of practical interest, the Hamiltonian embedding involves weak coupling to a heat bath R whose thermal equilibrium is not noticeably perturbed by S. Strictly speaking, as long as S + R is finite in its number of degrees of freedom, the Hamiltonian nature entails quasi-periodic rather than truly irreversible temporal behavior. However, for practical purposes, quasi-periodicity and true irreversibility cannot be distinguished in the “effectively” dissipative systems S in question. The equilibration time(s) imposed on S by the coupling to R are typically large compared to the time scales τR of all intrabath processes probed by the coupling; in such situations, to which discussions will be confined, the dissipative motion of S

F. Haake, Quantum Signatures of Chaos, Springer Series in Synergetics, 3rd ed., C Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-05428-0 8, 

279

280

8 Dissipative Systems

acquires a certain Markovian character. Then, the density operator (t) of S obeys a “master” equation of the form (t) ˙ = l(t)

(8.2.1)

with a suitable time-independent generator l of infinitesimal time translations. The dissipative motion described by (8.2.1) must, of course, preserve the Hermiticity, positivity, and normalization of . Now, I shall proceed to sketch how the master equation (8.2.1) for the density operator  of S can be derived from the microscopic Hamiltonian dynamics [3, 4] of S + R. The starting point is the Liouville–von Neumann equation for the density operator W (t) of S + R, ˙ (t) = − i [H, W (t)] ≡ L W (t). W 

(8.2.2)

Here H denotes the Hamiltonian which comprises “free” terms HS and HR for S and R, as well as an interaction part HSR , H = HS + HR + HSR .

(8.2.3)

A similar decomposition holds for the Liouvillian L . The formal integral of (8.2.2) reads W (t) = e Lt W (0).

(8.2.4)

To slightly simplify the algebra to follow, we assume stationarity with respect to HR and the absence of correlations between S and R for the initial density operator, W (0) = (0)R, [HR , R] = 0.

(8.2.5)

Inasmuch as the bath is macroscopic, it is reasonable to require that R = Z R−1 e−β HR , TrR {R} = 1.

(8.2.6)

The formal solution (8.2.4) of the initial-value problem (8.2.2, 8.2.5) suggests the definition of a formal time-evolution operator U (t) for S, (t) = U (t)(0), U (t) = TrR {e Lt R}.

(8.2.7)

Due to the coupling of S and R, this U (t) is not unitary. Assuming that U (t) possesses an inverse, we arrive at an equation of motion for (t),

8.2

Hamiltonian Embeddings

281

(t) ˙ = l(t)(t),

(8.2.8)

l(t) = U˙ (t)U (t)−1 ,

in which l(t) may be interpreted as a generator of infinitesimal time translations. In general, l(t) will be explicitly time-dependent but will approach a stationary limit l = lim U˙ (t)U (t)−1

(8.2.9)

t→∞

on the time scale τR characteristic of the intrabath processes probed by the coupling. On the much larger time scales typical for S, the asymptotic operator l generates an effectively Markovian process. Only in very special cases can l be constructed rigorously [5, 6]. A perturbative evaluation with respect to the interaction Hamiltonian HSR is always feasible and in fact quite appropriate for weak coupling. Assuming for simplicity that TrR {HSR R} = 0,

(8.2.10)

one easily obtains the perturbation expansion of l as  l = LS +

∞

  dt TrR L SR e(L R +L S )t L SR e−(L R +L S )t R

(8.2.11)

0

where third- and higher order terms are neglected (Born approximation). The few step derivation of (8.2.11) uses the expansion of e Lt in powers of L SR , e Lt = e(L R +L S )t +



t





dt e(L R +L S )t L SR e(L R +L S )(t−t ) + . . . ,

(8.2.12)

0

the identities TrR {L S ( . . . )} = L S TrR {( . . . )}, TrR {L R ( . . . )} = 0, and the assumed properties (8.2.5) and (8.2.10) of R and HSR . Of special interest for the remainder of this chapter will be an application to spin relaxation. Therefore, we shall work out the generator l for that case. The spin or angular momentum is represented by the operators Jz , J± which obey [J+ , J− ] = 2Jz , [Jz , J± ] = ±J± .

(8.2.13)

The free spin dynamics may be generated by the Hamiltonian HS = ω Jz

(8.2.14)

and for the spin bath interaction, we take   HSR =  J+ B + J− B †

(8.2.15)

282

8 Dissipative Systems

where B and B † are a pair of Hermitian-conjugate bath operators. We shall speak of the observables entering the interaction Hamiltonian as coupling agents. The Hamiltonian of the free bath need not be specified. By inserting HSR from (8.2.14) in the second-order term of l given in (8.2.11), we obtain 

∞

l (2)  = 0





dt B(t)B † eiωt [J− , J+ ] ∞

+

 dt B † B(t) eiωt [J+ ,  J− ] + H.c..

(8.2.16)

0

Interestingly, the c-number coefficients appearing here are the Laplace transforms of the bath correlation functions B(t)B †  and B † B(t), taken at the frequency ω of the free precession (apart from the imaginary unit i). The correlation functions refer to the reference state R of the bath which is assumed to be the thermal equilibrium state given in (8.2.6); their time dependence is generated by the free-bath Hamiltonian HR . For the sake of simplicity, I have assumed, in writing (8.2.16), that B(t)B(0) = B † (t)B † (0) = 0. The Laplace transforms of the bath correlation functions are related to their Fourier transforms by $∞ 0

dt eiωt f (t) = f (ω) =

1 2

f (ω) +

$ +∞ −∞

i P 2π

$ +∞ −∞

f (ν) dν ω−ν ,

dteiωt f (t),

(8.2.17)

where f (t) stands for either B(t)B †  or B † B(t) and P denotes the principal value. Furthermore, since one and the same bath Hamiltonian HR generates the time dependence of B(t) and determines the canonical equilibrium (8.2.6), the well-known fluctuation-dissipation theorem holds in the form [7] $ +∞ −∞ $ +∞ −∞

dteiωt B(t)B †  = 2κ(ω) [1 + n th (ω)] dteiωt B † B(t) = 2κ(ω)n th (ω)

(8.2.18)

where  2κ(ω) =

+∞

 dteiωt [B(t), B † ]

(8.2.19)

−∞

is a real Fourier-transformed bath response function and n th (ω) =

1 eβω − 1

(8.2.20)

is the average number of quanta in an oscillator of frequency ω at temperature 1/β. Combining (8.2.11), (8.2.14), (8.2.15), (8.2.16), (8.2.17), (8.2.18), and (8.2.19) we obtain the master equation

8.2

Hamiltonian Embeddings

283

 ˙ = l = −i (ω + δ + δth )Jz − δ Jz2 ,  +κ(1 + n th ) {[J− ,  J+ ] + [J− , J+ ]} +κn th {[J+ ,  J− ] + [J+ , J− ]}

(8.2.21)

with the “frequency shifts” δ=

1 P π



+∞

dν −∞

κ(ν) 1 , δth = P ω−ν π



+∞

dν −∞

n th (ν)κ(ν) . ω−ν

(8.2.22)

This master equation was first proposed to describe superfluorescence in [8] and put to experimental test in that context in [9]. The first commutator in (8.2.21) describes a reversible motion according to a Hamiltonian (ω + δ + δth )Jz − δ Jz2 ; the first term of this generates linear precession around the z-axis at a shifted frequency ω + δ + δth , whereas the second term yields a nonlinear precession around the z-axis. The remaining terms in the master equation are manifestly irreversible. In view of the temperature dependence of n th , the stationary solution ¯ of the master equation (8.2.21) must be separately stationary with respect to both the reversible and the irreversible terms. While the reversible terms admit any function of Jz , the irreversible ones single out ¯ = Z S−1 e−β HS ,

(8.2.23)

i.e., the unperturbed canonical operator with the temperature ∝ 1/β equalling that of the bath. The stationarity of this ¯ may be checked by a little calculation with the help of the identity J± ex Jz = e±x e−x Jz J±

(8.2.24)

but in fact follows from very general arguments [10]: Due to the structure of (8.2.21), ¯ must be independent of δ and κ, i.e. of zeroth order in HSR . In the limit of vanishing coupling, however, the angular momentum and the bath become statistically independent and their total energy tends to the sum of the respective unperturbed parts. The only function of HS or, equivalently, Jz meeting these requirements is the exponential (8.2.23). The master equation obviously respects the conservation law for the squared angular momentum, as indeed it must since [HS + HSR + HR , J 2 ] = 0, J 2 = j( j + 1) = const.

(8.2.25)

Once the quantum number j is fixed, the Hilbert space is restricted to 2 j + 1 dimensions, and the density operator is representable by a (2 j + 1) × (2 j + 1) matrix. A convenient basis is provided by the joint eigenvectors | jm of J 2 and Jz pertaining to the respective eigenvalues j( j + 1) and m. By employing

284

8 Dissipative Systems

J± | jm =

&

( j ∓ m) ( j ± m + 1) | j, m ± 1,

(8.2.26)

the master equation (8.2.21) is easily rewritten as a set of coupled differential equations for the matrix elements mm (t). Since no phase is distinguished for the polarizations J± , there is a closed set of “rate equations” for the probabilities mm (t) ≡ m (t),  ˙ m (t) = 2κ(1 + n th ) ( j − m)( j + m + 1)m+1 − ( j − m + 1)( j + m)m  +2κn th ( j − m + 1)( j + m)m−1 − ( j − m)( j + m + 1)m . (8.2.27) These actually form a master equation of the Pauli type with energy-lowering transition rates w(m + 1 → m) = 2κ(1 + n th )( j − m)( j + m + 1) and upward rates w(m → m + 1) = 2κn th ( j − m)( j + m + 1). As revealed by the factor 1 + n th , downward transitions of the angular momentum are accompanied by spontaneous or induced emissions of quanta ω into the bath. Upward transitions, on the other hand, require the absorption of quanta ω from the bath and thus are proportional to n th . Of course, neither absorption nor induced emission can take place when the heat bath temperature is too low, i.e., when kB T  ω. In this latter limit, the angular momentum keeps dissipating its energy, i.e., emitting quanta into the bath, until it settles into the ground state, m = − j. It is as well to note in passing that the master equation (8.2.21) or (8.2.27) is nothing but a slightly fancy version of Fermi’s golden rule. Indeed, the transition rates w(m + 1 → m) and w(m → m + 1) may be calculated directly with that rule; then, the constant κ(ω) ≡ κ appears as proportional to the number of bath modes capable, by resonance, of receiving quanta ω emitted by the precessing angular momentum. I should also add a word about the limit of validity of the master equation (8.2.21). Due to the perturbative derivation, we are confined to the case of small coupling, κ, δ  ω,

(8.2.28)

in which the free precession is only weakly perturbed. Moreover, the damping constant κ, in energy units, must be smaller than the bath temperature, κ  kB T.

(8.2.29)

To explain this latter restriction, it is useful to consider (8.2.20) and to replace the free-precession frequency ω by a complex variable z. The function n th (z) has imaginary poles at βz μ = 2π iμ, μ = 0, ±1, ±2, . . . . Then, one may reconstruct the correlation functions B(t)B †  and B † B(t) by inverting the Fourier transforms (8.2.18). Closing the frequency integrals in the upper half of the z plane, as is necessary for t > 0, one encounters thermal transients eizμ t = e−|zμ |t ; these transients

8.3

Time-Scale Separation for Probabilities and Coherences

285

must decay much faster than e−κt or else the assumed time-scale separation for the bath and the damped subsystem is violated, and the asymptotic generator l becomes meaningless. A limit of special relevance for the remainder of this chapter is that of low temperatures, n th  1, and large angular momentum, j  1. In this quasi-classical low-temperature regime, classical trajectories become relevant as a reference behavior about which the quantum system displays fluctuations. Classical dynamics may be obtained from (8.2.21) by first extracting equations of motion for the mean values  J(t) and then factorizing as Jz J±  = Jz J±  etc. In the limit of negligible n th and with the help of a spherical-coordinate representation, Jz  = j cos Θ, J±  = j sin Θe±iφ ,

(8.2.30)

classical dynamics turns out to be that of the overdamped pendulum, ˙ = Γ sin Θ, φ˙ = 0, Θ Γ = 2κ j.

(8.2.31)

Evidently, the classical limit must be taken as j → ∞ with constant Γ . Then, the angle Θ relaxes toward the stable equilibrium Θ(∞) = π according to tan

Θ(0) Θ(t) = eΓ t tan . 2 2

(8.2.32)

8.3 Time-Scale Separation for Probabilities and Coherences Off-diagonal density matrix elements between states differing in energy by not too many quanta display lifetimes of the same order of magnitude, under the dissipative influence of heat baths, as diagonal elements. A typical example is provided by the damped angular momentum considered above, when the quantum number j is specified as j = 1/2. The damping rates for the off-diagonal element 1/2,−1/2 and the population probability 1/2,1/2 are easily found from (8.2.21), respectively, as γ⊥ = κ(1 + 2n th ) γ = 2κ(1 + 2n th ),

(8.3.1)

i.e., they are indeed of the same order of magnitude. A different situation arises for large quantum numbers. Coherences between mesoscopically or even macroscopically distinct states usually have lifetimes much shorter than those of occupation probabilities. I propose to illustrate that phenomenon of “accelerated decoherence” for an angular momentum subject to the low-temperature version (n th = 0) of the damping process (8.2.21); it suffices to consider the dissipative part of that process whose generator Λ is defined by

286

8 Dissipative Systems

˙ = Λ = κ



J− ,  J+ ] + [J− , J+ ]} ;

(8.3.2)

the classical limit of that process was characterized above in (8.2.31) and (8.2.32). For reasons to be explained presently, it is convenient to start with a superposition of two coherent states (7.6.12), |  = c|Θφ + c |Θ φ 

(8.3.3)

where the complex amplitudes c and c are normalized as |c|2 + |c |2 = 1. In the limit of large j, the two component states may indeed be said to be macroscopically distinguishable; the superposition is often called a Schr¨odinger cat state, reminiscent of Schr¨odinger’s bewilderment with the notorious absence of quantum interference with macroscopically distinct states from the classical world. In fact, the phenomenon of accelerated decoherence that we are about to reveal explains why such interferences are usually impossible to observe. The rigorous solution exp{Λt}|  | of the master equation (8.3.2) originating from the initial density operator |  | for the cat state (8.3.3) can be constructed without much difficulty; see [11–15]. However, we reach our goal more rapidly by considering the piece Θφ,Θ φ (t) = exp{Λt}|ΘφΘ φ |

(8.3.4)

and estimating its lifetime from the initial time rate of change of its norm NΘφ,Θ φ (t) = TrΘφ,Θ φ (t)† Θφ,Θ φ (t) .

(8.3.5)

By differentiating w.r.t. time, setting t = 0, inserting the generator Λ, and using (7.6.17), we immediately get   N˙ Θφ,Θ φ (t) = −(2κ j) j sin2 Θ + sin2 Θ − 2 cos(φ − φ ) sin Θ sin Θ

  (8.3.6) + 12 (1 + cos Θ)2 + (1 + cos Θ )2 . Now, the announced time-scale separation may be read off: The probabilities described by Θ = Θ , φ = φ have the classical rate of change Γ = 2κ j; in contrast, for sin Θ = sin Θ and/or φ = φ , i.e., coherences, the first square bracket in the foregoing rate does not vanish and carries the “acceleration factor” j. Inasmuch as j is large, we conclude that the cat state (8.3.3) rapidly (on the time scale 1/jΓ ) decoheres to a mixture, 

  c|Θφ + c |Θ φ  cΘφ| + c Θ φ | −→ |c|2 |ΘφΘφ| + |c |2 |Θ φ Θ φ | ,

(8.3.7)

while the weights of the component states begin to change only much later at times of the order 1/Γ .

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Time-Scale Separation for Probabilities and Coherences

287

An exception of the general rule of accelerated decoherence is highly interesting: Coherences between coherent states which lie symmetrically to the “equator” (Θ = π − Θ) on one and the same “great circle” (φ = φ) are exempt from the rule, simply because sin Θ = sin(π − Θ). These give rise to long-lived Schr¨odinger cat states (8.3.3) [14]. A deeper reason for the immunity of these exceptional cat states will appear once we have clarified the distinction of the coherent states for the process under consideration. To that end, we must come back to the coupling agents J± entering the interaction Hamiltonian (8.2.15). These happen to have coherent states as approximate eigenstates in the semiclassical limit of large j, J± |Θφ ≈ je±iφ sin Θ |Θφ

if

tan2 Θ  1/j.

(8.3.8)

That property is worth checking since the notion of an approximate eigenvector is not met all too often. But indeed, it is easily seen, again with the help of (7.6.17), that the two vectors J± |Θφ and je±iφ sin Θ |Θφ have norms with a relative difference of order 1/j and√ comprise an angle (in Hilbert space) in between themselves which is of order 1/ j, provided that the coherent state in question does not appreciably overlap the polar ones at Θ = 0 and Θ = π . The approximate eigenvalues e±iφ sin Θ are doubly degenerate, and the two pertinent approximate eigenstates have the same azimuthal angle φ whereas the polar angles are symmetric about π/2. The Schr¨odinger cat state with equatorial symmetry cannot be rapidly decohered by the damping mechanism in consideration since its two component states are precisely such a doublet. To appreciate the foregoing reasoning better, we may think of the unitary motion generated by some Hamiltonian that has a degenerate eigenvalue E. An initial superposition of states from the degenerate subspace will subsequently acquire the overall prefactor exp(−iEt/) but will not undergo any internal change. Or, to stay with dissipative motions, we may consider an interactive Hamiltonian HSR = X B with Hermitian coupling agents X and B for the system S and the reservoir R, respectively. Then, the Fermi golden rule type arguments of Sect. 8.2 yield a master equation with the dissipative generator   Λ = (κ/x02 ) [X,  X ] + [X , X ]

(8.3.9)

where κ is a rate constant and x 0 is a constant of the same dimension as the coupling agent X . Obviously now, occupation probabilities of eigenstates |x of the coupling agent X are not influenced at all by the damping: Λ|xx| = 0. Coherences are 2  affected, however, since Λ|xx | = κ (x − x )/x0 |xx |, and the proportionality of the decoherence rate to (x − x )2 /x02 signals accelerated decoherence. Acceleration would fail only for superpositions of eigenstates |x, α belonging to one degenerate eigenvalue x. In principle, such states could still be very different, even macroscopically, as suggested by the toy example X = Y 2 for which the eigenstates | ± y of Y form a doublet with common eigenvalue y 2 of X .

288

8 Dissipative Systems

The phenomenon considered here is universal. The unobservability of superpositions of different pointer positions in measurement devices finds its natural explanation here and so does the reduction of superpositions to mixtures in microscopic objects coupled to macroscopic measurement or preparative apparatuses [16–19]. Examples of current experimental interest include superconducting quantum interference devices (SQUIDS), Bloch oscillations in Josephson junctions, and optical bistability. The common goal of such investigations is to realize the largest possible distinction, hopefully mesoscopic, between two quantum states that still gives rise to detectable coherences in spite of weak dissipation. The first actual observations of accelerated decoherence were reported for photons in microwave cavities in [20]. A negative implication of the phenomenon in question also deserves mention. The classical alternatives of regular and chaotic motion can be reflected, as shown in Chap. 7, in different appearances of the sequence of quantum recurrences in the time evolution of certain expectation values. Quantum recurrence events of whatever appearance, however, are constructive interference phenomena requiring preservation of coherences between successive events. Inasmuch as recurrence events involve coherences between classically distinct states, rather feeble damping suffices to destroy the phase relations necessary for substantial constructive interference.

8.4 Dissipative Death of Quantum Recurrences We proceed to a quantitative analysis of recurrences in the presence of feeble damping [21–23]. Due to the finiteness of its Hilbert space, the kicked top will be a convenient example to work with again. The main conclusions, however, will be of quite general validity. The clue to the treatment lies in the fact that very weak dissipation suffices to suppress recurrence events. Thus, the damping may be considered a small perturbation of the unitary part of the dynamics. Furthermore, under conditions of classical chaos ideas borrowed from random-matrix theory allow a full implementation of first-order perturbation theory. The kicked top to be employed here must be described now by a dissipative map for the density operator (t) after the tth kick, (t + 1) = eΛ e L (t)

(8.4.1)

where e L represents the unitary motion discussed in previous chapters, e L  = FF † F = e−i p Jz e−iλJy /(2 j+1) . 2

(8.4.2)

For the sake of convenience, the axis of the nonlinear precession is now taken to be the y-axis and that of the linear precession to be the z-axis. The nonunitary factor eΛ describes the dissipative part of the evolution. To keep the situation simple, we

8.4

Dissipative Death of Quantum Recurrences

289

will assume low temperatures, n th ≈ 0, and neglect the frequency shift terms, δ = 0. Then, the generator Λ is obtained from (8.2.21) as Λ =

Γ ([J− ,  J+ ] + [J− , J+ ]) 2j

(8.4.3)

except that here the classical damping constant Γ = 2 jκ is used to facilitate comparison with the classical behavior (8.2.31) and (8.2.32). The free-precession term in (8.2.21) is not included in Λ but is written instead as a separate factor in the unitary evolution operator e L ; no approximation is involved in that separation since Λ[Jz , ] = [Jz , Λ]. Thus, the whole dissipative quantum map (8.4.1) may be said to allow linear precession around the z-axis with concurrent damping, whereas the nonlinear precession prevails during a separate phase of the driving period. Note that the time t = 0, 1, 2, . . . and the coupling constants Γ, λ, p are all taken as dimensionless and that  = 1. In the weak-damping limit, the generator of the map can be approximated as eΛ e L = (1 + Λ + . . . )e L .

(8.4.4)

With the eigenstates of the unitary Floquet operator F denoted by |μ, the dyadic eigenvectors of e L read |μν| and the corresponding unimodular eigenvalues exp (−iφμ + iφν ). To first order in Λ, the perturbed eigenvalue, for μ = ν, is λμν = e−i(φμ −φν ) (1 − Λμν ) ≈ e−i(φμ −φν )−Λμν , μ = ν,

(8.4.5)

where the real damping constant   Λμν = −Tr (|μν|)† Λ|μν| Γ = (−2μ|J− |μν|J+ |ν + ν|J+ J− |ν + μ|J+ J− |μ) 2j

(8.4.6)

is the μν element of the dyad −Λ|μν|; the latter is the infinitesimal increment of |μν| generated by Λ during the time interval dt, divided by dt. With Λ expressed in the | j, m basis, the damping constant Λμν takes the form

Λμν =

+j   a(m, n) |μ| j, m|2 |ν| j, n|2 m,n = − j

−b(m, n)μ| j, m j, m + 1|μν| j, n + 1 j, n|ν ,

(8.4.7)

290

8 Dissipative Systems

Γ [( j + m)( j − m + 1) + ( j + n)( j − n + 1)] 2j Γ b(m, n) = [( j − m)( j + m + 1)( j − n)( j + n + 1)]1/2 . j

a(m, n) =

Once the eigenvectors |μ of the Floquet operator F are known, the sums in (8.4.7) can be evaluated. The classical dynamics implied by (8.4.1), (8.4.2), and (8.4.3) in the limit j → ∞ is chaotic on the overwhelming part of the sphere ( J/j)2 = 1 for λ ≈ 10, p ≈ π/2, Γ ≈ 10−4 , the case to be studied in the following. The strange attractor accommodating the chaotic trajectories covers the classical sphere almost uniformly. Of course, for dissipation to become manifest in the classical dynamics, a time of the order 1/Γ = 104 must elapse, and the strangeness of the attractor is revealed only on even larger time scales. In the situation of global classical chaos chosen here, the Floquet operator F may be considered a random matrix drawn from Dyson’s circular orthogonal ensemble. Then, the second term in Λμν , as given by (8.4.7), is, for j  1, unable to make a non-zero contribution to the sum over m, n since the individual eigenvector components are random in sign. The remaining term in Λμν may be simplified as

Λμν =

+j   Γ  ( j + m)( j − m + 1) |μ| j, m|2 + |ν| jm|2 . 2 j m =−j

(8.4.8)

The damping constants Λμν now appear to be random numbers. Due to the sum √ over m, however, the relative root-mean-square deviation from the mean is ∼ 1/ j and to within this accuracy the Λμν are in fact all equal to one another and to their ensemble mean (Problem 8.8). Therefore, they must also equal their arithmetic mean,

Λμν = =

2 j+1  Γ ( j + m)( j − m + 1) |μ| j, m|2 j(2 j + 1) μ = 1 m

 Γ ( j + m)( j − m + 1) j(2 j + 1) m

= 23 Γ ( j + 1) ≈ 23 Γ j, μ = ν.

(8.4.9)

In short, the random numbers Λμν are self-averaging in the limit of large j. The result just obtained is in fact quite remarkable. It is independent of the coupling constants λ and p and even comes with the claim of validity for all tops with global chaos in the classical limit which are weakly damped according to (8.4.3). Furthermore, since Λμν turns out to be independent of μ and ν for μ = ν, all off-diagonal dyadic eigenvectors |μν| of e L display the same attenuation. Finally,

8.4

Dissipative Death of Quantum Recurrences

291

the proportionality of Λμν to j reflects the the phenomenon of accelerated decoherence. The life time of the coherences Λμν is ∼ 1/Γ j while classically the damping is noticeable only on a time scale larger by a factor j. For perturbation theory to work well, the damping constants Λμν must be small. As to how small, we may argue that our first-order estimate should not be outweighed by second-order corrections. In the latter, small denominators of order 1/j 2 appear since 2 j(2 j + 1) nonvanishing zero-order eigenvalues φμ − φν lie in the accessible (2π )-interval. Thus, we are led to require that

Γ 

1 . j3

(8.4.10)

A word about the diagonal dyads |μμ| may be in order. They are all degenerate in the conservative limit, because they are eigenvectors of e L with eigenvalue unity. The fate of that eigenvalue for weak damping must be determined by degenerate perturbation theory, i.e., by diagonalizing Λ in the (2 j + 1)-dimensional space spanned by the |μμ|, a task in fact easily accomplished. The elements of the (2 j + 1) × (2 j + 1) matrix in question read Tr {|μμ|(Λ|νν|)}  Γ  δμν μ|J+ J− |μ − |μ|J− |ν|2 =− j

 Γ |μ|m|2 ( j + m)( j − m + 1) δμν =− j m  1/2 − ( j + m)( j − m + 1)( j + m )( j − m + 1) mm

 × ν| j, m − 1 j, m|μμ| j, m  j, m − 1|ν .

(8.4.11)

They are quite similar in structure to the damping constants Λμν given in (8.4.7). As was the case for the latter, random-matrix theory can be invoked in evaluating the various sums in (8.4.11), provided there is global chaos in the classical limit. Due to the sums over m and m , all matrix elements (8.4.11) are again self-averaging, i.e., have root-mean-square deviations √ from their ensemble mean that are smaller than the mean by a factor of order 1/ j. In the ensemble mean, only the terms with m = m in the double sum contribute. For the off-diagonal elements, μ = ν, the probabilities | jm|μ|2 and | j, m − 1|ν|2 may be considered independent, and therefore both ensembles average to 1/(2 j + 1) ≈ 1/2 j. For the diagonal elements, μ = ν, the contribution of the second sum in (8.4.11) need not be considered further since it is smaller than the first by a factor of order 1/j.

292

8 Dissipative Systems

Thus, the matrix to be diagonalized reads   2 1 − δμν μ |(Λ|νν|)| μ = − Γ j δμν − . 3 2j

(8.4.12)

Its diagonal elements precisely equal the eigenvalues −Λμν given in (8.4.9) while its off-diagonal elements are smaller by the factor 1/2 j. Interestingly, in spite of their relative smallness, the off-diagonal elements may not be neglected since probability conservation requires that 

2 j+1

μ |(Λ|νν|)| μ = 0 ;

(8.4.13)

μ=1

indeed, μ|(Λ|νν|)|μdt is the differential increment, due to the damping, in the probability of finding the state |μ a time span dt after the state |ν had probability one. The matrix (8.4.12) is easily diagonalized. It has the simple eigenvalue zero and the 2 j-fold eigenvalue (−2Γ j/3). The corresponding eigenvalues of the map eΛ e L are λ = 1 simple, λ = e−2Γ j/3 2 j-fold.

(8.4.14)

Clearly, the simple eigenvalue unity pertains to the stationary solution of the quantum map (8.4.1). Special comments are in order concerning the equality of the 2 j-fold eigenvalue in (8.4.14) and the modulus of those considered previously in (8.4.5) and (8.4.9). Even though this equality implies one and the same time scale for the relaxation of the coherences μ|(Λ|μν|)|ν and the probabilities μ|(Λ|νν|)|μ, there is no contradiction with the time-scale separation discussed in Sect. 8.3; the latter refers to coherences  j, m|| j, m  and probabilities  j, m|| j, m. The eigenstates of J 2 and Jz form a natural basis when the damping under consideration is the only dynamics present, whereas the eigenstates |μ of J 2 and the Floquet operator F are the natural basis vectors when the damping is weak and chaos prevails in the classical limit. The equality of all damping rates encountered here suggests an interpretation in which, under conditions of global classical chaos, each eigenstate |μ and thus also each probability μ|(Λ|νν|)|μ comprises coherences between angular momentum eigenstates | j, m and | j, m  where |m − m | takes values up to the order j. Such an interpretation is indeed tantamount to the applicability of random-matrix theory, which assigns, in the ensemble mean, one and the same value 1/(2 j + 1) ≈ 1/2 j to all probabilities for finding Jz = m in a state |μ. This may be viewed as a quantum manifestation of the assumed globality of classical chaos: The strange attractor covers the classical sphere ( J/j)2 = 1 uniformly.

8.4

Dissipative Death of Quantum Recurrences

293

Global uniform support is evident in the stationary solution of the quantum map (8.4.1): Indeed, it is easily verified that the eigenvector of the matrix (8.4.12) pertaining to the simple vanishing eigenvalue has all components equal to one another. Therefore, the unique density operator invariant under the map (8.4.1) is the equipartition mixture of all 2 j + 1 Floquet eigenstates |μ,

¯ =

2 j+1  1 |μμ|. 2 j + 1 μ=1

(8.4.15)

These results for the eigenvalues of the map eΛ e L enable strong statements about the time dependence of observables. For instance, all observables with a vanishing stationary mean display the universal attenuation law O(t) = e−2Γ jt/3 O(t)Γ = 0 .

(8.4.16)

The angular momentum components Ji provide examples of such observables. For instance,

Jz (∞) = =

2 j+1  1 μ|Jz |μ 2 j + 1 μ=1 2 j+1 + j   1 m |m|μ|2 2 j + 1 μ=1 m =−j

+j  1 = m = 0. 2 j + 1 m =−j

(8.4.17)

Figure 8.1 compares the ratio Jz (t)/Jz (t)Γ = 0 , calculated by numerically iterating the map (8.4.1), for Γ = 1.25 × 10−4 and Γ = 0 with the random-matrix prediction (8.4.16). The agreement is quite impressive even though the numerical work was done for j as small as 40. An implication of the above concerns the fate of quantum recurrences under dissipation. Quasi-periodic reconstructions of mean values in the conservative limit must suffer the attenuation described by (8.4.16). The mean frequency of large fluctuations for Γ = 0 roughly equals the quasi-energy spacing 2π/(2 j + 1). Therefore, the critical damping above which recurrences are noticeably suppressed can be estimated as π/j ≈ 2Γ j/3, i.e., Γcrit ≈

1 . j2

(8.4.18)

294

8 Dissipative Systems

Fig. 8.1 Ratio of the time-dependent expectation value Jz (t) with and without damping for the kicked top (8.4.1) and (8.4.2). Parameters were chosen as j = 40, p = 1.7, λ = 10. The coherent initial state was localized at Jx /j = 0.43, Jy /j = 0, Jz /j = 0.9. The smooth curve represents the prediction (8.4.16) of random-matrix theory. The optimal fit for the decay rate to the numerical data is 0.61 jΓ whereas random-matrix theory predicts (2/3) jΓ

For Γ < Γcrit , the damping constants 2Γ j/3 are smaller than even a typical nearest neighbor spacing 2π/(2 j + 1) of conservative quasi-energies. Then, the quasi-periodicity of the conservative limit must still be resolvable, of course. Figure 8.2 displays the fate of recurrences in Jz (t) for the map (8.4.1), (8.4.2), and (8.4.3) under conditions of global classical chaos. The survival condition Γ < Γcrit for recurrences is clearly visible. The amplitude of the quasi-periodic temporal fluctuations of Jz (t) displayed in Fig. 8.2 calls for comment. The ordinates in that figure are scaled in the expectation that the variance of the temporal fluctuations behaves as   2 1/2 1 1  Jz  − Jz  ∼√ σ = j j

(8.4.19)

$T where the bar denotes a time average, Jz  = T1 0 dtJz (t) with T > ∼ j. This behavior is indeed borne out in Fig. 8.2 and somewhat more explicitly in Fig. 8.3. The scaling (8.4.19) can be understood on the basis of random-matrix theory. Since the damping is taken care of by the factor exp (−2Γ jt/3) according to (8.4.16), the argument may be given for Γ = 0; it uses the scaling Jz ∼ j and the eigenvector statistics from Sect. 4.9. Since similar arguments have been expounded before, this particular one is left to the reader as Problem 8.11. As a further check of random-matrix theory the damping constants Λμν of (8.4.7) and the eigenvalues of the matrix (8.4.11) were evaluated numerically, using the numerically determined eigenstates |μ of the Floquet operator given in (8.4.2). The calculations were done for various pairs of p, λ, all corresponding to global classical chaos and for j = 5, 10, 20, 40, 80. The arithmetic mean of all

8.4

Dissipative Death of Quantum Recurrences 1.0 /j .5

295

j = 10

j = 10

Γ = 0.

Γ =5 · 10−4

.6 .4 .2 0 -.2 -.4 -.6

j = 20

j = 20

Γ = 0.

Γ =5 · 10−4

.4

j = 40

j = 40

–.4

Γ =0.

Γ =5 · 10−4

.3 .2 .1 0 –.1 –.2 –.3

j = 80

j = 80

Γ =0.

Γ =5 · 10−4

0 –0.5 –1.0

.2 0 –.2

0

100 200 300 400 500

t

0

100 200 300 400 500 60

t

√ Fig. 8.2 Time-dependent expectation value Jz / j for the kicked top (8.4.1) and (8.4.2) without damping (left column) and with Γ = 5 × 10−4 for several values of j. Initial conditions and coupling constants p, λ are as in Fig. 8.1. √ The influence of j on the lifetime of recurrences is clearly visible. The mean Jz  is referred √ to j in the expectation that the strength of the temporal fluctuations of Jz /j is of the order 1/ j; see also Fig. 8.3

(2 j + 1)2 − (2 j + 1) off-diagonal damping constants so determined is√independent of p and λ and matches the random-matrix result to within less than 1/ j. The variation with j of the relative root-mean-square deviations from the mean is consistent √ with a proportionality to 1/ j; see Fig. 8.4.

296

8 Dissipative Systems

σ .1

0.5

10

20

50

j

100

Fig. 8.3 Variance of the temporal fluctuations of Jz  according to (8.4.19). Dynamics and parameters are as √in Figs. 8.1 and 8.2. The straight line represents the prediction of random-matrix theory, σ ∼ 1/ j. The four points were obtained by numerically iterating the map (8.4.1, 8.4.2) and performing the time average indicated in the text; linear regression on the four points suggests an exponent −0.51

σ .1

5

10

20

j

50

100

Fig. 8.4 Check on the equality of all “off-diagonal” damping constants Λμν in the weak-damping limit Γ j  1 of the kicked top (8.4.1) and (8.4.2) with p = 1.7, λ = 6. The plot shows the ¯ μν , variation with j of the root-mean-square deviation of the Λμν from their arithmetic mean Λ 1/2  2 ¯ μν ) ] /Γ j. The five points, based on j = 5, 10, 20, 40, 80, yield, by linear σ = [(Λμν − Λ regression, the power law σ ∼ j −0.50 which is precisely the prediction of random-matrix theory (straight line)

8.5 Complex Energies and Quasi-Energies Dissipative quantum maps of the form (t + 1) = M(t)

(8.5.1)

8.5

Complex Energies and Quasi-Energies

297

have generators M that are neither Hermitian nor unitary. Their eigenvalues e−iφ are complex numbers that, for stable systems, are constrained to have moduli not exceeding unity, |e−iφ | ≤ 1.

(8.5.2)

Indeed, an eigenvector of M with Im {φ} > 0 would grow indefinitely in weight with the number of iterations of the map. In the conservative limit, all eigenvalues actually lie on the circumference of the unit circle around the origin of the complex plane. Their phases φ are differences of eigenphases of the unitary Floquet operator, φμν = φμ − φν . If the Hilbert space is N dimensional, N of the conservative φμν vanish, and the remainder falls into N (N − 1)/2 pairs ±|φμ − φν |. Only N of these pairs refer to adjacent quasi-energy levels, i.e., give nearest neighbor spacings. As the damping is increased, the eigenvalues e−iφ tend to wander inward from the unit circle and eventually can no longer be associated with the pairs of conservative quasi-energies from which they originated; such an association is possible only within the range of applicability of a perturbative treatment of the damping, such as that given in the last section. In general, only one eigenvalue, that pertaining to the stationary solution of the map (8.5.1), is excepted from the inward migration and actually rests at unity. Figure 8.5 depicts the inward motion of the e−iφ for a kicked top. We shall refer to the complex φ as complex or generalized quasi-energies. Given that, for conservative maps in N -dimensional Hilbert spaces with N  1, statistical analyses of the Floquet spectrum yield important characteristics and, especially, the possibility of distinguishing “regular” and “chaotic” dynamics, the question naturally arises whether statistical methods can be usefully applied to dissipative maps as well [24]. The density of eigenvalues is worthy of particular attention. In the conservative limit, the quasi-energies tend to be uniformly distributed along the circumference of the unit circle. A linear distribution of uniform density along a circle of smaller radius will still arise for weak damping, as long as first-order perturbation theory is still reliable and provided that random-matrix theory is applicable in zeroth order, i.e., provided chaos prevails in the classical limit. Indeed, as shown in the preceding section, the generalized quasi-energies φμν begin their inward √ journey with uniform radial “speed” ∂φ/∂Γ ; the relative spread of speed is ∼ 1/ j. However, for large damping, a two-dimensional surface distribution tends to arise in place of the one-dimensional line distribution. Numerical results like those displayed in Fig. 8.5 suggest isotropy but radial nonuniformity for that surface distribution. To render density fluctuations in different parts of the spectrum meaningfully comparable, the coordinate mesh within the unit circle must in general be rescaled to achieve a uniform mean density of points throughout. This is similar in spirit to the unfolding of real energy spectra discussed in Sect. 4.8. Now, fluctuations may

298

8 Dissipative Systems

Im e−iϕ 1.0

0.5 0.0

–0.5

–1.0 –0.1

–0.5

0.0

0.5

1.0 Re e−iϕ

Fig. 8.5 Inward migration from the unit circle in the complex plane of the eigenvalues of the dissipative map (8.4.1) and (8.6.1) for a kicked top with j = 6, p = 2, λ0 = 8, λ1 = 10 as the damping constant Γ is increased from 0 to 0.4 in steps of 0.005. A crossing of two eigenvalue “lines” in this picture does not in general correspond to a degeneracy since the crossing point may have different values of Γ on the two lines. The reflection symmetry about the real axis is a consequence of Hermiticity conservation for the density operator by the dissipative quantum map; see Sect. 10.5

be characterized by, e.g., the distribution of nearest neighbor spacings; when the spacing between two points in the plane is defined as their Euclidean separation, a nearest neighbor can be found for each point and a spacing distribution established. All of the above considerations require only slight modifications for damped systems under temporally constant external conditions. In such cases, the density operator obeys a master equation of the form (t) ˙ = l(t), l = L + Λ,

(8.5.3)

where the generator l of infinitesimal time translations generally has a conservative part L and a damping part Λ. The eigenvalues of l, which are more analogous to the exponents iφ than to the eigenvalues e−iφ of discrete-time maps, must have negative real parts for stable systems; distributed along the imaginary axis in the conservative limit, they tend to spread throughout the left half of the complex plane once the damping becomes strong. In view of the conservative limit and the analogy to periodically driven systems, it is convenient to denote the eigenvalues of l by −iE and to regard the E as complex or generalized energies. It should be kept in mind, though, that for vanishing damping, the eigenvalue in question becomes the difference of two eigenenergies of the Hamiltonian, E μν = E μ − E ν , and the corresponding eigen-dyad of l becomes |μν| with |μ and |ν energy eigenstates.

8.6

Different Degrees of Level Repulsion for Regular and Chaotic Motion

299

8.6 Different Degrees of Level Repulsion for Regular and Chaotic Motion Now, we consider strong damping outside the range of applicability of first-order perturbation theory. For greater flexibility, the kicked-top dynamics of Sect. 8.4 will be generalized slightly to allow two separate conservative kicks per period F = e−i p Jz e−iλ0 Jz /2 j e−iλ1 Jy /2 j . 2

2

(8.6.1)

The damping generator Λ in the map (8.4.1) is kept unchanged, i.e., given by (8.4.3). Now, the map eΛ e L must be diagonalized numerically. As in the zerodamping case, the angular-momentum basis | jm proves convenient for that purpose, especially since the dissipative part eΛ of the map can be given in closed form as the tetrad   (eΛ )mn, pq =  jm| eΛ | j p jq| | jn,

(8.6.2)

which represents the exact solution of the master equation ˙ = Λ at t = 1 originating from an arbitrary initial dyad | j p jq|. For the explicit form of the tetrad, the reader is referred to [8, 11]. Figure 8.6 displays the eigenvalues of the map eΛ e L which were obtained by numerical diagonalization for j = 15, Γ = 0.07. The coupling constants p, λ0 , λ1 in the Floquet operator were chosen to yield either regular classical motion (Fig. 8.6a; p = 2, λ0 = 11.7, λ1 = 0) or predominantly chaotic classical motion (Fig. 8.6b; p = 2, λ0 = 11.7, λ1 = 10). Mere inspection of the two annular clouds suggests a lesser inhibition with respect to close proximity under conditions of classically regular motion (Fig. 8.6a).

Fig. 8.6 Complex eigenvalues of the dissipative map (8.4.1) and (8.6.1) for j = 15, Γ = 0.07, p = 2, λ0 = 11.7 under conditions of (a) classically regular motion (λ1 = 0) and (b) chaos (λ1 = 10). Note again the reflection symmetry about the real axis. The inhibition to close proximity is clearly lower in the regular than in the chaotic case

300

8 Dissipative Systems

To unfold the spectra, a local mean density was determined around each point as ¯ = n/π dn2 where dn is the distance to the nth nearest neighbor; after making sure that the precise value of n does not matter, the choice n = 10 was made as a compromise respecting both limits in 1  n  (2 j + 1)2 . Then, the distance to the first neighbor was rescaled as & S = d1 ¯

(8.6.3)

in analogy to (4.8.11). (Note that in (4.8.11), ¯ is normalized as a probability density and refers to a distribution of points along a line.) Figure 8.7 depicts the spacing staircases I (S) for the two clouds of Fig. 8.6. Clearly, linear repulsion [i.e., a quadratic rise of I (S) for small S] is obtained when the classical dynamics is regular, whereas cubic repulsion [I (S) ∝ S 4 ] prevails under conditions of classical chaos. The distinction between linear and cubic repulsion of the complex levels and its correlation with the classical distinction between predominantly regular motion and global chaos does not seem to be a peculiarity of the dynamics chosen. At any rate, for the model considered, linear repulsion prevails not only in the strictly integrable case λ1 = 0 but also for all λ1 in the range 0 ≤ λ1 ≤ 0.2 which corresponds to the predominance of regular trajectories on the classical sphere; cubic repulsion, on the other hand, arises not only for λ1 = 6 but was also found for larger values of λ1 . These results certainly suggest universality of the two degrees of level repulsion, but they cannot be considered conclusive proof thereof. An interesting corroboration and, in fact, extension comes from the recent observation that the damped periodically driven noisy rotator displays linear and cubic repulsion of the complex I(S) 1.0 .8 .6 .4 .2 .0

.2

.6

10

1.4

1.8

S

Fig. 8.7 Integrated spacing distributions I (S) for two different universality classes. One curve in each of the two pairs refers to the map (8.4.1) and (8.6.1) with j = 10, Γ = 0.1, p = 2, and 100 different values of λ0 from 10 ≤ λ0 ≤ 12. The “regular” case has λ1 = 0 and the “chaotic” one λ1 = 8. The second line in the regular pair represents the Poissonian process in the complex plane according to (8.7.4); in the “chaotic” pair, the spacing staircase of general non-Hermitian matrices of dimension 2 j + 1 = 21 appears. The insert reveals linear and cubic repulsion

8.6

Different Degrees of Level Repulsion for Regular and Chaotic Motion

301

Floquet eigenvalues of its Fokker-Planck equation, depending on whether the noiseless deterministic limit is, respectively, regular or chaotic [25]. The reader will have noticed that Fig. 8.7 depicts a pair of curves for each of the “regular” and “chaotic” cases. One curve in a pair pertains to the map in consideration while the other represents a tentative interpretation which we shall discuss presently. In intuitive generalizations of the conservative cases, the interpretations involve a Poissonian random processin the plane (rather than along a line) for the regular limit and a Gaussian ensemble of random matrices restricted neither by unitarity nor by Hermiticity in the chaotic limit. Figure 8.7 suggests that both interpretations work quite well. It would be nice to have an exactly solvable damped system with generic spectral fluctuations, i.e.; linear repulsion. A good candidate seems to be the map eΛ e L with F = e−i p Jz e−iλJz /2 j 2

Λ =

Γ 2j

([Jz ,  Jz ] + [Jz , Jz ]) +

γ j3



 [Jz2 ,  Jz2 ] + [Jz2 , Jz2 ] .

(8.6.4)

Its “eigen-dyads” | j, m j, m | are built by the eigenvectors of J 2 and Jz . The eigenvalues can be read off immediately as

λ

mm



λ = exp −i (m − m ) p + (m − m ) 2j  Γ γ −(m − m )2 − (m 2 − m 2 )2 3 . 2j j

2

2



(8.6.5)

A peculiarity of the dynamics (8.6.4) is that the damping generator does not permit transitions between the conservative Floquet eigenstates | j, m but only phase relaxation in the coherences between such states. Consequently, the (2 j + 1)-fold degeneracy of the conservative eigenvalue unity, common to all diagonal dyads | jm jm|, is still present in the λmm for γ , Γ > 0. One might expect generic

Fig. 8.8 Integrated spacing distribution I (S) for the eigenvalues (8.6.5) of the classically integrable dissipative map (8.6.4) with j = 30, p = 17.3, λ = 24.9, Γ = 0.012, γ = 0.011. The staircase is hardly distinguishable from the prediction for the two dimensional Poissonian process, Sect. 10.7

302

8 Dissipative Systems

spacing fluctuations since both the real and the imaginary part of ln λmm depend on the two quantum numbers m ± m . Thus, the whole spectrum may be looked upon as a superposition of many effectively independent ones. This expectation is indeed borne out by the spacing staircase of Fig. 8.8.

8.7 Poissonian Random Process in the Plane Imagine that N points are thrown onto a circular disc of radius R, with uniform mean density and no correlation between throws. The mean area per point is π√R 2 /N , and the mean separation s¯ between nearest neighbors is of the order R/ N . The distribution of nearest-neighbor spacings and the precise mean separation s¯ are also readily determined. The probability P(s)ds of finding the nearest neighbor of a given point at a distance between s and s + ds equals the probability that one of the (N − 1) points is located in the circular ring of thickness ds around the given point and that all (N − 2) remaining points lie beyond this: P(s)ds = (N − 1)

2πs ds π R2

 1−

π s2 π R2

N −2 .

It is easy to check that this distribution is correctly normalized, The mean spacing reads

(8.7.1) $R 0

   1 R √ s¯ = √ N (N − 1)2 dt t 2 (1 − t 2 ) N −2 . N 0

ds P(s) = 1.

(8.7.2)

√ After rescaling the integration variable as t = τ/ N , the square bracket in (8.7.2) is seen, as N → ∞, to approach the limit √

 2 0

N

√ N   ∞ π τ2 2 dτ τ 2 1 − →2 dτ τ 2 e−τ = . N 2 0

(8.7.3)

To find the asymptotic spacing distribution for large N , the spacing must be √ referred to its mean s¯ = R π/4N . After introducing the appropriately rescaled variable S = s/¯s and performing the limit as in (8.7.3), the distribution (8.7.1) is turned into P(S) = 12 π Se−π S

2

/4

.

(8.7.4)

Amazingly, (8.7.4) gives Wigner’s conjecture exactly for the spacing distribution in the GOE and COE discussed in Chap. 4. As was shown there, the Wigner distribution is rigorous for the GOE of 2 × 2 matrices but is only an approximation, albeit a rather good one, for N × N matrices with N > 2. In the present context, the

8.8

Ginibre’s Ensemble of Random Matrices

303

Wigner distribution arises as a rigorous result in the limit N → ∞. According to Figs. 8.7 and 8.8, this distribution is reasonably faithful to the spacing distributions of the respective regular dynamics. The question arises why the Poissonian random process in the plane should so accurately reproduce the spacing fluctuations of generic integrable systems with damping. To achieve an intuitive understanding, the reader may recall from Chap. 5 the most naive argument supporting the analogous statement for Hamiltonian dynamics: the spectrum of an integrable Hamiltonian with f degrees of freedom may be approximated by torus quantization. Its levels are labelled by f quantum numbers, and the spectrum may be thought of as consisting of many independent subspectra. An analogous statement can be made about the spectrum of the map eΛ e L with the conservative Floquet operator F from (8.6.1) and the damping generator Λ from (8.4.3), provided λ1 = 0 in F to make the classical limit integrable. As is easily checked, this map has the symmetry 

 Jz , eΛ e L  = eΛ e L Jz , 

(8.7.5)

for arbitrary . It follows that the map in question does not couple dyads | j, m + k j, m| with different values of k. Indeed, by using | j, m + k j, m| for  in (8.7.5) and writing eΛ e L | j, m + k j, m| =



Cmmk k | j, m + k  j, m |,

m k





mk mk = 0, i.e., Cmk ∝ δk k . Conone obtains from (8.7.5) the identity (k − k)Cmk sequently, k is a good quantum number for the map in question and the cloud of eigenvalues depicted in Fig. 8.6a may be segregated into many subclouds, one for each value of k. When the map defined by (8.4.3) and (8.6.1) is turned into a classically nonintegrable map by allowing λ1 > 0, the symmetry (8.7.5) is broken. Consequently, now the previously segregated (λ1 = 0) subspectra begin to interact. It is still not counterintuitive that the spacing distribution remains Poissonian as long as λ1 is small (0 ≤ λ1 ≤ 0.2); experience with conservative level dynamics suggests that a large number of close encounters of levels must have taken place before strong level repulsion can arise throughout the spectrum.

8.8 Ginibre’s Ensemble of Random Matrices By dropping the requirement of Hermiticity, Ginibre [26] was led from the Gaussian unitary ensemble to a Gaussian ensemble of matrices with arbitrary complex eigenvalues z. The joint probability density for the N eigenvalues of such N × N matrices looks similar to (4.3.15) for the GOE,

304

8 Dissipative Systems

 N  1 ... N  N −1 , 2 2 P(z 1 , . . . , z N ) = N |z i − z j | exp − |z i | , π i< j i =1

(8.8.1)

but is to be normalized by integrating each z i over the complex plane. The goal of this section is to extract the distribution of nearest-neighbor spacings from (8.8.1) and to compare it with the numerically obtained spacing distribution of the classically chaotic top discussed in Sect. 8.6. The calculations to follow fall in several separate parts, each of which merits interest in its own right. Some of them are alternatives to Mehta’s [27] and Ginibre’s classic treatments. First, we shall determine the normalization factor in (8.8.1), then proceed to evaluating the reduced joint densities P j (z 1 , z 2 , . . . , z j ) with j = 1, 2, 3, . . . , and finally use the results obtained to find the spacing distribution P(S).

8.8.1 Normalizing the Joint Density The product of differences in (8.8.1) is related to a Vandermonde determinant,   1 z 1 z 12 . . . z 1N −1    N −1 1, ... N  k−1  1 z 2 z 22 . . . z 2  = . (z i − z j ) = det z i ..  ,  .. .  i< j  1 z z 2 . . . z N −1  N N N

(8.8.2)

which was already encountered in Sect. 4.12. Then, the squared modulus of the product in (8.8.2) takes the form 1, ... N

⎛ |z i − z j |2 = det ⎝

N 

⎞ ∗ k−1 ⎠ . z i−1 j zj

(8.8.3)

j =1

i< j

It will be convenient for the following to represent again an n × n determinant as the Gaussian integral over anticommuting variables (4.13.1),  det A =

⎛ ⎝

1, ... n j

⎞ dη∗j dη j ⎠ exp

 −

1 ... n

 ηi∗ Aik ηk

(8.8.4)

ik

introduced in Sect. 4.13. We employ this identity to the determinant in (8.8.3) for which the exponential exp (−η∗ Aη) in (8.8.4) may be simplified slightly. Since pairs ηi∗ ηk commute among themselves, the general representation yields

8.8

Ginibre’s Ensemble of Random Matrices

⎛ det ⎝

N 

⎞ ∗ k−1 ⎠ z i−1 j zj

j =1



305

⎛

⎞

⎝

dη∗j dη j ⎠

1, ... N

=

1, ... N

j

 exp −

1 ... N

j

 ∗ k−1 ηi∗ ηk z i−1 j zj

.

(8.8.5)

ik

Moreover, when each of the N exponentials on the r.h.s. is expanded in a Taylor series, second- and higher order terms vanish identically. Indeed, 



2 ηi∗ ηk z i−1 z ∗ k−1

ik

=



ηi∗ η j ηk∗ ηl z i−1 z ∗ j−1 z k−1 z ∗ l−1 = 0

(8.8.6)

i jkl

since the product of the four Grassmann variables is antisymmetric in j and l and the product of the four ordinary numbers is symmetric. It follows that ⎞ ⎛ N  ∗ k−1 ⎠ z i−1 det ⎝ j zj j =1

 =  =

⎛ ⎝

1, ... N

⎞ dη∗j dη j ⎠

j

⎛

⎞

1, ... N

 1−



j

 ∗ k−1 ηi∗ ηk z i−1 j zj

ik

  1, ... N 1, ... N  i−1 ∗ k−1 ∗ ∗ ⎠ ⎝ dη j dη j ηi ηk z j z j − j

j

(8.8.7)

ik

where the last step is based on the vanishing of any Grassmann integral over a constant integrand. With the help of this integral representation of the product of differences in (8.8.1), the calculation of the normalization factor N becomes an easy matter. When the z integrals are done first, each of them yields a factor 

  1 2 ηi∗ ηk z i−1 z ∗ k−1 = ηi∗ ηi (i − 1)! d 2 z e−|z| π ik i

(8.8.8)

and the normalization factor N is determined by  N =

⎛

,

⎝

j

⎞dη∗j dη j ⎠ −

N 

.N ηi∗ ηi (i − 1)!

.

(8.8.9)

i =1

For non-zero contributions to arise, each of the N identical factors in the integrand must provide a different pair ηi∗ ηi ; there are N ! such contributions, and they are all numerically equal since the pairs ηi∗ ηi , η∗j η j commute,

306

8 Dissipative Systems

N = N !(−1) N

N ,

⎞ . ⎛ N , , ⎝ dη∗j dη j η∗j η j ⎠ = (i − 1)! j!.

i =1

(8.8.10)

j =1

j

Thus, the correctly normalized joint probability density of all N eigenvalues reads  1 ... N , e−|zk |2 , |z i − z j |2 . P ({z}) = π k! k i< j

(8.8.11)

8.8.2 The Density of Eigenvalues The probability density for a single eigenvalue is obtained by integrating the joint density over all but one variable,  P1 (z 1 ) =

d 2 z 2 . . . d 2 z N P(z 1 , . . . , z N ) ⎛

=⎝

1, ... N

⎞−1 j!⎠

j

×

 1, ... N

e−|z1 | π

2

 2, ... N k

e−|zk | d zk π

2



2

 1 ... N   ,  ∗ ∗ i−1 ∗ n−1 dηl dηl ηi ηn z m z m − . m

l

(8.8.12)

in

Here again we employ the representation (8.8.4) and (8.8.7) for the product of differences in the joint distribution (8.8.1). After doing the N − 1 integrals over the z k , the density P1 (z) reads ⎛ P1 (z) = ⎝

1, ... N

-

⎞−1 j!⎠

j

× −



−|z|2

e



π

⎛

1, ... N

⎝

j

dη∗j dη j ⎠

−



 ηi∗ ηk z i−1 z ∗ k−1

ik

. N −1 ηl∗ ηl (l − 1)!

⎞

.

(8.8.13)

l

Since N − 1 of the factors in the integrand involve pairs ηl∗ ηl with coinciding indices, the remaining factor must have this property as well for the 2n-fold Grassmann integral to pick up a nonzero contribution,

8.8

Ginibre’s Ensemble of Random Matrices

⎛ P1 (z) = ⎝

1, ... N

⎞−1 j!⎠

j

× −

1 ... N

−|z|2

e



π

307

⎛

1, ... N

⎝

⎞ dη∗j dη j ⎠

−

j

1 ... N

 ηi∗ ηi |z|2(i−1)

i

. N −1 ηl∗ ηl (l − 1)!

.

(8.8.14)

l

For every fixed value of the summation index i in the first factor of the integrand, there are (N − 1)! ways of assigning indices to the remaining pairs η∗ η such that each of the naturals 1, 2, . . . , N appears once as an index on a pair η∗ η; thus all factorials in the normalization factor are cancelled except for 1/[(i − 1)!N ], P1 (z) =

1 −|z|2 e N −1 (|z|2 ), e Nπ

(8.8.15)

where en (x) denotes the nth order polynomial in x equalling the first n + 1 terms of the Taylor expansion of ex about the origin, en (x) = 1 +

x xn + ... . 1! n!

(8.8.16)

It is immediately obvious from (8.8.15) and (8.8.16) that P1 (z) approaches the constant 1/N π in the limit N → ∞ with z held fixed. Conversely, when N is kept constant and z grows indefinitely, P1 (z) vanishes. The transition region between the two limiting cases lies at |z|2 ≈ N . To investigate the transition it is convenient to note that ∂ en (x) = en−1 (x) ∂x

(8.8.17)

and therefore, e−x en (x) =



∞

dy x

y n −y e . n!

(8.8.18)

For large √ n, the integrand in (8.8.18) has a sharp maximum at y = n; its width is of the order n. When x lies to the left of the maximum by several times the width, the integral deviates only slightly from unity; for x well to the right of the maximum, however, the integral is exponentially small since it collects only contributions from the wing of the integrand. For x near the maximum, the asymptotic large-n behavior √ reveals itself when the variable ξ = (x − n)/ 2n is introduced and the limit n → ∞ taken with of Stirling’s formula. Setting f (ξ ) = en (x)e−x , one finds √ the−ξhelp 2

and thus, f (ξ ) = (1/ π )e en (x)e−x ≈

  1 x −n . erfc √ 2 2n

(8.8.19)

308

8 Dissipative Systems

Interestingly, this asymptotic formula still encompasses the correct limits 1 and 0 for x well below √ and well above n, respectively. Most importantly, since the relative width ∼ n/n of the transition vanishes with n → ∞, the function en (x)e−x behaves essentially like the unit step function Θ(n − x). √ To sum up, as N → ∞, the eigenvalues z tend to cover the circular disc of radius N around the origin of the complex plane √ with uniform probability density 1/π N . Needless to say, by rescaling as z → z N , the circle in question is contracted to the unit circle that was encountered above for complex matrices representing dissipative quantum maps.

8.8.3 The Reduced Joint Densities By integrating P(z 1 , . . . , z N ) over N − n variables, the joint probability density of n eigenvalues is obtained. Instead of (8.8.13) for n = 1, now, ⎛ Pn (z 1 , . . . , z n ) = ⎝

1, ... N

⎞−1 j!⎠

j

e−|z1 | e−|zn | ... π π 2

2



⎛

⎞

⎝

dη∗j dη j ⎠

1, ... N j

5 N −n .  -1 ... n  ,   i−1 ∗ k−1 ∗ ∗ ηi ηk zl zl ηm ηm (m − 1)! . − − × l

m

ik

(8.8.20) As before, each of the naturals 1, 2, . . . N must appear precisely once as the index of an η∗ and an η. For a given set of n indices {i} on the η∗ and the identical set {k} (apart from ordering) on the η in the square bracket, there are (N − n)! possibilities of using up the remaining naturals + for the (N − n) curly brackets in (8.8.20). The latter then provide the factor j∈{i} (−η∗j η j )( j − 1)!. With the N − n pairs η∗j η j integrated out, the reduced density becomes Pn (z 1 , . . . , z n ) =

−|z 1 |2

(N − n)! e N! π ×

1 ... N

...

−|z n |2

e

π



 (−1)n

⎛

⎞

⎝

dη∗j dη j ⎠

1, ... n j

ηi∗1 ηk1 . . . ηi∗n ηkn z 1i1 −1 z 1∗ k1 −1 . . .

{i},{k}

 ×z nin −1 z n∗ kn −1 /(i 1 − 1)! . . . (i n − 1)! .

(8.8.21)

The equality of the sets {i} and {k} can be realized in n! different ways, corresponding to the n! permutations {Pi} of the {i}. For each such permutation, the 2n integrals can be carried out to yield the factor (−1)n (−1)P where (−1)P is the signature of the permutation. The square bracket in (8.8.21) becomes

8.8

Ginibre’s Ensemble of Random Matrices

[] =

1 ... N {i}

309

z 1i1 −1 z nin −1  (−1)P (z 1∗ )−1+Pi1 . . . (z n∗ )−1+Pin . ... (i 1 − 1)! (i n − 1)! P

At this point, the restriction that all the i be different becomes superfluous since the sum over the permutations P vanishes if any of the i coincide. Instead of summing over the permutation in the exponents, one may sum over the permutations of indices on the z ∗ : 

1 ... N

∗ i n −1 ∗ i 1 −1 (z n z Pn ) (z 1 z P1 ) ... (i 1 − 1)! (i n − 1)! {i} P       ∗  ∗ ∗ e N −1 z 2 z P2 . . . e N −1 z n z Pn (−1)P e N −1 z 1 z P1 =

[] =

(−1)P

P

 = det e N −1 (z i z k∗ ) .

(8.8.22)

Thus, the reduced density reads ⎛ Pn (z 1 , . . . , z n ) =

(N − n)! ⎝ N!

1, ... n

−|z j |2

e

π

j

⎞

 ⎠ det e N −1 (z i z k∗ ) .

(8.8.23)

This includes (8.8.15) as the special case n = 1 and even implies a new representation of the full joint density (8.8.1) for n = N . Similarly to the behavior of P1 (z 1 ), the joint distribution Pn (z 1 . . . z n ) vanishes in the limit N → ∞ if at least one of the variables z lies outside the circle of radius √ N , |z|2 > N . When all the z lie within that circle, on the other hand, the truncated exponentials approach the full ones such that Pn (z 1 . . . z n ) = π −N

 (N − n)! ∗ det e(zi −zk )zk . N!

(8.8.24)

Obviously, homogeneity within the circle |z i |2 < N is established since the asymptotic distribution (8.8.24) is invariant under a common shift, z i → z i + δ, of all variables.

8.8.4 The Spacing Distribution It will be convenient to determine the spacing distribution P(s) from the probability H (s) that an eigenvalue has its nearest neighbor further away than s, i.e., P(s) = −

d H (s). ds

(8.8.25)

310

8 Dissipative Systems

We shall refer to H (s) as the hole probability. The more familiar integrated spacing distribution I (s), i.e., the probability that an eigenvalue has its nearest neighbor closer than s, is related to H (s) by I (s) = 1 − H (s). With  . . .  denoting the average over all N eigenvalues, and the weight is the joint density (8.8.1), the hole distribution reads H (s) = Θ (|z 2 − z 1 | − s) Θ (|z 3 − z 1 | − s) . . . Θ (|z N − z 1 | − s) . (8.8.26) It is left to the reader in Problem 8.15 to consider the case N = 2 and to verify that  2 2 H2 (s) = e1 s2 e−s /2 , (8.8.27) 2 P2 (s) = 12 s 3 e−s /2 or, with the mean spacing rescaled to unity,  P2 (S) = (34 π 2 2−7 )S 3 exp −(32 π 2−4 )S 2 .

(8.8.28)

In the following, arbitrary values of N will be admitted initially, but eventually the limit N  1 will be taken. To that end, we rewrite (8.8.26) as 8 H (s) =

N , 



1 − Θ s − |z 1 − z j | 2

2



9

j =2

= 1+

N −1  (−1)ν ν =1

ν!

2 ... N n 1 =n 2 = ... =n ν

   × Θ s − |z 1 − z n 1 |2 . . . Θ s 2 − |z 1 − z n ν |2



2

(8.8.29)

and invoke the joint density of ν + 1 eigenvalues (8.8.23), H (s) − 1 =

 N −1  (−1)ν

d 2 z 1 −|z1 |2 e ν!N |z1 |< N π ν =1   d 2 z 2 −|z2 |2 d 2 z ν+1 −|zν+1 |2 × ... e e π |z 2 −z 1 | ∼ t , the result (8.11.31) may be expanded in ∗ ∗ powers of (t − t )/t . The first few terms of that expansion, √  π π erfc 2 2    t − t∗ π t − t∗ 2 + ∗ + + ... , t 4 t∗ 

δ pt2  ≈ 2l 2 | ln d|

1−

(8.11.33)

∗ should describe the earliest manifestations of damping for t < ∼ t . Figure 8.13 indeed 2 reveals a range of time with a quadratic increase of δ pt .

8.12 Problems 8.1 For j = 1/2 determine the time-dependent solution of the master equation (8.2.21). 8.2 Let the bath be a collection of harmonic oscillators with the free Hamiltonian HR =



ωi bi+ bi ,

i

and assume that the angular momentum couples to the bath variable

336

8 Dissipative Systems

B=g



bi

i

and its conjugate as

HSR = (J+ B + J− B † ).

Show that Fermi’s golden rule yields the transition rates W (m → m + 1) and W (m + 1 → m). 8.3 Let an angular momentum be coupled to a heat bath as HSR = Jz (B + B † ). Show that the master equation in the weak-coupling limit is ˙ = −i[ω Jz + δ Jz2 , ] +κ(1 + 2n th ) ([Jz ,  Jz ] + [Jz , Jz ]) . Discuss the lifetimes of populations and coherences in the Jz representation. 8.4 Show that the master equation for the damped harmonic oscillator reads (t) ˙ = −i(ω + δ)[b† b, (t)] (8.12.1)     +κ(1 + n th ) [b, b† ] + [b, b† ] + κn th [b† , b] + [b† , b] . Give the stationary solution and the expectation values b(t) and b† (t)b(t). In the low-temperature limit (n th = 0) also give the probability of finding m quanta in the oscillator at time t > 0, provided there were m 0 initially. Recall from your † elementary quantum mechanics that the √m-quantum state is defined by b b|m = m|m with m = 0, 1, 2 . . . and b|m = m|m − 1. 8.5 Show that the master equation (8.3.2) for the damped spin generally gives no time-scale separation for probabilities and coherences in the Jz basis. Use the initial rate of change of the norm of the temporal successor of | j, m j, m |. Note that one does not have a right to expect accelerated decoherence on that basis since J− is not among the coupling agents! 8.6 In the low-temperature limit, rewrite the master equation (8.2.21) as an equation of motion for the weight function P(Θ, φ, t) in the representation of (t) by a diagonal mixture of coherent angular momentum states,  (t) =

sin Θ dΘ dφ P(Θ, φ, t)|Θ, φΘ, φ|.

See the definition (7.6.12). Consult Ref. [38, 39] for help.

8.12

Problems

337

8.7 Return to Problem 8.4 for the damped harmonic oscillator. Now use Glauber’s † coherent states [40] |β, defined by b|β = β|β or |β = exp(−|β|2 /2) × eβb |0, to show that coherences between different such states decay faster than probabilities β||β. Consider the norm of eΛt |ββ|. For the first report of accelerated decoherence for the damped harmonic oscillator, you may want to check ([41]). 8.8 Use the distribution P(y) from (4.9.13) to show that the root-mean-square devi√ in (8.4.8) is proportional to j and that the ation from the mean of the Λμν given√ relative deviation is proportional to 1/ j. 8.9 Calculate the damping constants Λμν analogous to (8.4.9) and the eigenvalues corresponding to (8.4.11) for the damping generator of Problem 8.3. 8.10 Show that Jx,y (∞) = 0 under conditions of global classical chaos. Proceed similarly to (8.4.17). % 8.11 Calculate (Jz  −  J¯ z )2 /j for Γ = 0 and the initial state chosen from the classically chaotic regime. ( . . . ) means averaging over time. 8.12 Find the nearest neighbor spacing distribution for the d-dimensional Poissod nian process, Pd (s) = αds d−1 e−αs , and determine the parameter α. 8.13 Find the kth nearest neighbor spacing distribution for the d-dimensional Poissonian process, Pdk (s) =

α k d (k−1)d −αs d e . s (k − 1)!

8.14 Normalize the joint distribution of the eigenvalues for Dyson’s circular ensembles using Grassmann integration. 8.15 Establish the spacing distribution (8.8.27) for Ginibre’s ensemble of 2 × 2 matrices with the help of (8.8.11) and (8.8.26). 8.16 Argue that a finite-size approximation to the hole distribution is obtained as H (s) =

N −1  ,

en (s 2 )e−s

2



.

n=1

8.17 Establish the correction −s 2 ln s in (8.8.43) with the help of the EulerMcLaurin summation formula.

338

8 Dissipative Systems

8.18 Show that (8.8.1) is equivalent to Di jkl = D ∗jilk . 8.19 Recall how for [M, A] = 0 with A antiunitary and A2 = 1, a representation can be constructed whose basis vectors are A invariant and which makes M a real matrix. 8.20 Show that the real eigenvalues of the generator D repel linearly. 8.21 Show that P(S) = S 3 /(8λ4 )K 0 (S 2 /4λ2 ) for complex symmetric 2×2 matrices with a Gaussian element distribution W (x) = (λ/π ) exp (−λ|x|2 ), λ = λ∗ , where K 0 is the modified Bessel function of order zero. 8.22 Calculate the normalization coefficient c in (8.10.27).   determined in Problem 7.4, determine 8.23 Using the moments ξ 2n  = (λ/2)2n 2n n the deviation from Gaussian behavior of P( p), as given by (8.11.9). 8.24 Specify the frequency dependence of the bath response function κ(ω) yielding the master equation (8.11.16); also give the condition on the bath temperatures implicit in (8.11.16). 8.25 As an alternative to the damping mechanism treated in the text, use I+ = I− = cos Θ instead of (8.11.15). Specialize again to low temperatures, and employ the Born and Markov approximation in constructing the generator Λ. Which choice of κ(ω) secures  p˙  =  p ln d?

References 1. U. Weiss: Quantum Dissipative Systems (World Scientific, Singapore, 3rd edition 2008) 2. D. Braun: Dissipative Quantum Chaos and Decoherence Springer Tracts in Modern Physics, Vol. 172 (Springer, Berlin, 2001) 3. R. Graham, F. Haake: Quantum Statistics in Optics and Solid State Physics, Springer Tracts in Modern Physics, Vol. 66 (Springer, Berlin, Heidelberg, 1973) 4. H. Spohn: Rev. Mod. Phys. 53, 569 (1980) 5. F. Haake: Z. Phys. B48, 31 (1982) 6. F. Haake, R. Reibold: Phys. Rev. A32, 2462 (1985) 7. L.P. Kadanoff, G. Baym: Quantum Statistical Mechanics (Benjamin, New York, 1962) 8. R. Bonifacio, P. Schwendimann, F. Haake: Phys. Rev. A4, 302, 854 (1971) 9. M. Gross, S. Haroche: Physics Reports 93, 301 (1982) 10. L.D. Landau, E.M. Lifschitz: Course of Theoretical Physics, Vol. 5, Statistical Physics (Pergamon, London, 1952) 11. P.A. Braun, D. Braun, F. Haake, J. Weber: Euro. Phys. J. D2, 165 (1998) 12. P.A. Braun, D. Braun, F. Haake: Euro. Phys. J. D3, 1(1998) 13. D. Braun, P.A. Braun, F. Haake: Physica D131, 265 (1999)

References

339

14. D. Braun, P.A. Braun, F. Haake: Opt. Comm. 179, 195 (2000); and In P. Blanchard, D. Giulini, E. Joos, C. Kiefer, I.O. Stamatescu (eds.) Decoherence: Theoretical, Experimental, and Conceptual Problems (Springer, Berlin, 2000) 15. D. Braun: Chaos 9, 760 (1999) 16. A.J. Leggett: Prog. Th. Phys. Suppl. 69, 80 (1980) 17. A.O. Caldeira, A.J. Leggett: Ann. Phys. (NY) 149, 374 (1983) 18. F. Haake, D.F. Walls: Phys. Rev. A36, 730 (1987) 19. W.H. Zurek: Phys. Rev. D24, 1516 (1981); D26, 1862 (1982); Phys. Today 44, 36 (1991); Prog. Th. Phys. 89, 281 (1993) 20. M. Brune, E. Hagley, J. Dreyer, X. Maˆıtre, A. Maali, C. Wunderlin, J.M. Raimond, S. Haroche: Phys. Rev. Lett. 77, 4887 (1996); S. Haroche: Phys. Today 51, 36 (1998) 21. R. Grobe, F. Haake: Z. Phys. B68, 503 (1987) and Lect. Notes Phys. 282, 267 (1987); 22. R.R. Puri, G.S. Agarwal: Phys. Rev. A33, 3610 (1986) 23. S.M. Barnett, P.L. Knight: Phys. Rev. A33, 2444 (1986) 24. R. Grobe, F. Haake, H.-J. Sommers: Phys. Rev. Lett. 61, 1899 (1988) 25. L.E. Reichl, Z.-Y. Chen, M. Millonas: Phys. Rev. Lett. 63, 2013 (1989) 26. J. Ginibre: J. Math. Phys. 6, 440 (1965) 27. M.L. Mehta: Random Matrices (Academic, New York 1967; 2nd edition 1991; 3rd edition Elsevier 2004) 28. M. Abramowitz, I.A. Stegun: Handbook of Mathematical Functions (Dover, New York 1970) 29. N. Lehmann, H.-J. Sommers: Phys. Rev. Lett. 67, 941 (1991) 30. R. Grobe, F. Haake: Phys. Rev. Lett. 62, 2889 (1989) 31. R. Grobe: Ph.D. Thesis, Essen (1989) 32. L. Onsager: Phys. Rev. 37, 405 (1931) and Phys. Rev. 38, 2265 (1931) 33. G.S. Agarwal: Z. Phys. 258, 409 (1973) 34. H.J. Carmichael, D.F. Walls: Z. Phys. B23, 299 (1976) 35. G.M. Zaslavski: Phys. Lett. 69A, 145 (1978) 36. G.M. Zaslavski, Kh.-R.Ya. Rachko: Sov. Phys. JETP 49, 1039 (1979) 37. T. Dittrich, R. Graham: Ann. Phys. 200, 363 (1990) 38. F.T. Arecchi, E. Courtens, G. Gilmore, H. Thomas: Phys. Rev. A6, 2211 (1972) 39. R. Glauber, F. Haake: Phys. Rev. A13, 357 (1976) 40. R.J. Glauber: Phys. Rev. 130, 2529 (1963); 131, 2766 (1963) 41. D.F. Walls, G. Milburn: Phys. Rev. A31, 2403 (1985)

Chapter 9

Classical Hamiltonian Chaos

9.1 Preliminaries This chapter will present classical Hamiltonian mechanics to the extent needed for the semiclassical endeavors to follow. I can confine myself to the bare minimum since many excellent texts on classical chaos are available [1–5]. Readers with a good command of nonlinear dynamics might want to right away start with Sect. 9.14 where I begin expounding the fact that long periodic orbits of hyperbolic systems are not independent individuals but rather come in closely packed bunches. Different orbits in each bunch are topologically distinct but can be nearly indistinguishable in action, and for that reason have strong quantum signatures. The origin of orbit bunching can be seen in close self-encounters which a long orbit is bound to undergo. The orbit stretches involved in close self-encounters can be “switched” to form “partner orbits”. The classical bunching phenomenon was discovered in the course of semiclassical attempts to explain universal fluctuations in quantum energy spectra, the theme of the next chapter. It has turned out to be the key to a semiclassical understanding of universal features in quantum transport and of dynamical localization as well. Certain subtle issues related to extremal properties of the action, the so called focal points and Lagrangian manifolds could have been made part of the present chapter. They are, however, deferred to the semiclassical considerations of the next chapter since that context will provide a good motivation.

9.2 Phase Space, Hamilton’s Equations and All That At issue are dynamics with f degrees of freedom whose phase space is spanned by f coordinates q1 , . . . , q f and as many momenta p1 , . . . , p f . Configuration space is spanned by the f coordinates q. A Hamiltionian function H (q, p) generates the time evolution through Hamilton’s equations q˙ =

∂H , ∂p

p˙ = −

∂H ∂q

F. Haake, Quantum Signatures of Chaos, Springer Series in Synergetics, 3rd ed., C Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-05428-0 9, 

(9.2.1)

341

342

9 Classical Hamiltonian Chaos

or, more compactly1 ,  X˙ = Σ∂ X H = F(X ),

Σ=

 0 1 . −1 0

(9.2.2)

The matrix Σ is antisymmetric and squares to minus the 2 f × 2 f unit matrix, ˜ = −Σ, Σ

Σ 2 = −1.

(9.2.3)

When sets of phase space points are imagined transported a` la Hamilton the picture of a flow emerges. That flow is free of sources, divF(X ) = ∂X Σ∂ X H = 0.

(9.2.4)

A solution of Hamilton’s equations yields the current point X t as a function of the initial point X 0 and may be written as X t = Φt (X 0 ).

(9.2.5)

It will sometimes be convenient to represent the flow by a nonlinear operator Φt and write X t = Φt X 0 . Hamiltonian time evolution has an underlying group structure in the sense Φt+τ = Φt Φτ and Φ−t Φt = 1. For two “phase space functions” f (q, p), g(q, p) one defines the Poisson bracket { f, g} =

∂g ∂ f ∂ f ∂g − ∂ p ∂q ∂ p ∂q

(9.2.6)

which might be written as { f, g} = (∂X g)Σ∂ X f . Coordinates and momenta are said to form canonical pairs in the sense { pi , q j } = δi j ,

{ pi , p j } = {qi , q j } = 0.

(9.2.7)

Any reparametrization of phase space, q, p → Q, P, for which the P, Q are again canonical pairs such that {Pi (q, p), Q j (q, p)} = δi j , {Pi (q, p), P j (q, p)} = {Q i (q, p), Q j (q, p)} = 0 is called a canonical transformation. Time evolution can be looked upon as a canonical transformation inasmuch as the basic brackets (9.2.7) hold for the time dependent coordinates and momenta at any time, { pi (t), q j (t)} = δi j (while in general, of course, { pi (t), q j (t )} = δi j for t = t ). Hamilton’s equations (9.2.1) are form invariant under canonical transformations. A phase space function f (q, p, t) with an explicit time dependence (beyond the impicit one through the time dependent coordinates and momenta) obeys the evolution equation

1

Here and below, it is best to read X =

q  p

, ∂X =

∂q  ∂p

, and ∂X = (∂q , ∂ p ).

9.3

Action as a Generating Function

343

df ∂f = {H, f } + , dt ∂t

(9.2.8)

of which Hamilton’s equations (9.2.1) are special cases. A further special case is d H/dt = ∂ H/∂t; for autonomous dynamics the latter identitiy implies energy conservation. Unless explicitly noted otherwise I shall confine the subsequent discussion to that case. The non-integrable “chaotic” dynamics at issue here do not allow for solutions of Hamilton’s equations in terms of “quadratures” (i.e., explicit integrals). They have no or fewer than f independent constants of the motion. Chaotic behavior of autonomous dynamics requires at least two freedoms since energy is conserved.

9.3 Action as a Generating Function Any canonical transformation can be induced by a generating function. In particular, the transition along the classical trajectory from the inital coordinate q to the final coordinate q during the time span t is generated by a function S(q, q , t) which is the action, i.e. the time integral of the Lagrangian evaluated along the classical trajectory in question, S(q, q , t) =



t

˙ )). dτ L(q(τ ), q(τ

(9.3.1)

0

To show this and to get the final and initial momenta p, p we look at infinitesimal deflections in configuration space and the pertinent action 



t

S(q + δq, q + δq, t) =

˙ ) + δ q(τ ˙ )) dτ L(q(τ ) + δq(τ ), q(τ (9.3.2)    t ∂L ∂L ˙ ) . = S(q, q , t) + dτ δq(τ ) + δ q(τ ∂q ∂ q˙ 0 0

Integrating by parts the second term in the integral and realizing that (i) Lagrange’s equations dtd ∂∂ qL˙ − ∂∂qL = 0 hold along the classical trajectory and (ii) the momentum is given by p = ∂ L/∂ q˙ we get t  S(q + δq, q + δq , t) − S(q, q , t) = p(τ )δq(τ ) = pδq − p δq

0

(9.3.3)

and thus the final and initial momenta as p=

∂ S(q, q , t) , ∂q

p = −

∂ S(q, q , t) . ∂q

(9.3.4)

The partial derivative ∂ S/∂t is obtained when the action S(q, q , t +δt) is considered for the classical trajectory from q to q during the inifinitesimally longer time

344

9 Classical Hamiltonian Chaos

span t + δt. That trajectory must run close to the original one and we may expand similarly as above, S(q, q , t + δt) =



t+δt

˙ ) + δ q(τ ˙ )) dτ L(q(τ ) + δq(τ ), q(τ

(9.3.5)

0

t  ˙ + p(τ )δq(τ ) . = S(q, q , t) + δt L(q(t), q(t)) 0

˙ But this time we have δq(0) = δq(t + δt) = 0 and thus δq(t) = −q(t)δt such that we can read off the time derivative of the action as ∂ S(q, q , t) ˙ = L(q(t), q(t)) − pq˙ = −H (q, p) = −E. ∂t

(9.3.6)

Instead of fixing the time span t we may consider the transition q → q with prescribed energy E. That transition is generated by the Legendre-transformed action (called Hamilton’s principal function by some authors) S0 (q, q , E) = S(q, q , t) + Et, ∂ S0 (q, q , E) ∂ S0 (q, q , E) . , p = − p= ∂q ∂q

(9.3.7)

In the foregoing definition of the energy dependent action S0 (q, q , E) the time argument of S(q, q , t) must be fixed according to ∂ S0 (q, q , E) = t(q, q , E); ∂E the latter identity follows from (9.3.6) through

(9.3.8)

∂t ∂ S ∂t = −E . ∂t ∂ E ∂E

9.4 Linearized Flow and Its Jacobian Matrix To study what happens near a given trajectory X t = Φt (X 0 ) it is appropriate to linearize the flow as δ Xt =

∂Φt (X 0 ) δ X 0 = Mt (X 0 )δ X 0 . ∂ X0

(9.4.1)

An initial point deflected by δ X 0 from X 0 ends up with a deflection δ X t from X t at time t. A deflection δ X from X lives in a linear vector space called the tangent space at X . The 2 f × 2 f matrix Mt (X 0 ) can be looked upon as the Jacobian matrix of the canonical transformation X 0 → X t .

9.4

Linearized Flow and Its Jacobian Matrix

345

We may equivalently linearize the equations of motion δ X˙ =

 ∂ F(X )  δX ∂ X  X =Φt X 0

(9.4.2)

and are led to the Jacobian matrix as a time-ordered exponential 



t

Mt (X 0 ) = exp 0

  ∂ F  dτ . ∂ X  X =Φτ X 0 )

(9.4.3)

+

The latter representation entails the group-type property Mt+τ (X ) = Mt (Φτ X )Mτ (X )

(9.4.4)

as well as the evolution equation  ∂ F  ˙ Mt (X 0 ) = ∂X 

Mt (X 0 ).

(9.4.5)

X =Φt X 0

The Jacobian determinant of a Hamiltonian flow equals unity due to the absence of sources, see (9.2.4) as is clear from the following little calculation. The fore˙ t Mt−1 = d Tr ln Mt and thus going evolution equation (9.4.5) entails divF = Tr M dt $t $t Tr ln Mt = 0 dτ divF. An arbitrary flow therefore has det Mt = exp 0 dτ divF while for a Hamiltonian one with divF = 0 we indeed infer det Mt (X ) = 1.

(9.4.6)

The determinant det Mt is the Jacobian of the canonical transformation X 0 → X t . Its constant value unity therefore means that the inifinitesimal volume element is constant in time, d2 f X0 = d2 f Xt .

(9.4.7)

In fact, the volume element is invariant under any canonical transformation, as can be shown with the help of the generating function S(q, q ); see Problem 9.1. It is worth noting that the linearized flow inherits Hamiltonian character from the original non-linear one2 . The transition δ X 0 → δ X t may thus be seen as a canonical transformation, and that observation will be important in Sect. 9.7.

2 The time dependent Hamiltonian for the linearized flow is obtained from the original H by expanding around the chosen trajectory and dropping terms of higher than second order in ΔX .

346

9 Classical Hamiltonian Chaos

9.5 Liouville Picture The flow of sets of phase-space points mentioned above is worthy of further comments. Each point X fully specifies the state of the system under study. A set of points can be imagined to represent an ensemble of replicas of the system. A density ρ(X, t) in phase space reflects the dynamics of the ensemble as time evolves3 . The integral of ρ(X, t) over all of phase space remains constant in time since time evolution neither adds nor takes away any system from the ensemble,  d 2 f Xρ(X ) = const ;

(9.5.1)

if the constant value of the integral is chosen as unity ρ becomes a probability density, but other normalizations can be useful as well. The evolution equation of the density ρ is most easily found by momentarily assuming an ensemble of N replicas at the points X μ (t), μ = 1 . . . N such that ρ(X, t) =

N 

δ(X − X μ (t))

(9.5.2)

μ=1

becomes a number density of replicas in phase space. Taking the time derivative and using Hamilton’s equations one gets Liouville’s theorem, the evolution equation in search, ρ˙ + {H, ρ} = ρ˙ + ∂X Fρ = 0,

(9.5.3)

which has the form of a continuity equation for the flow. Since the “velocity field” F(X ) is devoid of sources (see (9.2.4)) the set of replicas behaves like an incompressible gas. Indeed, divF = 0 allows to write Liouville’s theorem in the form ˜ X ρ = 0 which reveals time independence of ρ in a locally co-moving frame ρ˙ + F∂ of coordinates, X = X − Ft, t = t. An important conclusion from Liouville’s theorem is a generalization of the constancy in time of the normalization integral (9.5.1) to any 2 f dimensional subvolume V of phase space, irrespective of the shape,  V(t)

d 2 f Xρ(X, t) = const ;

(9.5.4)

the constancy of the density ρ(X, t) in locally co-moving coordinates is at the basis of that invariance, together with the constancy of each differential volume element d 2 f X , see (9.4.7). 3 The notion of ensembles is of particular importance for the statistical treatment of many-particle sytems.

9.6

Symplectic Structure

347

An interesting special case arises when the initial density ρ(X, 0) is chosen as the “characteristic function” of V(0) which vanishes outside and equals unity everywhere inside. Time evolution turns the density into the characteristic function of V(t) such that  d 2 f X = const. (9.5.5) V(t)

The foregoing integral is known as one of Poincar´e’s integral invariants. The reader is kindly $ invited to slightly modify of the argument just given to show that the integral V d 2 f X is in fact invariant under any canonical transformation.

9.6 Symplectic Structure Three infinitesimally close phase-space points X, X + δ X , and X + δ X define a parallelogramm in the tangent space at X . The quantity δ X Σδ X = δqδp − δpδq

(9.6.1)

 (which for f freedoms can be read as i [δqi δpi −δpi δqi ] ) is called symplectic area element4 .Alluding to area in that name is appropriate since in the ith phase plane

the contribution δqi δpi −  which equals the area of the parallelogramm  δpi δqi arises spanned by the vectors

δqi δpi

and

δqi

δpi

(see Fig. 9.1).

The symplectic area element is conserved. The column vector dtd δ X = F(X + δ X ) − F(X ) = (δ X · ∂ X )F(X ) = (δ X · ∂ X )Σ∂ X H and its transpose are involved in the proof of that conservation law. The antisymmetry (9.2.3 ) of the “fundamental X = −(δ X · matrix Σ of the symplectic structure” yields the “row vector” dtd δ 2  ∂ X )∂ X Σ H . Due to Σ = −1 the announced conservation law then indeed results, d  X (δ X · ∂ X )∂ X H = 0. δ X Σδ X = (δ X · ∂ X )(∂X H · δ X ) − δ dt

(9.6.2)

p δp δq

Fig. 9.1 Conserved symplectic area element spanned by two vectors in tangent space to phase space

4

δp δq  q

The symplectc area element can also be written as the antisymmetric wedge product δ X ∧ δ X .

348

9 Classical Hamiltonian Chaos

A (second, beyond (9.5.5)) Poincar´e integral invariant arises when the symplectic area element is integrated over a two dimensional surface S with a closed boundary γ . The resulting area  S

δ X Σδ X =

 Si

i

(dqi dpi − dpi dqi )

(9.6.3)

receives an additive contribution from each phase plane wherein the projections Si and γi appear. Instead of the parallelogram-shaped symplectic area elements dqi dpi − dpi dqi , the rectangular elements dqi dpi may be collected. The symplectic area on S then takes the form   7

 dqi dpi = dqi pi ; (9.6.4) δ X Σδ X = S

i

Si

i

γi

the last member of the foregoing chain of equations is obtained through Stoke’s theorem, with the integrand pi (qi ) representing the closed curve γi . Now, when all points on the surface S and thus all points on the contour γ move with the Hamiltonian flow the integral remains invariant, due to the conservation of the differential symplectic area element. The integral invariant (9.6.4) is not only invariant under time evolution but in fact under general canonical transformations. It plays an important role in advanced presentations of analytical mechanics [6]. It will come up in the action difference for pairs of orbits differing in close encounters, (see Sect. 9.15.2).

9.7 Lyapunov Exponents Initially close-by phase-space trajectories of chaotic dynamics in general separate exponentially as time elapses. In other words, the pertinent initial-value problem (i.v.p.) can be said to be exponentially unstable. That property is in fact often taken as the paradigm of chaos. The present section is devoted to a quantitative discussion. The Jacobian matrix Mt (X ) allows to associate exponential growth rates alias Lyapunov exponents with 2 f different directions e in the tangent space at every phase-space point X , λ(X, e) = lim

|t|→∞

1 ln Mt (X )e . t

(9.7.1)

Inasmuch as the Hamiltonian flow leaves the generator d F(X )/d X of the linearized flow bounded the foregoing limit must exist, with λ < ∞. We label the 2 f distinguished directions and the pertinent Lyaplounov exponents by an index as ei , λi (X ) = λ(X, ei ); several directions may yield the same λ. To identify the direction e1 pertaining to the largest Lyapunov exponent λ1 one must vary a trial direction e until the rate λ1 is obtained both for large positive and

9.7

Lyapunov Exponents

349

negative times. By further variations of e another direction e2 must be found where either λ1 is again in effect for t → ±∞ (then λ1 would be degenerate) or the next largest exponent λ2 is. So proceeding stepwise one can establish all λ’s and the 2 f distinguished directions. It is well to note that the different directions are in general not mutually orthogonal. Further insight into good strategies for finding the λ’s and e’s will arise below. Writing out the norm in the above definition (9.7.1) as Mt (X )e

2

t (X )Mt (X )e = e˜ M

(9.7.2)

we see that the Lyapunov exponents may also be obtained through the eigenvalues t (X )Mt (X ) as σi (t, X ) of the non-negative real symmetric matrix M λi (X ) = lim

t→∞

1 ln σi (t, X ). 2t

(9.7.3)

t Mt must not be confused with the The mutually orthogonal eigenvectors of M vector fields ei (X ) distinguished by the different Lyapunov exponents. In the direction e (X ) ∝ X˙ = Σ∂ X H along a trajectory, the Lyapunov exponent vanishes. To see that, we may differentiate Hamilton’s equations (9.2.2) w.r.t. time and conclude that δ X = τ X˙ solves the linearized evolution equation (9.4.1). With δ X 0 = τ X˙ 0 , δ X t = τ X˙ t we thus have X˙ t = Mt (X 0 ) X˙ 0 and5 1 1 X˙ t F(X t ) ln = 0. = lim ln ˙X 0 t→∞ t t→∞ t F(X 0 )

λ(X 0 , e (X 0 )) = lim

(9.7.4)

A second vanishing Lyapunov exponent arises in the direction perpendicular to the energy shell, e E ∝ ∂ X H . Continuous symmetries and the pertinent conservation laws would entail further vanishing λ’s. In analogy with the conserved energy two neutral directions come with a conserved quantity C, and these can be shown to be ∂ X C and Σ∂ X C [2]. In particular, for integrable dynamics where trajectories wind around f dimensional tori such that initially close points stay close forever, no exponential divergence is possible and all λ’s vanish. For chaotic Hamiltonian flows the Lyapunov exponents arise in pairs ±λ. This important fact follows from the Hamiltonian character of the linearized flow (9.4.1). Consequently, the time evolution is again a canonical transformation. Parametrizing the deflections δ X t by canonical pairs we have Poisson brackets {δpi (t), δq j (t)} =

5 For the Hamiltonian dynamics under exclusive consideration here, no trajectory can end in, start out from, or pass through a stationary point such that neither F(X 0 ) = 0 nor F(X t ) = 0 make for worry; isolated moments of time with F(X t ) = ∞ arise for elastic collisions of the hard-wall type but do not invalidate the conclusion.

350

9 Classical Hamiltonian Chaos

δi j , {δpi (t), δp j (t)} = 0, {δqi (t), δq j (t)} = 0, valid at all times. The latter equations can be written compactly as the matrix identity ˜ M =Σ MΣ

˜ =Σ; MΣ M

or

(9.7.5)

˜ MΣ M ˜ M = Σ and implies that an eigenvalue σi is accompanied an iterate reads M with its inverse 1/σi as another one, with eigenvectors u i and Σu i . The associated Lyapunov exponents therefore are ±λi . It follows that the number of vanishing Lya2 f punov exponents is even. We further conclude i=1 λi = 0, and that property of Hamiltonian flows may be seen as a manifestation of Liouville’s theorem (preservation of phase-space volumes, see Sect. 9.2). To find further properties of the Lyapunov exponents we must dig a little deeper into nonlinear dynamics.

9.8 Stretching Factors and Local Stretching Rates By its definition (9.4.1) the Jacobian Mt (X 0 ) tranports a deflection δ X 0 in the tangent space at X 0 to the deflection δ X t in the tangent space at X t = Φt X 0 . Writing ei (X 0 ) for the initial deflection and Λi ei (X t ) for the one at time t we may note Mt (X 0 )ei (X 0 ) = Λi (t, X 0 )ei (X t ) ;

(9.8.1)

if we were to normalize as ei (X 0 ) = ei (X t ) = 1 the “stretching factor” Λi (t, X 0 ) would reflect all the stretching or shrinking incurred under the linearized flow. It is actually convenient to avoid the condition of unit length and stipulate the weaker condition that ei (X t ) does not grow exponentially in time, lim

t→∞

1 ln ei (X t ) = 0. t

(9.8.2)

The stretching factor Λi (t, X 0 ) then picks up all exponential growth associated with the direction ei , if there is any. We may choose 2 f different directions ei (X 0 ), i = 1 . . . 2 f , in general not orthogonal, to get the Lyapunov exponents as λi (X ) = λ(X, ei ) = lim

t→∞

1 ln Λi (t, X ) . t

(9.8.3)

A second set of 2 f vectors f i (X 0 ) then exists such that the e’s and f ’s form biorthogonal pairs at each phase-space point X ˜f i (X )e j (X ) = δi j

2f  i=1

ei (X ) ˜f i (X ) = 1.

(9.8.4)

9.8

Stretching Factors and Local Stretching Rates

351

A “cocycle decomposition” of Mt (X ) can now be written as Mt (X ) =

2f 

ei (Φt X )Λi (t, X ) ˜f i (X ).

(9.8.5)

i=1

The group-type property (9.4.4) of the Jacobian entails one for the stretching factors, Λi (t + τ, X ) = Λi (t, Φτ X )Λi (τ, X ).

(9.8.6)

An important conclusion from the foregoing group-type property and the relation (9.8.3) between the Lyapunov exponents and the stretching factors is λi (X ) = λi (Φτ X ), i.e. the constancy of all Lyapunov exponents along each trajectory. Different trajectories may have different λ’s, though. To reveal the final property of the Lyapunov exponents of relevance here we differentiate the definition (9.8.1) of the stretching factor w.r.t. time to get the evolution equation ˙ i (t, X ) = χi (Φt X )Λi (t, X ) Λ

(9.8.7)

with the “local stretching rate”   ∂ F(X ) ∂ei (X ) ei (X ) − F(X ) . χi (X ) = ˜f i (X ) ∂X ∂X

(9.8.8)

In contrast to the Lyapunov exponents the local stretching rate can and in general does vary along a trajectory. Upon formally $ t integrating the evolution equation of the stretching factor we have Λi (t, X ) = exp 0 dτ χi (Φτ X ) and thus find the Lyapunov exponent as a time average of the local stretching rate along a trajectory, 1 t→∞ t



t

λi (X ) = lim

dτ χi (Φτ X ).

(9.8.9)

0

For ergodic dynamics time averages equal ensemble averages (over the energy shell), and we can conclude that in that case all endless trajectories have the same Lyapunov exponents. Periodic orbits may still retain their individual λ’s; for growing periods, however, these λ’s can be expected to approach those of infinite trajectories. We shall come back to the effective ergodicity of long periodic orbits in Section 9.12 below. A final comment is in order on the 2 f vector fields ei (X ) giving rise to the various Lyapunov exponents. The ei (X ) can be grouped into “stable directions” (λi < 0), “unstable directions” (λi > 0) and “neutral directions” (λi = 0). The three groups span the stable, unstable, and neutral subspaces of the tangent space at each phasespace point X .

352

9 Classical Hamiltonian Chaos

9.9 Poincar´e Map Laying the ground for modern nonlinear dynamics Henri Poincar´e introduced a description of Hamiltonian flows in terms of a discrete map in a subspace of phase space which I propose to first consider for autonomous flows on the (2 f −1) dimensional energy shell H = E. A second (2 f − 1) dimensional hypersurface intersecting the energy shell in a 2 f − 2 dimensional manifold is to be chosen such that it is pierced by all trajectories of interest. That latter “Poincar´e surface of section” P may be parametrized by ( f − 1) canonical pairs {qi , pi ; i = 1, . . . , f − 1} which I summarily denote by x (see Fig. 9.2). For example, for f = 2 and a Hamiltonian of the form H = p12 /2m + p22 /2m + V (q1 , q2 ) the energy shell % may be spanned by q1 , p1 , q2 with the missing momentum given by p2 = ± 2m(E − V (q1 , q2 )) − p12 . Fixing, say, the plane q2 = 0 to accompany the surface H = E we have the two dimensional Poincar´e section q1 , p1 . For f > 2 I may similarly set q f = 0, p f = % spanned by the pair  f −1 2 ± 2m(E − V (q)) − i=1 pi . It is convenient to place P transversal to the flow such that it does not contain the direction e along the flow. If there are no constants of the motion other than the energy, P then contains no neutral directions at all. In case there are further constants of the motion it is convenient to also exclude the pertinent pairs of neutral directions from P [2]. An endless trajectory pierces through the Poincar´e section over and over again, each time in a different point x. For periodic orbits, of course, only a finite number of distinct piercing points arise. Of exclusive interest for us are piercings with the same sign of p f , say the positive one, since with that restriction a point x in P is one-to-one with a phase-space point and thus defines a phase-space trajectory. The n-th such piercing of a trajectory through P, denoted by xn , uniquely fixes the

Fig. 9.2 Poincar´e section in three-dimensional energy shell, pierced by periodic orbit

9.9

Poincar´e Map

353

subsequent one, xn+1 , as well as the “first-return time” T (xn ). The points and times of piercings are related by the Poincar´e map xn+1 = φ(xn ),

tn+1 = tn + T (xn ).

(9.9.1)

The Jacobian matrix of the Poincar´e map, m(x) =

∂φ(x) ∂x

(9.9.2)

is 2( f − 1) × 2( f − 1) if the energy is the only conserved quantity. Inasmuch as P is parametrized by canonical pairs of variables the Poincar´e map amounts to a canonical transformation and therefore is “area” preserving, det m = 1.

(9.9.3)

The linearized Poincar´e map, δxn+1 = m(xn )δxn , allows to determine 2( f − 1) Lyapunov exponents of endless trajectories through λi (x0 ) = lim

1

n→∞ tn

ln m(xn )ei .

(9.9.4)

For periodic orbits the Jacobian m is customarily called monodromy matrix and provides a convenient starting point for getting the Lyapunov exponents. If x ∈ P is a point of an orbit γ with the primitive period Tγ the eigenvalues Λi of m(x) yield the λ’s as λi = (1/Tγ ) ln Λi , without any limiting procedure. It may be well to summarize at this point the important properties of the eigenvalues Λi of the monodromy matrix m: (i) They are real or come in pairs of mutual complex conjugates (since m is a real matrix.) (ii) They come in pairs of mutual inverses. (iii) Inasmuch as the Poincar´e section contains no neutral direction unity does not qualify as an eigenvalue. (iv) Complex eigenvalues are unimodular and refer to stable subspaces; for f = 2 the underlying orbit is stable, and such behavior is forbidden for hyperbolic dynamics. The Poincar´e surface of section allows to numerically test for integrability vs chaos in the case f = 2 where P is two dimensional. A constant of the motion independent of the energy would intersect P in a one dimensional curve. A sequence of piercings xn would “fill” that curve. In the non-integrable case where no such constant of the motion exists the xn will not be confined to a curve but rather explore a whole area in P. In generic systems one meets with islands of regular motion surrounded by a chaotic “sea” and in such cases P can accommodate sequences xn of both types, curve filling and aerea exploring (see Fig. 9.3).

354

9 Classical Hamiltonian Chaos

Fig. 9.3 Piercings of regular (a) and chaotic (b) orbit through Poincar´e section

a

b

p2

q1 p1

9.10 Stroboscopic Maps of Periodically Driven Systems Periodic driving makes for explicitly time dependent Hamiltonians, H (t) = H (t + T ),

(9.10.1)

where T is the period of the driving. A stroboscopic description with the strobe period T is then indicated and amounts to looking at the continuous flow X t = Φt (X 0 ) ≡ Φt X 0 only at the discrete times t = nT, n = 0, 1, 2, . . .. The map X nT = ΦT X (n−1)T = ΦnT X 0

(9.10.2)

still defines a group since Φ(n+m)T = ΦnT ΦmT and ΦnT Φ−nT = 1 and again is a canonical transformation. The Jacobian matrix MnT of the strobe map defines a linearized discrete map, δ X nT = MnT (X 0 )δ X 0 ,

MnT (X 0 ) =

∂ΦnT (X 0 ) , ∂ X0

(9.10.3)

and again satisfies the group-type relation M(m+n)T (X 0 ) = MnT (X mT )MmT (X 0 ).

(9.10.4)

As well as area preservation, det MnT = 1. Everything said above about Lyapunov exponents can be transcribed. Important differences lies in the absence of energy conservation and of the two neutral directions e E , e . As a consequence, periodic driving enables single-freedom systems to behave chaotically.

9.12

The Sum Rule of Hannay and Ozorio de Almeida

355

Should there be other constants of the motion the stroboscopic map can be reduced to the corresponding subspace of the phase space, as for flows. A popular special case is periodic kicking involving Dirac deltas as H (t) = H0 + H1 +∞ n=−∞ δ(t − nT ). Examples are the kicked rotator and the kicked top. When dealing with maps below I shall suppress the strobe period T as X nT → X n , ΦnT → Φn , MnT → Mn ; in brief, the dimensionless integer time n will be employed as well as dimensionless Lyapunov exponents, λT → λ. I should mention that there are area preserving maps which cannot be understood as stroboscopic descriptions of Hamiltonian flows. The baker map and the cat map are well known examples, see, e.g. Ref. [3]. Their Jacobian matrices still have the properties just mentioned for stroboscopic maps.

9.11 Varieties of Chaos Generic systems have mixed phase spaces where islands of regular motion are surrounded by “chaotic seas”. Upon changing a suitable control parameter periodic orbits “are born” or “die out” at bifurcations. For ergodic dynamics time averages along endless trajectories equal ensemble averages with suitable invariant measures. Mixing dynamics have equilibrating phase-space densities, on the energy shell or, if more conserved quantities exist, the correspondingly reduced submanifold: A cloud of points on the accessible manifold spreads out uniformly. Correlation functions of observables tend to factorize as lim|t−t |→∞ A(t)B(t ) → AB with stationary means A, B. Mixing implies ergodicity but not vice versa. I shall be mostly concerned with hyperbolic systems which have non-zero Lyapunov exponents everywhere or at least almost everywhere. Ergodicity and mixing are then granted (unless the phase space falls into disjoint regions).

9.12 The Sum Rule of Hannay and Ozorio de Almeida Ergodicity implies that almost all endless trajectories visit everywhere in the accessible part of phase space. For autonomous flows with only the energy conserved the energy surface is then covered uniformly by a typical trajectory as

lim

T →∞

1 T



T

dt δ(X − Φt X 0 ) =

0

1 δ(H (X ) − H (X 0 )) Ω

(9.12.1)

$ where Ω denotes the volume of the energy shell, Ω = d 2 f X δ(H (X ) − H (X 0 )). For stroboscopic (or other area preserving) maps without any conserved quantities ergodicity means similarly

356

9 Classical Hamiltonian Chaos

lim

N →∞

N 1  1 δ(X − Φn X 0 ) = N n=0 Ω

(9.12.2)

but now Ω is the full phase space volume which is assumed finite. As nonadmissible appear initial points X 0 on periodic orbits since a finite-period orbit can certainly not visit everywhere. However, the set of all periodic orbits with periods in a window [T, T + ΔT ] might be expected to cover the available space uniformly as T → ∞. The sum rule of Hannay and Ozorio de Almeida [7, 8] substantiates that expectation in a most useful way, by identifying the appropriate weighting of each orbit. The following derivation of the HOdA sum rule assumes isolated periodic orbits. For pertinent mathematical work see Refs. [9, 10].

9.12.1 Maps Inasmuch as the ergodic property (9.12.2) holds for almost all initial points (excluded should remain short periodic orbits) and for all points X in phase space I may take the freedom to set X = X 0 and to let the sum over n start at some value N0 overwhelmingly larger than all characteristic times of the dynamics at hand. Indeed then, the delta function δ(X 0 − Φn X 0 ) peaks at the periodic points on period-n orbits or on orbits whose primitive periods n/r fit an integer number r of times in n. With the promise to presently deliver justification I discard such repeated shorter orbits and write δ(X 0 − Φn X 0 ) =

(period n) n   p

i=1

1 p δ(X 0 − X i ) ; | det(M p − 1)|

(9.12.3)

p

here M p = ∂∂ XXn0 is the 2 f × 2 f Jacobian matrix of the period-n orbit p, customarily p called monodromy matrix, which is the same on all n points X i on that orbit. On inserting the foregoing identity in the ergodicity property (9.12.2) and integrating over phase space I get lim

N →∞

N 1  N n=N



(period n)

0

p

n = 1. | det(M p − 1)|

(9.12.4)

 Apart from fluctuations the quantity p | det(Mn p −1)| must itself approach unity as the period n grows. Doing away with such fluctuations by an average over a sufficiently large window Δn of periods with 1  Δn  n I get 8(period n)  p

n |M p − 1|

9 Δn

1 = Δn

(periods ∈ [n,n+Δn])

 p

n ∼1 | det(M p − 1)|

9.12

The Sum Rule of Hannay and Ozorio de Almeida

357

and, by finally dividing out n, the sum rule of Hannay and Ozorio de Almeida, 1 Δn

(periods ∈ [n,n+Δn])

 p

1 1 ∼ . | det(M p − 1)| n

(9.12.5)

An immediate consequence of the HOdA sum rule is exponential proliferation of periodic orbits with growing period. Indeed, for large periods n the monodromy matrix is dominated by an exponential as | det(M p − 1)| ∼ enλ where λ stands for the sum of all positive Lyapunov exponents. The sum rule thus implies #{period-n orbits} ∼

enλ . n

(9.12.6)

In fact, that exponential proliferation yields the selfconsistent justification of dropping r -fold repetions of orbits of primitive period n/r with r = 2, 3, . . . from the above sums: these shorter orbits are exponentially outnumbered by the orbits with primitive period n while the determinant det(M p − 1) keeps growing like eλn for all period-n orbits, primitive or not.

9.12.2 Flows For Hamiltonian flows, the HOdA sum rule looks rather the same as for maps, 1 ΔT

(periods ∈ [T,T +ΔT ])

 p

1 1 = | det(m p − 1)| T

for

ΔT  T,

(9.12.7)

but the monodromy matrix m p of the orbit p is now 2( f − 1) × 2( f − 1) and refers to a Poincar´e map on the energy shell, see (9.9.1) and (9.9.2). The derivation is a little trickier but proceeds in the same spirit as for maps. It is well to realize that the phase-space point Φt X 0 never leaves the energy shell in which the initial point X 0 lies. The ergodicity property (9.12.1) can thus be reformulated for the on-shell flow. To that end I choose the energy E as one phase-space coordinate and write X = E, x with suitable 2 f − 1 coordinates x on the energy shell. With the on-shell flow denoted by xt = ΦtE x0 , ergodicity according to (9.12.1) implies lim

T →∞

1 T



T 0

dt δ(x − ΦtE x0 ) =

1 , Ω

(9.12.8)

where E = H (X 0 ) is understood. Now proceeding as for maps I set x = x0 and integrate over the energy shell. On the r.h.s. of (9.12.8) that integral  yields unity while on the l.h.s. a sum of contributions of periodic orbits arises, 1 = p I p . To pick up the contribution I p of an orbit p the energy-shell integral may be confined to a tube

358

9 Classical Hamiltonian Chaos

τ p enclosing p; due to the delta function in the integrand the tube may be chosen so thin that no other orbit can squeeze in. Each tube can be parametrized by 2( f − 1) transverse coordinates x⊥ (with x⊥ = 0 on the orbit) and a longitudinal one along the flow. The latter coordinate must be a time-like one6 and will be called t ; it runs from zero up to the period T p . Within the tube τ p the flow ΦtE shifts the longitudinal coordinate t to (t + t)modT p while the action on x⊥ can be approximated linearly,   ∂ x⊥t  ΦtE (t , x⊥ ) = (t + t)modT p , x⊥ .  ∂ x⊥ x⊥ =0

(9.12.9)

The 2 f − 1 dimensional delta function in the integral over the tube τ p thus includes a T p -periodic temporal delta function, δ(x − ΦtE x) = =

  ∂ x⊥t  δ(t − t − t + r T p )δ x⊥ − x  ⊥ ∂ x⊥ x⊥ =0 r=1

∞ 

∞ 

δ(t − r T p ) δ(x⊥ − m rp x⊥ ) ;

(9.12.10)

r=1



∂ x⊥t  ∂ x⊥ x =0 ⊥

here, the Jacobian matrix

is evaluated at successive completions of the

orbit p where it becomes a power of the monodromy matrix m p . As for maps I discard repetitions and keep δ(t − T p ). The tube integral now results as  I p = δ(t − T p )



Tp

dt

d 2 f −2 x⊥ δ(x⊥ − m p x⊥ ) =

0

T p δ(t − T p ) . | det(m p − 1)|

(9.12.11)

The time average demanded on the l.h.s. of (9.12.1) remains to be done. By there setting the lower limit of the time integral to some value T0 much in excess of all characteristic times of the dynamics I again exclude short periodic orbits, lim

T →∞

1 T



T

dt T0

 T p δ(t − T p ) = 1. | det(m p − 1)| p

(9.12.12)

As for maps I now argue that the integrand must approach unity for growing time t, fluctuations apart. To get rid of those latter it suffices to do a time average over some finite interval ΔT ,

Imagine phase space parametrized by a canonical pair p , q for momentum and coordinate along the flow, together with f −1 further pairs making up x⊥ ; then replace p by E and realize ∂∂pE = q˙ to conclude dp dq = d Edt ; here dt equals the time differential dt, but it is well to distinguish the phase-space coordinate confined as 0 ≤ t < T p from the time t which never ends. 6

9.13

Propagator and Zeta Function

>

359

? T p δ(t − T p ) 1 = | det(m p − 1)| ΔT

(T 0. They may be imagined ordered as 0 < Re γ1 < Re γ2 < . . .. The leading one

360

9 Classical Hamiltonian Chaos

determines the fraction of the volume of a subspace σ of the energy shell that has not escaped by the time t as  σ

d 2 f −1 x d 2 f −1 x0 δ(x − ΦtE x0 )



−1 dx

σ

∼ e−γ1 t .

(9.13.4)

I follow Cvitanovi´c and Eckhardt [5, 12] to express the trace in terms of periodic orbits. To pick up the contribution of the (r -fold traversal of the) pth primitive orbit, I do as in the previous section and restrict the integration range to a narrow toroidal tube around that orbit. The tube integral has already been calculated in the previous section and now yields the trace 

d 2 f −1 x P(x, t|x) =



Tp

p



∞  r=1

δ(t − r T p ) | det(M rp − 1)|

(9.13.5)

∞ ∂  e−ikr T p dk e ; ∂k p r =1 r | det(M rp − 1)| −∞

i = 2π

∞

ikt

here I have taken the freedom to rename the monodromy matrix as m p → M p , following common practice. The shorthand Z (s) = exp −

∞  p

r=1

esr T p r | det(M rp − 1)|

. (9.13.6)

offers itself now. It is known as the dynamical zeta function of the flow and allows us to write the trace under study as  d

2 f −1

i x P(x, t|x) = 2π



+i∞

−i∞

ds e−st

Z (s)  m i e−γi t . = Z (s) i

(9.13.7)

But the foregoing identity means that the resonances γi can be determined from the zeros of Z (s), just as if the latter were something like a secular determinant for the classical propagator; this is why Z (s) is called a zeta function. To facilitate comparison with the quantum zeta function to be met in Sect. 10.5. I reexpress the classical zeta function as a product over periodic orbits. For maximal convenience I do that for f = 2. Readers interested in arbitrary f will find pleasure in generalizing themselves or may consult [5, 13]. We must recall that for Hamiltonian flows of two-freedom systems, the stability matrix has two eigenvalues that are mutual inverses. For hyperbolic dynamics with no stable orbits, these eigenvalues are real and may be denoted as Λ p and 1/Λ p , where |Λ p | > 1. The inverse-determinant weight of the pth orbit may be expanded as

9.13

Propagator and Zeta Function

361

−1 −2 | det(M rp − 1)|−1 = |(1 − Λrp )(1 − Λ−r = |Λ p |−r (1 − Λ−r p )| p )

=

∞  ∞ 

j+k)r |Λ p |−r Λ−( . p

(9.13.8)

j=0 k=0

Once this expansion is inserted in the zeta function (9.13.6), the sum over repetitions of the pth orbit yields a logarithm, ln Z (s) =

∞ ∞   p

j=0 k=0

 ln 1 −



esTp j+k

|Λ p |Λ p

.

(9.13.9)

Choosing l ≡ k + j and k as summation variables I can do the sum over k as well, and upon reexponentiating get the desired product form of the zeta function Z (s) =

∞ ,, p l=0



esTp 1− |Λ p |Λlp

l+1 .

(9.13.10)

Now, it is easy to see that the zeta function has a simple zero at s = 0. Since the small-s behavior must be tied up with long orbits, we may, looking at ln Z (s) sT p |Λ p |−1 Λ−l as given by (9.13.10), expand ln(1 − esTp |Λ p |−1 Λ−l p ) ≈ −e p and then keep only the leading term of the l-sum to get s→0

ln Z (s) −→ −

 ∞  esTp esT dT ≈− ≈ ln(sT0 ) ; |Λ p | T T0 p

(9.13.11)

here I have invoked the HOdA sum rule in the form (9.12.1) to recast the sum over long orbits into an integral, with T0 some reference time. We may imagine that T0 is the period upward of which the HOdA sum rule begins to apply and thus T0  TH ; the precise value does not matter. At any rate, the claim Z (s) ∝ s for s → 0 is borne out. Of course, the simple zero of Z (s) at s = 0 reflects the simple unit eigenvalue of the propagator corresponding to the stationary eigenfunction. In Chap. 10.5 I shall discuss a quantum zeta function related to the spectral determinant of a quantum Hamiltonian. The classical Z (s) as the exponentiated periodic-orbit sum (9.13.6) will find the quantum counterpart (see (10.5.10)) ∞   e irS p (s)/ % , ζ (s) ∝ exp − r p r=1 r | det(M p − 1)|

(9.13.12)

where S p is the action of the pth orbit with the so called Maslov phase included. Even the infinite-product representation (9.13.10) will have a quantum analogue. Both the classical and quantum zeta functions relate the spectrum of an evolution operator to periodic orbits. The differences between the two functions reflect the

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9 Classical Hamiltonian Chaos

fact that probabilities are added in classical mechanics while in quantum mechanics probability amplitudes are superimposed, hence the square root of the inverse determinant and the action dependent phase factor in the quantum zeta function. It is also worth noting that the periodic-orbit form of the classical zeta is exact while the quantum zeta in general is not. Special cases (like geodesic flows on surfaces of constant negative curvature [14]) apart, the quantum zeta is a semiclassial approximation with corrections of higher orders in Planck’s constant.

9.14 Exponential Stability of the Boundary Value Problem The exponential instability of the initial-value problem of hyperbolic dynamics dwelled on above may be considered the paradigm of chaos. I turn to a consequence which on first sight might look like a contradiction in terms. A boundary-value problem a` la Hamilton (b.v.p.) is defined by specifying initial and final positions q0 and qt (but no momentum) and asking for the connecting trajectory piece during a prescribed time span t. No solution need exist, and if one exists it need not be the only one. However, hyperbolicity forces a solution to be locally unique, unless q0 and qt happen to be conjugate points7 . Of foremost interest are time spans long compared to the inverse of the smallest positive Lyapunov exponent, for short t  1/λ (The present discussion need not be burdened by distinguishing between Lyapunov exponents and local stretching rates). Then, slightly shifted boundary points yield a new trajectory piece, with the help of the linearized flow (9.4.1). The new trajectory piece approaches the original one within intervals ; towards of duration ∼ 1/λ in the beginning and at the end, like the “inside” the distance between the perturbed and the original trajectory decays exponentially. That fact is most easily comprehended by arguing in reverse: only an exponentially small transverse shift of position and momentum at some point “deep inside” the original trajectory piece can, if taken as initial data, result in but slightly shifted end points. Indeed, then, we may speak of exponential stability of the boundary-value problem as a consequence of the exponential instability of the initial-value problem. A wealth of insights is opened by that stability of the b.v.p.. For instance, when beginning and end points for the boundary-value problem are merged, qt = q0 with fixed t, each solution in general has a cusp (i.e. different initial and final momenta) there. If the cusp angle is close to π one finds, by a small shift of the common beginning/end, a close-by periodic orbit smoothing out the cusp and otherwise hardly distinguishable from the cusped loop, as shown schematically in Fig. 9.4 Similarly, any piece of an infinite trajectory is closely approached by periodic orbits. To find one such orbit, one may proceed in three steps. First, a b.v.p. with any

7 A focal or conjugate point in configuration space allows for a continuous family of trajectories to fan out, each with a different momentum, which all reunite in another such point; we shall have to deal with conjugate points in the next chapter, see Sect. 10.2.2

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Sieber–Richter Self-Encounter and Partner Orbit

363

Fig. 9.4 Search for a periodic orbit near a closed loop

two points close to the ends of the trajectory piece and with equal duration yields a nearby trajectory, as shown above. Next, the two new boundary points may be varied until a large-angle selfcrossing of the resulting trajectory is found somewhere. Finally, the cusp can be smoothed away as explained above.8 The exponential stability of the boundary-value problem has become the clue to a world of phenomena related to close self-encounters of trajectories and long periodic orbits. As we shall see in the pages to follow, long periodic orbits are not mutually independent individuals but rather come in closely packed bunches.

9.15 Sieber–Richter Self-Encounter and Partner Orbit 9.15.1 Non-technical Discussion Figure 9.5 depicts a long periodic orbit in the so called cardioid billiard. That orbit bounces against the boundary in some dozens of distinct points. It appears to behave ergodically, i.e. to fill the area of the billiard densely and uniformly. Moreover, the depicted orbit crosses itself many times, in the two dimensional configuration space. The smaller the crossing angle the longer the two crossing orbit stretches remain close; if the closeness of those two stretches persists through many bounces we speak of a narrow 2-encounter, the “2” standing for two orbit stretches mutually close in configuration space. A periodic orbit with a small-angle self-crossing (a narrow 2-encounter with two encounter stretches connected by two “links”) is sketched in Fig. 9.6. In drawing 8 Henri Ponicar´e may have had in mind similar ideas when writing [15] “Etant donn´ees . . . une solution particuli`ere quelconque de ces e´ quations, on peut toujours trouver une solution p´eriodique (dont la p´eriode peut, il est vrai, eˆ tre tr`es longue), telle que la diff´erence entre les deux solutions soit aussi petite que l’on veut, pendant un temps aussi long qu’on le veut. D’ailleurs, ce qui nous rend ces solutions p´eriodiques si pr´ecieuses, c’est qu’elles sont, pour ainsi dire, la seule brˆeche par o`u nous puissions essayer de p´en´etrer dans une place jusqu’ici r´eput´ee inabordable”.

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9 Classical Hamiltonian Chaos

Fig. 9.5 Cardioid billiard with near ergodic periodic orbit

such a configuration-space cartoon some artist’s licence is taken, inasmuch as all other self-crossings and all bounces against the boundary are dispensed with. The cartoon first appeared in a paper by Aleiner and Larkin [16] on weak localization in disordered electronic systems. It was fully exploited for chaotic dynamics by Sieber and Richter [17] who proved the following facts for the Hadamard– Gutzwiller model (geodesic flow on a compact surface of constant negative curvature of genus 2, a hyperbolic system with time reversal invariance): (i) The equations of motion allowing for the self-crossing orbit also allow for a partner orbit which has the crossing replaced by a narrowly avoided crossing, as shown by the dashed line in Fig. 9.6. Away from the crossing or avoided crossing, the partner orbit is exponentially close to the original one; (ii) the action difference for the two orbits vanishes in proportion to the squared crossing angle, ΔS ∝  2 ; (iii) In the limit of large periods T , the number of partners of an orbit related to small-angle

Fig. 9.6 Cartoon of Sieber–Richter pair of orbits. One orbit has small-angle crossing which the other narrowly avoids. Difference between orbits grossly exaggerated. Time reversal invariance required

9.15

Sieber–Richter Self-Encounter and Partner Orbit

365

crossings/avoided crossings has a principal term easily accessible through ergodicity (∝ T 2 sin ), with a relative correction (λT )−1 ln  2 ; (iv) the orbit pairs in question yield the key to the semiclassical explanation of universal spectral fluctuations, and therefore the phenomenon of partner formation in close self-encounters deserves thorough discussion here. The existence of the partner avoiding the crossing follows for general hyperbolic dynamics from the exponential stability of the boundary value problem mentioned above: We may slightly shift apart beginning and end for each loop while retaining , with nearly no change for those loops away from two junctions, as the junctions; by tuning the shifts we can smooth out the cusps in the junctions, , and thus arrive at the partner orbit with an avoided crossing and reversed sense of traversal of one loop. The existence of the partner orbit may also be seen as a consequence of the shadowing theorem [18]. The Sieber–Richter pair of orbits of Fig. 9.6 exists only for time reversal invariant dynamics: The arrows indicate opposite sense of traversal of the left loop which is practically identical otherwise for the two orbits, away from the self-encounter. For a very close 2-encounter, neither orbit in the Sieber–Richter pair can exist without time reversal invariance, because each is required to have two long encounter stretches running mutually close, but with opposite senses of traversal. Somewhat loosely speaking, one may say that the two encounter stretches are nearly “antiparallel”; in phase space, they are “nearly identical up to time reversal”. A long orbit can (and does) also display “parallel” 2-encounters and these again give rise to partner orbits, even for dynamics without time reversal invariance. Figure 9.7 depicts the interesting variant of partner formation then arising. The orbit with two crossing stretches (full line in the figure) may be said to consist of two loops, each of which has a cusp angle close to π. The partner (dashed in the figure) therefore decomposes into two shorter periodic orbits each of which smoothes out the cusp of the close-by loop of the original orbit. Such composite orbits (which together closely “shadow” a periodic orbit) are called pseudo-orbits in the physics literature. The existence of the pseudo-orbit as a partner of the self-crossing orbit in Fig. 9.7 follows from the above discussion of the exponential stability of the boundary-value problem.

Fig. 9.7 Cartoon of simplest pseudo-orbit, partner of orbit with parallel crossing. Time reversal invariance not required

366

9 Classical Hamiltonian Chaos

Intuitive as the phenomenon of partner formation in close self-encounters may now appear, the way towards that insight has been long and difficult. Action correlations between orbits have long been hunted for, mostly in the quest for a semiclassical explanation of universal spectral fluctuations in quantum chaos [19, 20]. Only in 2001 Sieber and Richter found their “figure-eight pair” as the first manifestation of close self-encounters providing the general mechanism of creating correlated orbits, for the Hadamard–Gutzwiller model. Their reasoning was rapidly extended to general chaotic two-freedom systems [21] and eventually to more than two degrees of freedom [22, 23]. Self-encounters with more than two stretches close and the related orbit bunches (first pairs, later quadruplets) were identified in Refs. [23–28]; see Sect. 9.16 below. Extensions to encounters of endless trajectories were investigated in the context of (universal quantum features in) chaotic transport processes [29–35]. A general discussion of orbit bunches can be found in [36]. No genuinely classical “applications” of bunches of trajectories or orbits are known as yet but such might come with time. Readers wishing to think further in that direction are invited to look at Problem 9.5.

9.15.2 Quantitative Discussion of 2-Encounters Poincar´e section, encounter stretches, and links. For theoretical purposes it is best to characterize close self-encounters in phase space or rather the energy shell, for autonomous flows [37–39]. For simplicity I shall assume two degrees of freedom, but all ideas hold for f > 2 as well [22]. For each phase-space point X a Poincar´e section P can be defined transverse to the orbit. Given hyperbolicity, P is spanned by one stable direction es and one unstable direction eu , and the normalization eu ∧ es ≡ eu Σes = 1

(9.15.1)

can be adopted. Considering a neighboring trajectory whose piercing through P is, reckoned from X , δ X = ses + ueu

(9.15.2)

and imagining P moving with the flow, we see the components s, u change as u(t) = Λ(t, X )u(0),

s(t) = s(0)/Λ(t, X ),

Λ(t, X ) ∼ eλt

(9.15.3)

where Λ(t, X ) > 1 is the local stretching factor; as indicated, for large times that factor grows exponentially with the Lyapunov exponent as the rate. To parametrize a 2-encounter within an orbit γ I put P at some point X 1 on one of the two encounter stretches, and choose the moment of that “first” passage as time zero, t1 = 0. The second encounter stretch pierces through P at the time t2 in a point X 2 . For a parallel 2-encounter X 2 is very close to X 1 while for an antiparallel

9.15

Sieber–Richter Self-Encounter and Partner Orbit

367

one the time reverse T X 2 is close to X 1 ; in a unified notation I shall write Y2 . The small difference Y2 − X 1 can be decomposed in terms of the stable and unstable directions at X 1 , Y2 − X 1 = (u 2 − u 1 )eu (X 1 ) + (s2 − s1 )es (X 1 ),  parallel encounter X2 Y2 = T X 2 antiparallel encounter.

(9.15.4) (9.15.5)

When P is moved through the encounter along with the flow the components s2 − s1 , u 2 − u 1 change according to (9.15.3). However, since Hamiltonian time evolution amounts to a canonical transformation the symplectic surface element ΔS = (u 2 − u 1 )eu (X 1 ) ∧ (s2 − s1 )es (X 1 ) = (u 2 − u 1 )(s2 − s1 )

(9.15.6)

remains constant and therefore the precise location of P within the encounter is immaterial. The invariant surface element ΔS will turn out to determine all properties of the encounter of interest. To simplify the notation I shall mostly elevate the first piercing point X 1 to the origin of unstable and stable axes in P such that the invariant surface element takes the form if u 1 = s1 = 0.

ΔS = u 2 s2

(9.15.7)

The definition of the self-encounter can now be completed by setting a bound c > 0 to the components s, u as P is moved, |u 2 |, |s2 | ≤ c.

(9.15.8)

The encounter thus begins when |s2 | falls below the bound and ends when |u 2 | rises above (see Fig. 9.8). The bound must be small enough for the encounter stretches to allow for the mutually linearized treatment of (9.15.3) and should roughly exhaust the range of linearizability, but otherwise the precise value is of no importance. Which of the two partner orbits is called γ and then used to define beginning and end of the encounter is, of course, also immaterial.

s

s

s u

Fig. 9.8 Beginning and end of a 2-encounter

u

u

368

9 Classical Hamiltonian Chaos

The bound c and the surface element ΔS = u 2 s2 are two invariant parameters characterising the 2-encounter under consideration. The encounter becomes closer as ΔS/c2 decreases, and very close encounters enjoy the inequality ΔS  c2 . The orbit pieces in between encounter stretches will be referred to as links. Encounter duration. The duration tenc of an encounter can be estimated due to the asymptotic exponential growth of the stretching factor: The time ts = λ−1 ln(c/|s2 |) passes in between the beginning of the encounter and the piercing through P and the time tu = (1/λ) ln(c/|u 2 |) between the piercing and the encounter end, such that the encounter duration tdec = tu + ts =

1 c2 ln λ |u 2 s2 |

(9.15.9)

is determined by the symplectic surface element ΔS. Partner orbit. The partner orbit γ coming with the 2-encounter is also uniquely fixed by the stable and unstable components s2 , u 2 . Figure 9.9 again depicts the orbits γ , γ in configuration space (part a), as well as the Poincar´e section (part b). a) left port 1

right port 1 X1

Y2’ Y1’

Y2 left port 2

b)

right port 2

stable

s2

Y2’

Y2

X1

Y1’

unstable

u2 Fig. 9.9 Piercings X 1 , Y2 of γ (full line) and piercings Y1 , Y2 of γ (dashed line) for a Sieber/Richter pair, depicted (a) in configuration space, with arrows indicating the momentum, and (b) in the Poincar´e section P parametrized by stable and unstable coordinates. The symplectic area of the rectangle is the action difference ΔS

9.15

Sieber–Richter Self-Encounter and Partner Orbit

369

In (a), the configuration-space piercing points are decorated with arrows indicating momenta for the quadruple of points mutually close in phase space, X 1 , T X 2 for one orbit and X 1 , T X 2 for the other. In (b), the Poincar´e section P is shown with the latter four points   0 , X1 = 0

  u2 T X2 = , s2

X 1

  u2 = , 0

T

X 2

  0 = . s2

(9.15.10)

Putting the origin of the stable and unstable axes at X 1 is a convenient choice. Now, X 1 and X 1 must have practically coinciding stable components s1 = 0 ≈ s1

since when going backwards in time from those piercings we see the two orbits approach one another. Conversely, starting from X 2 and X 2 we see mutual approach of the two trajecories when going forward in time and conclude that those piercings must have very nearly the same unstable components; the time reverses then again share the same stable components s2 ≈ s2 . Similarly, X 1 and T X 2 must practically coincide in their unstable components u 1 ≈ u 2 . Finally, the unstable component of T X 2 practically agrees with the vanishing one of X 1 . The approximate equalities just invoked cannot be strict equalities since the mutual approaches of the two orbits invoked in each of the four cases eventually end somewhere within the links, to give way to divergence towards reentering the encounter; however, the error made for X 1 , T X 2 when  taking ≈→= is exponentially small, of the order exp −λ(t2 −t1 ) = exp − λt2 . Action difference of partner orbits. Next, I propose to show that the action difference surface element ΔS = u 2 s2 . of the two partner orbits γ , γ equals the invariant C That surface element arises as the line integral dq p along the four edges of the parallelogram of Fig. 9.9b, say X 1 → X 1 → T X 2 → T X 2 within P. The action difference Sγ − Sγ can be seen as the sum of four pieces. To characterize them, a pair of exponentially close configuration-space points, ql on γ and ql

on γ , is chosen somewhere in the middle of the left link and similarly a pair qr , qr

in the right link, see Fig. 9.9. The actions Sγ and Sγ then read  Sγ =

q1

ql

 Sγ =



dq p(q, ql ) +

q2

dq p(qr , q) +

q1

q1

ql



qr

dq p(q, ql ) + 

qr

qr

q1

dq p(qr , q) +



ql

dq p(q, qr ) +



q2

qr

dq p(ql , q)

q2

dq p(q, qr ) +



ql

q2

dq p(ql , q).

In each of the foregoing integrals the path of integration is uniquely fixed (as a piece of γ or γ ) by the lower and upper limit; the integrand is a unique function of the integration variable q, given the fixed second argument (ql or qr on γ , ql or qr

on γ ). One of the four pieces of the action difference arises from the first terms in Sγ and Sγ , and there I identify the points ql and ql , accepting an exponentially small error,

370

9 Classical Hamiltonian Chaos

 ΔS

(1)



q1

≈



q1

dq p(q, ql ) −

dq p(q, ql ) =

ql

q1

q1

ql

dq p(q, ql ).

(9.15.11)

A contour integral along an arbitrary path from q1 to q1 results in the second of the foregoing equalities since the integrand p(q, ql ) is a unique function of q; that ∂ S(q, ql ) of the generating function for function is in fact the gradient p(q, ql ) = ∂q the canonical transformation {ql , pl } → {q, p} to which the time evolution along the trajectory starting at X l amounts. The path of integration may be chosen to run in P since the end points X 1 = {q1 , p1 } and X 1 = {q1 , p1 } lie therein; as Fig. 9.9 reveals the path then runs along the unstable axis. For the next piece of the action difference I combine the second term in Sγ with the third in Sγ and set qr = qr , again tolerating an exponentially small error,  ΔS

(2)

qr



q1 qr



≈

dq p(qr , q) −

 =

dq p(qr , q) −

q1

 =

q2

q2

dq p(q, qr ) qr qr q2

dq p(qr , q)

dq p(qr , q)

q1

where for the second line the sense of traversal was reverted in the integral between qr and q2 , by interchanging the integration limits and replacing p(q, qr ) → − p(qr , q). The contour integral in the last line may be evaluated along any path from q1 to q2

since p(qr , q) is a unique function of q. Viewed in phase space, the path starts at X 1 and leads to T X 2 , two points on P; the path may thus be chosen to remain within P and then runs along the stable axis as shown in part b of Fig. 9.9. The reader is kindly invited to reason similarly to get the remaining two pieces  ΔS

≈

(3)



q2

dq p(q, qr ) −

qr



qr

q1

dq p(qr , q) =

q1

dq p(qr , q)

q2

with the final path in P from T X 2 to X 1 and  ΔS (4) ≈

ql



dq p(ql , q) −

q2



ql

q2

dq p(ql , q) =

q2

q2

dq p(q, ql )

(9.15.12)

with the final path in P from T X 2 to T X 2 . The sum of the four pieces now gives the action difference as the closed-contour integral in P 7 Sγ − Sγ =

dq p

(9.15.13)

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Sieber–Richter Self-Encounter and Partner Orbit

371

around the parallelogram spanned by the sequence of points X 1 , X 1 , T X 2 , T X 2 shown in part b of Fig. 9.9. That integral is a canonically invariant quantity; it is the Poincar´e integral invariant discussed in Sect. 9.6. The evaluation in the simplest case of two degrees of freedom is an elementary task since the coordinates u, s form a canonical pair whereupon the line integral just gives the enclosed area. Moreover, the parallelogram is appropriately depicted as a rectangle in Fig. 9.9 since the normalization (9.15.1) absorbs the sine of the angle enclosed by eu and es . Indeed, then, the action difference in search is given by the symplectic surface element characterizing the 2-encounter, Sγ − Sγ = ΔS = u 2 s2 .

(9.15.14)

That result must remain correct for f > 2, with u 2 s2 to be interpreted as a sum of terms, Csimply because the Poincar´e invariant (9.15.13) is such a sum, C f − 1 f −1 dq p = i=1 dqi pi . In view of generalizations to follow the following remark is in order. Without the identification of X 1 with the origin in P the action difference would have come out in the form (9.15.6), i.e. as Sγ − Sγ = ΔS = (u 2 − u 1 ) (s2 − s1 ). Phase-space densities. I finally go for the expected number of close 2-encounters within a periodic orbit, given hyperbolic dynamics. Of interest is the limit of very large periods T , larger by far than any other characteristic time of the classical dynamics. In particular, the Lyapunov time 1/λ, typical encounter durations tenc , and the time scale tmix on which ergodicity and mixing emerge will be assumed dwarfed by T . The term “expected” means an average over the set of all period-T orbits; within that set, the number of 2-encounters fluctuates. Moreover, I shall focus on very narrow 2-encounters for which the encounter time obeys 1/λ, tmix  tenc  T.

(9.15.15)

More detailed and important information is contained in the expected number n T (ΔS)dΔS of 2-encounters with symplectic area elements in the differential interval [ΔS, ΔS + dΔS]. That quantity is in fact the most important statistical characteristic of 2-encounters since it equals the expected number of partner orbits of a given orbit, the partners “generated” in 2-encounters and differing in action from the given orbit by ΔS. I start from the probability for a Poincar´e section P pierced by an orbit γ in a point X 1 at time zero to contain a second piercing X 2 of γ such that T X 2 − X 1 (for an antiparallel self-encounter) or X 2 − X 1 (for a parallel self-encounter) appears with stable and unstable coordinates in the differential box [u, u + du] × [s, s + ds] (with s, u inside the rectangle |s| ≤ c, |u| ≤ c defining the encounter) and a time delay t2 − t1 = t2 in the interval [t, t + dt]. The encounter duration is then fixed near tdec = λ−1 ln(c2 /|us|). Due to the assumption (9.15.15) the second piercing is

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9 Classical Hamiltonian Chaos

statistically independent of the first, and the probability in question is given by the Liouville measure on the energy shell9 dudsdt , Ω  = [u, u + du] × [s, s + ds],  parallel enc. X2 Y2 = T X 2 antiparallel enc.

prob{Y2 − X 1 ∈ , t2 ∈ [t, t + dt]} =

(9.15.16)

where Ω is the volume of the energy shell. I shall show presently that the delay t2 can range in an interval of length T − 2tenc ; anticipating that restriction I proceed to the probability of the differential rectangle containing the second piercing after whatever delay, prob{T X 2 − X 1 ∈ } =

duds(T − 2tenc ) . Ω

(9.15.17)

The probability for the randomly chosen Poincar´e section P to intersect a 2encounter characterized by the invariant symplectic surface element ΔS = us is now obtained by integrating as 



(T − 2tenc ) . Ω −c −c (9.15.18) 1 The factor 2 prevents an overcounting of each 2-encounter, necessary since each of the two encounter stretches can be chosen as the “first” and yields a separate parametrization of the same encounter. The double integral in the foregoing expression is easily evaluated by changing the integration variables to the product us and, say, u; the result reads prob{ΔS ≤ us ≤ ΔS + dΔS} =

1 dΔS 2

prob{ΔS ≤ us ≤ ΔS + dΔS} =

c

du

c

ds δ(ΔS − us)

λtenc (T − 2tenc ) dΔS Ω

(9.15.19)

where the encounter duration must be read as a function of the invariant surface element, tenc = (1/λ) ln(c2 /|ΔS|). On the other hand, that latter probability can be expressed in terms of the expected number of 2-encounters in γ with fixed symplectic surface element, NT (ΔS)dΔS, as the ratio of the expected cumulative duration of all such 2encounters to the orbit period T , prob{ΔS ≤ us ≤ ΔS + dΔS} =

2tenc NT (ΔS)dΔS ; 2T

(9.15.20)

9 To check normalization to unity by integrating over the energy shell would require an extension of the coordinates u, s beyond P.

9.16 l-Encounters and Orbit Bunches

373

here, as in (9.15.18), a factor 12 had to be worked in order to count each encounter once. Comparison of the two expressions yields the number density of 2-encounters with fixed ΔS in a period-T orbit, NT (ΔS) =

λT (T − 2tenc ) . Ω

(9.15.21)

As already announced in the beginning of this subsection, NT (ΔS)dΔS also has the meaning of the number of partner orbits γ of a given period-T orbit γ with the action difference Sγ − Sγ in the interval [ΔS, ΔS + dΔS]. The overall number of partners generated in 2-encounters with any action difference in the permissible interval [−c2 , c2 ] is obtained by integrating as 

c2

−c2

dΔS NT (ΔS) =

λT 2 c2  1  1− . Ω λT

(9.15.22)

In the quantum applications to be discussed in the next chapter it will be technically convenient to work with an auxiliary density wT (s, u) of which the above $$ c NT (ΔS) is the marginal NT (ΔS) = 12 −c duds wT (u, s)δ(ΔS − us), wT (u, s) =

T (T − 2tenc ) . tenc Ω

(9.15.23)

Note that to conform with established usage in the literature I have left out from wT the factor 12 which avoids overcounting. Somewhat loosely speaking one may interpret the latter bivariate quantity as twice the density of 2-encounters in the Poincar´e section. Both densities, NT (ΔS) and wT (u, s), have a leading term ∝ T 2 and a minute correction of relative order tenc /T . The latter will turn out of decisive importance for the semiclassical explanation of universal spectral fluctuations below. I finally fill the promise to show that the delay t2 of the second piercing ranges in an interval of length T − 2tenc . For the antiparallel encounter a` la Sieber and Richter we need to inspect Fig. 9.9a. Since the first encounter stretch is taken as oriented towards the right, the duration of the right link can be read off as t2 − 2tu and that of the left link as T − t2 − 2ts . These link durations must be non-negative, and therefore t2 is confined to the interval 2tu ≤ t2 ≤ T − 2ts ; since tu + ts = tenc the length of that interval comes out as anticipated. For the parallel 2-encounter of Fig. 9.7 the same requirement of non-negative link durations is easily seen to imply tenc ≤ t2 ≤ T − tenc and the same interval length arises.

9.16 l-Encounters and Orbit Bunches Qualitative discussion. The 2-encounters of an orbit just discussed are but a special case of self-encounters. Any number of stretches can run mutually close and when that number is l it is appropriate to speak of an l-encounter [23–28].

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9 Classical Hamiltonian Chaos

Outside an l-encounter, an orbit has l links. Of interest are again orbits with very large periods T . With appeal to the exponential stability of the boundary value problem discussed in Sect. 9.14 we can again ascertain the existence of partner orbits γ

of an orbit γ . The links of γ and γ are nearly identical in the following sense: Reckoned from each end, each link of γ approaches its correspondent of γ expovisualizes that behavior. nentially on the time scale 1/λ; the cartoon To obtain a partner γ from γ the l encounter stretches are slightly deformed to differently connect links. The number of different connections is l! such that an l-encounter may be said to “generate” a bunch of altogether l! orbits; in other words, an orbit γ has l! − 1 partners. However, as is already obvious from Fig. 9.7 for l = 2, γ may be a decomposing pseudo-orbit. Moreover, some partners are formed without the active participation of all l encounter stretches, inactive being stretches which connect the same links as in γ . Equally contained in the number l! − 1 of partners are orbits for which the l-encounter under discussion acts like a collection of effectively independent encounters whose cumulative number of stretches is l. The interesting question as to how many of the l! − 1 partners are (i) genuine, i.e. non-decomposing orbits and (ii) generated such that all l encounter stretches are active without effectively forming a collection of distinct “sub-encounters” will be addressed below10 . Poincar´e section. To parametrize an l-encounter a Poincar´e section P transverse to the orbit at an arbitrary point X 1 (passed at time t1 ) inside one of the encounter stretches. The other stretches pierce through P at times t j ( j = 2, . . . , l) in points X j . All l piercings are mutually close in configuration space while the momenta are either close to or nearly opposite to p1 ; in a unified notation, I shall write Y j for X j in the first case and for T X j in the second one, so as to have all Y j close to X 1 . The small differences can be decomposed in terms of the stable and unstable directions at X 1 ; with the choice u 1 = s1 = 0 the differences read Y j − X 1 = u j eu + s j es .

(9.16.1)

To define the encouner the components are restricted by |s j |, |u j | ≤ c.

(9.16.2)

Encounter duration. As in (9.15.9) we employ the time spans (1/λ) ln(c/|u j |) for the unstable components and (1/λ) ln(c/|s j |) for the stable components to reach c, reckoning from P towards the future and past, respectively. The encounter duration tenc = tu + ts thus generalizes to

10

In permutation-theory parlance, condition (ii) distinguishes l-encounters for which the permutation 1, 2 . . . , l → i 1 , i 2 , . . . , il of left-port labels to right-port labels is a single cycle, rather than a collection of separate cycles; the cycle structure of permutations will become an issue presently; see Fig. 9.10.

9.16 l-Encounters and Orbit Bunches

tu = min j

375

 c 1 ln , λ |u j |

tenc = tu + ts =

ts = min j

 c 1 ln , λ |s j |

2

1 c ln ; λ maxi {|u i |} max j {|s j |}

(9.16.3)

Due to (9.15.3) the duration tenc again remains invariant as P is shifted. Partner formation. The initial and final points of each encounter stretch will be called “entrance” and “exit” ports. If all encounter stretches are almost parallel, - the entrance ports lie all on the same side of the encounter and the exit as in ports on the other side. If, however, the encounter involves mutually time reversed - , it is useful to call “left” the ports on the side where the first stretches like  encounter stretch begins and “right” those on the other side. Obviously, that distinction between left/right and entrance/exit is a necessary one only for time reversal invariant dynamics; without T -invariance, entrance and left are synonymous, as are exit and right. The l stretches and their ports can be numbered in the order of their traversal by the orbit γ : the ith encounter stretch of γ connects the left port i to the right port i. A partner γ has the left port i connected to a different right port j. For γ and γ to be genuinely related by an l-encounter, the l reshufflings i → j must not separate in disjoint groups, or else we would face several distinct encounters at work in the formation of the partner, see Fig. 9.10 and the previous footnote. The points of piercing of a partner γ through P are fixed by those of γ , as already seen for l = 2 in the previous section. It is convenient to number the stretches of γ

by the labels of their left ports and to also employ that label for the Y j . With that choice taken, the ith stretches of γ and γ have (to exponential accuracy) the same stable components of Yi and Yi since upon continuing to the left we see exponential approach of γ and γ (or T γ ) far into the adjoining links, si = si

(common left port i).

(9.16.4)

Similarly, if the ith stretch of γ goes to the jth right port we observe subsequent exponential approach of γ and γ (or T γ ) towards the right and conclude equality

Fig. 9.10 (a) Connections between left and right ports in partner orbit γ related to γ in 6encounter; permutation 123456 → 264513 is single 6-cycle. (b) 6-encounter splits into three pieces; γ thus in effect related to γ in three 2-encounters; permutation 123456 → 214365 has three 2-cycles

376

9 Classical Hamiltonian Chaos

Fig. 9.11 Piercing points of γ ,γ differing in a 3-encounter; inside γ , left ports 1,2,3 are connected to right ports 2,3,1, respectively

stable

s3

Y 3’

Y3

Y2

s2

X1

Y’2

unstable

Y’1

u2

u3

of the unstable components of Yi and Y j , u i = u j

(common right port j).

(9.16.5)

Figure 9.11 depicts the locations of the Yi in P for a 3-encounter. Action difference. The action difference Sγ − Sγ is uniquely determined by the u i , si , i = 2, . . . , l. The above reasoning for l = 2 can be taken over in l − 1 successive steps. Each step interchanges the right ports of two encounter stretches, as in a 2-encounter. Only the last, (l − 1)th, step produces the partner γ while each previous step introduces an intermediate “step”-partner. Each step contributes to the action difference an amount of the type (9.15.14) or rather (9.15.6). It is again convenient to label stretches and piercing points according to the left port. The only difference to the above treatment of 2-encounters is that the jth step has the jth piercing point u j , s j of γ as the reference point whose components vanish only for j = 1; the action difference according to (9.15.6) must therefore be employed with appropriate labels. The stepwise procedure is illustrated in Fig. 9.12. The port-to-port stretches of an orbit γ within a 4-encounter are depicted in part (a) of the figure and the reconnections for a partner γ in part (d). Three steps lead from (a) to (d). In the jth step, the left ports j and j + 1 are respectively connected to the right ports j + 1 and 1.

Fig. 9.12 Stepwise transition from γ (depicted in a) to γ (shown in d) within 4-encounter; each step interchanges right ports of two encounter stretches

9.16 l-Encounters and Orbit Bunches

377

Recalling that the stable and unstable coordinates of piercing points are respectively determined by the left and right ports, we get the separations between the intermediate piercings u j+1 − u 1 , s j+1 − s j and thus the contribution of the jth step t the action difference as the product (u j+1 − u 1 )(s j+1 − s j ). The overall action difference for the example under consideration is the sum Sγ − Sγ =

3  (u j+1 − u 1 )(s j+1 − s j ).

(9.16.6)

j=1

All reconnections for l = 4, and in fact for any l can be treated analogously. A coordinate transformation is useful, {u i , si , i = 2, . . . , l} → {u˜ i , s˜i , i = 1, . . . , l − 1, such that the ith pair u˜ i , s˜i relates to the ith step and the product u˜ i s˜i becomes the contribution of that step to the action difference Sγ − Sγ =

l−1 

u˜ i s˜i .

(9.16.7)

i=1

For the example considered the transformation reads u˜ j = u j+1 − u 1 , s˜ j = s j+1 − s j ). In all cases, the transformation is linear and volume preserving. Due to the simple form (9.16.7) of the general overall action difference it is convenient to redefine the encounter region by bounding the new parameters rather than the old ones as |u˜ i |, |˜si | ≤ c. Phase-space densities. I again invoke the Liouville measure for the probability to find the l −1 points Y j for the piercings j = 2, . . . , l relative to an arbitrarily chosen reference one, X 1 at t1 = 0, in a differential box  j = [s j , s j + ds j ] × [u j , u j + du j ]

(9.16.8)

and the respective delays in [t j , t j + dt j ], l   , du j ds j dt j  prob Y j − X 1 ∈  j , jth delay ∈ [t j , t j + dt j ] j = 2, . . . , l = . Ω j=2



(9.16.9) The piercing times are ordered as t1 < t2 < . . . < tl < t1 + T where T again denotes the orbit period. Moreover, the links in between the l encounter stretches must have positive lengths. With those restrictions we need to integrate over the l − 1 delays in order to get the piercing probability accumulated over all permissible delays, ⎛ ⎞  l ,   du ds j j  ⎝ ⎠ d l−1 t prob Y j − X 1 ∈  j j = 2, . . . , l = Ω j=2 

(9.16.10)

378

9 Classical Hamiltonian Chaos

To do the (l −1)-fold time integral it is convenient to shift the integration variables so as to have zero as the common lower limit of the integration ranges. The restrictions of the range then take the simple form 0 ≤ t2 ≤ t3 ≤ . . . tl ≤ T − ltenc . Since l−1 enc ) the integrand is time independent the integral equals (T −lt , the volume of the (l−1)! (l − 1) dimensional simplex with edge length T − ltenc . The piercing probability accumulated over all permissible delays thus reads l   (T − ltenc )l−1 ,  du j ds j . prob Y j − X 1 ∈  j j = 2, . . . , l = (l − 1)!Ω l−1 j=2



(9.16.11)

The reader will recall the next step in the above reasoning for 2-encounters, where the probability to find a randomly chosen P intersecting a 2-encounter with the symplectic area element ΔS was constructed, irrespective of the position of P inside the encounter and of which encounter stretch is chosen as the reference stretch. The appropriate generalization to an l-encounter requires the separation of the 2(l − 1) parameters s, u into 2l − 3 new variables (Δ1 , Δ2 , . . . , Δ2l−3 ) which describe the internal configuration of the encounter independent of the location of P and the choice of the reference stretch, and a single variable fixing the location of P along the orbit; clearly, for l = 2 there is just a single “internal” variable Δ1 ≡ ΔS. It is useful to at least momentarily think in terms of such Δ’s now and realize that the variables u, s uniquely determine the Δ’s, say as Δ = σ (u, s). A probability density p(Δ) can then be defined by integrating over u, s such that the Δ’s are confined to a small interval,  p(Δ)d 2l−3 Δ =

d l−1 ud l−1 s δ(Δ−σ (u, s))

Δ 1 as functions of q1α , q0 and taking the partial derivative with respect to q1α (at constant q0 ) in the equation of motion (10.2.7). We conclude that Dnα = det G (n,α) =

  ∂qn  m ∂qn  = ; ∂q1α q0 τ ∂ p0α q0

(10.2.17)

the last member of the foregoing chain results from p0α = (m/τ )(q1α − q0 ). With the help of (10.2.5), we may express the inverse determinant by a second derivative of the action as m ∂ m ∂ 2 S (n,α) =− (q α − q0 ) = − . ∂qn ∂q0 τ ∂qn 1 τ Dnα

(10.2.18)

Replacing the determinant Dnα with the mixed second derivative of S (n,α) in the semiclassical propagator (10.2.13) and (10.2.14) we get the latter in the Van Vleck form3

Formally, the contribution of the αth classical path can be seen as arising from a Gaussian integral of the type (4.13.22) after analytic continuation to purely imaginary exponents; that continuation, achieved by the Fresnel integral (10.2.12), brings about the modulus operation and the ν α phase factors e−iπ/2 in (10.2.13). 2

This is equally valid in continuous time; if we let n → ∞, τ → 0 with nτ = t fixed, S (n,α) (q, q ) → S α (q, q , t).

3

388

10 Semiclassical Roles for Classical Orbits



q|U |q  ∼ n

 α

6

    1  ∂ 2 S (n,α)  i S (n,α) (q, q )/ − ν α π/2 , e i2π  ∂q∂q 

(10.2.19)

first spelled out for quantum maps by Tabor [18]. A simple prescription for determining the Morse index ν α derives from the recurα relative to sion relation (10.2.16) for the determinants Diα . A change of sign of Di+1 α Di signals that the number of negative eigenvalues is larger by one for G (i+1,α) than for G (i,α) . This follows from the fact that G (i,α) is obtained from G (i+1,α) by discarding the ith row and column. A well-known corollary of Courant’s minimax theorem [19] then says4 that the i eigenvalues of G (i+1,α) form an alternating sequence with the i −1 eigenvalues of G (i,α) . Therefore, the number ν α of negative eigenvalues of G (n,α) must therefore equal the number of sign changes in the sequence of determinants Diα from D1α to Dnα . Clearly, the prescription just explained relies on the kinetic energy to be a positive quadratic form in the momenta and distinguishes the coordinate representation; for more general situations see Ref. [20]. I leave to the reader as Problem 10.2 the generalization of the Van Vleck formula to f degrees of freedom. Actually, most applications of maps studied thus far are confined to f = 1, and reasonably so since periodically driven single-freedom systems can display chaos. It is only for autonomous dynamics (alias flows), to be looked at in Sect. 10.3.2, that nonintegrability requires f > 1. The very appearance (10.2.19) of the Van Vleck propagator suggests validity beyond the limitations of its derivation given here. It is indeed not difficult to check that the simple product structure (10.2.2) of the Floquet $ τ operator can be abandoned in favor of arbitrary periodic driving, U → (exp{−i 0 dt H (t)/})+ where the timedependent Hamiltonian need not even be the sum of a kinetic and a potential term. According to its property (10.2.17), the determinant Dnα characterizes the stability properties of the classical path {qiα }i=0,1,...,n through δqn ≈ Dnα δq1 . We may define a Lyapunov exponent for the αth path between q0 and qn by λα = lim

n→∞

1 ln |Dnα |. n

(10.2.20)

A path hopping about in a chaotic region will have positive λα such that the determinant in question will grow exponentially, Dnα ∼ eλα n ; a regular path, on the other hand, will have a vanishing Lyapunov exponent; its Dnα is uncapable of exponential growth but might oscillate or grow like a power of n. For a thorough discussion of such different behaviors the reader may consult any textbook on nonlinear dynamics; here, I must confine myself to the few hints to follow. For a rough qualitative characterization, we may replace all curvatures V

(qn ) along the path by a suitable average V

(qnα ) = mωα2 . Within this “harmonic” When the eigenvalues of G (i+1,α) are ordered as g1(i+1) ≤ g2(i+1) ≤ . . . ≤ gi(i+1) and those of G (i,α) (i) as g1(i) ≤ g2(i) ≤ . . . ≤ gi−1 the corollary to the minimax theorem in question yields g1(i+1) ≤ g1(i) ≤ (i+1) (i) (i) g2 ≤ g2 ≤ . . . gi−1 ≤ gi(i+1) . 4

10.2

Van Vleck Propagator

389

approximation dnα = 2 − ωα2 τ 2 = d is independent of n, whereupon the recursion relation (10.2.16) allows for solution by exponentials a n , whose bases are determined by the quadratic equation a = d − 1/a as a± = 1 − ωα2 τ 2 /2 ± & ωα2 τ 2 (ωα2 τ 2 − 4)/4). Accounting for D0 = 0, D1 = 1, we get the solution Dnα =

n n − a− a+ . a+ − a−

(10.2.21)

Three cases arise: (i) For 0 < ωα2 τ 2 < 4, the determinant oscillates as Dnα =

sin nφα sin φα

with

sin

φα 1 = |ωα τ |, 2 2

0 ≤ φα ≤ π.

(10.2.22)

The Morse index ν α is the integer part of nφα /π , the number of sign changes iφα , i = 1, 2, . . . , n. The boundary cases φα = 0 or π can in the sequence sin sin φα be included here as limiting ones and yield the subexponential growth Dnα = n with ν α = 0. Paths of this type are stable and are called elliptic. They exist where the curvature of the potential V (q) is positive and small. (ii) For ωα2 τ 2 < 0, the determinant grows exponentially, Dnα =

sinh n λ˜ α sinh λ˜ α

with

sinh

λ˜ α 1 = |ωα τ |, 2 2

λ˜ α > 0.

(10.2.23)

The Morse index vanishes since Dnα never changes sign. One speaks here of hyperbolic paths. They experience a potential with negative curvature. (iii) Finally, for ωα2 τ 2 > 4, the determinant grows exponentially in magnitude but alternates in sign from one n to the next, Dnα = (−1)(n−1)

sinh n λ˜ α sinh λ˜ α

with

cosh

λ˜ α 1 = |ωα τ |, 2 2

λ˜ α > 0. (10.2.24)

The Morse index is read off as ν α = n − 1. Such inverse hyperbolic paths lie in regions where the potential V (q) has a large positive curvature. In the two unstable cases the parameter λ˜ α approximates the Lyapunov exponent. The exponential growth of |Dnα | with the stroboscopic time n entails the amplitude Aαn in (10.2.13) and (10.2.14) to decay like e−λα n/2 for unstable paths, while stable paths do not suffer such suppression. It would be wrong to conclude, though, that unstable paths have a lesser influence on the propagator than stable ones since the latter are outnumbered by the former, roughly in the ratio eλn where λ is a Lyapunov exponent. The reader will recall the discussion of such exponential proliferation in Sect. 9.12. In the present context, the exponential proliferation threatens the semiclassical propagator (10.2.13) and (10.2.19) with divergence as n → ∞.

390

10 Semiclassical Roles for Classical Orbits

10.2.2 Flows As already mentioned, the Van Vleck propagator (10.2.19) also applies to the continuous time evolution generated by the Hamiltonian H = T + V : By taking the limit n → ∞, τ → 0 with the sproduct t = nτ fixed, we get the result anticipated in (10.2.1),  q|e−iH t/ |q  ∼ α

6

   1  ∂ 2 S α (q, q , t)  i S α (q, q , t)/ − ν α π/2 . (10.2.25) e i2π  ∂q∂q

Here, the discrete time index n was replaced with the continuous time t as an independent argument of the action of the αth classical path going from q to q during the time span t. As before, the action S α (q, q , t) is the generating function for the classical transition in question. To see what happens to the Morse index ν α in the limit mentioned (nothing, really, except for a beautiful new interpretation as a certain classical property of the αth orbit), a little excursion back to the derivation of the Van Vleck propagator of Sect. 10.1 is necessary. To avoid any rewriting, I shall leave the time discrete, ti = iτ with i = 1, 2, . . . , n and tn = t, and imagine n so large and τ so small that the discrete classical path from q0 = q to qn = q approximates the continuous path accurately. Recall that the Morse index was obtained as the number of negative eigenvalues of the matrix G (n,α) in the second variation of the action about the classical path, δ 2 S (n,α) ({φ}) =

n−1 m  (n,α) G φi φ j . τ i, j=1 i j

(10.2.26)

Hamilton’s principle characterizes the classically allowed paths as extremalizing the action. An interesting question, albeit quite irrelevant for the derivation of the equations of motion, concerns the character of the extremum. The interest in the present context hinges on its relevance for the value of the Morse index. Clearly, the action is minimal if the second variation δ 2 S (n,α) is positive, i.e., if the eigenvalues λ are all positive; in that case the Morse index vanishes. of the matrix G i(n,α) j If we stick to Hamiltonians of the form H = T + V and to sufficiently short time spans, minimality of the action does indeed hold, simply because locally any classical path looks like a straight line and resembles free motion. But for free motion, the eigenvalue problem in question reads 2Φi − Φi−1 − Φi+1 = λΦi for i = 1, 2, . . . , n − 1 where the boundary conditions are Φ0 = Φn = 0. The eigenvectors Φ can be constructed through an exponential ansatz for their components, Φi = a i , which yields the quadratic equation a + λ − 2 + 1/a = 0; the two roots a± obey a+ a− = 1; thus, each eigenvector is the sum of two exponentials, i i + c− a− with coefficients c± restricted by the boundary conditions Φi = c+ a+ n n n n + c− a − = 0; nontrivial solutions c± = 0 require a+ = a− and thus c+ + c− = c+ a+ 2n 2n a+ = a− = 1; now, both bases a± are revealed as unimodular, i.e., determined by

10.2

Van Vleck Propagator

391

a phase χ through a± = e±iχ ; then, the foregoing quadratic equation for the bases yields the eigenvalue λ = 2(1 − cos χ ) as bounded from below by zero; but zero is not admissible since a vanishing eigenvalue would entail a vanishing eigenvector; indeed, the eigenvectors are Φk ∝ eikχ − e−ikχ ∝ sin kχ and thus Φk ≡ 0 for χ = 0. To ease possible worry about the limit χ → 0 in the last conclusion, one should check that the normalization constant is independent of χ ; one easily finds Φk = (2/n)1/2 sin kχ. As required for minimality of the action for free motion, all eigenvalues λ of G turn out positive. But back to general classical paths! As we let the time t grow (always keeping a sufficiently fine gridding of the interval [0, t]), the action will sooner or later lose minimality (even though not extremality, of course). When this happens first, say for time tn = t c , an eigenvalue of G reaches zero and subsequently turns negative. The corresponding point q(t c ) on the classical path is called “conjugate” to the initial one, q(0) = q . More such conjugate points may and in general will follow later. At any conjugate point the number of negative eigenvalues of G will change by one. A theorem of Morse’s [21] makes the stronger statement that the number of negative eigenvalues of G for Hamiltonians of the structure H = T + V actually always keeps increasing by one as the configuration space path traverses a point conjugate to the initial one. Therefore, the Morse index equals the number of conjugate points passed.  To locate a conjugate point, one must look for a vanishing eigenvalue, i.e., solve j G i j Φ j = 0 which reads explicitly m (Φi+1 − 2Φi + Φi−1 ) = −V

(qi )Φi . τ2

(10.2.27)

This so-called Jacobi equation may be read as the Newtonian equation of motion (10.2.7) linearized around a classical path q(ti ) = qi from q0 = q to qn = q in discretized time. Here, of course, we must look for a solution Φi that satisfies the boundary conditions Φ0 = Φn = 0 and is normalizable as i Φi2 = 1. If such a solution exists, the “final” time tn = t is a t c , and the point q is conjugate to q

(see Fig. 10.1). A geometrical meaning of conjugate points is worth a look. It requires considering the family of classical paths qi ( p ) = q( p , ti ) with i = 1, 2, . . . originating from one and the same initial q0 ≡ q with different momenta p (see Fig. 10.2). Two such paths move apart as qi ( p + δp ) − qi ( p ) = ∂q∂i p( p ) δp . The “response function”

Fig. 10.1 Two consecutive focal (alias conjugate) points in configuration space: a bundle of trajectories fanning from the first reunites in the second

392

10 Semiclassical Roles for Classical Orbits

Fig. 10.2 Initial Lagrangian manifold L at q = 1, t = 0 and its time-evolved images L at t = 0.2 (left) and t = 2.0 (right) for the one-dimensional double-well oscillator with the potential V (q) = q 4 − q 2 . The directed thin lines depict orbits. The early-time L appears tilted clockwise against L and still rather straight, as is typical for systems with Hamiltonians H = p 2 /2m + V (q). At the larger time t = 2.0, L has developed “whorls” and “tendrils” due to the anharmonicity of V (q); moreover, four configuration-space caustics appear. This type of Lagrangian manifold is associated with the time-dependent propagator; it is transverse to the trajectories which carry it along as time evolves. Courtesy of Littlejohn [22]

J ( p , ti ) =

∂qi ( p ) ∂ p

(10.2.28)

measuring that divergence vanishes initially since the initial coordinate is fixed as independent of p . At the final time tn = t, however, the response function obeys  2 −1 ∂q( p ) ∂ S(q, q , t) =− J ( p , t) = ∂ p

∂q∂q



(10.2.29)

where the last member is given by (10.2.18) since p = mτ (q1 −q0 ); it reveals that the response function determines the preexponential factor in the Van Vleck propagator (10.2.25). Each path of the family considered obeys the equation of motion (10.2.7) which upon differentiation with respect to p yields m [J ( p , ti+1 ) − 2J ( p , ti ) + J ( p , ti−1 )] = −V

(qi )J ( p , ti ). τ2 The Jacobi equation (10.2.27) is met once more, and this time truly meant as a linearized equation of motion. If one wants to determine J ( p , ti ) from here, one needs a second initial condition that follows from the initial momentum as q1 = q0 + mτ p . Should the response function J ( p , ti ) vanish again at the final time tn = t, all member paths in the family would reunite in one and the same final point q( p , tn ) = q; moreover, that final point must be conjugate to q since, like

10.3

Gutzwiller’s Trace Formula

393

the eigenvector of the stability matrix G with vanishing eigenvalue, the response function J ( p , ti ) obeys the Jacobi equation and vanishes both initially and in the end. To underscore the geometrical image of the family fanning out from q and reconverging in q c , conjugate points are also called focal points. One more time anticipating material to be presented in the next section, I would like to mention yet another meaning of conjugate points. The propagator q|e−iH t/ |q  sharply specifies the initial coordinate as q but puts no restriction on the initial momentum. From the classical point of view, a straight line L (for systems with a single degree of freedom) in phase space that runs parallel to the momentum axis is thus determined. If every point on that line is dispatched along the classical trajectory generated by the classical Hamiltonian function H (q, p), a time-dependent image L of the initial L will arise. For very short times, L will still be straight but will appear tilted against L (see Fig. 10.2). At some finite time t c , when the potential energy has become effective, the image may and in general will develop a first caustic above the q-axis, i.e., a point with vertical slope, ∂ p/∂q = ∞ ⇔ ∂q/∂ p = 0; but inasmuch as all points on L can still be uniquely labelled by the initial momentum p on L , these caustics can also be characterized by ∂q/∂ p = 0, i.e., by the conjugate-point condition. Therefore, a point in configuration space conjugate to the initial q yields a caustic of L. Subsequent to the appearance of the first such caustic on L, the Van Vleck propagator can no longer consist of a single WKB branch in (10.2.25) but must temporarily comprise two such since there are two classical paths leading from the initial configuration-space point q to the final q within the time span t (see Fig. 10.2). Eventually, when further caustics have appeared on L, more WKB branches arise in the propagator, as labelled by the index α and summed over in (10.2.25). No essential difficulty is added for f degrees of freedom. Then the use of nonCartesian coordinates and replacement of the Newtonian form of the equation of motion by the Lagrangian or Hamiltonian may be indicated. The fact is worth mentioning that conjugate points may and often do acquire multiplicities inasmuch as (at most f ) eigenvalues λ may vanish simultaneously. Then, the Morse index counts the conjugate points with their multiplicities.

10.3 Gutzwiller’s Trace Formula Here, I shall derive the Gutzwiller type traces of quantum propagators. These are the principal ingredients for semiclassical approximations of quantum (quasi) energy spectra and of measures of spectral fluctuations. For maps, the relevant propagator is the time-dependent one, q|U n |q , while for autonomous flows, the tradition founded by Gutzwiller demands looking at the energy-dependent propagator alias resolvent, (E − H )−1 . The traces to be established will be semiclassical, i.e., valid only up to corrections of relative order 1/ and under additional restrictions to be revealed as we progress. The starting point is the Van Vleck propagator which itself is a sum of contributions from classical orbits. The latter structure is inherited by the

394

10 Semiclassical Roles for Classical Orbits

traces, and the contributing orbits are constrained to be periodic: Closure in configuration space is an immediate consequence of the traces being sums (or integrals) of diagonal elements q|U n |q or q|(E − H )−1 |q; then, periodicity in phase space results from a stationary-phase approximation in doing the integral which yields the condition p = p .

10.3.1 Maps As already explained in Chap. 4, the traces  tn =

+∞

d x x|U n |x 6    ∂ 2 S (n,α)  dx (n,α) α   e i{ S (x,x)/−ν π/2} ∼ √  

∂ x∂ x x=x

i2π α

(10.3.1)

−∞

are the Fourier coefficients of the density of levels (see (4.14.1), reproduced as (10.3.8) below) and building blocks for the secular coefficients through Newton’s formulae (see Sect. 4.15). Thus, their ( → 0)-approximants are the clues to a semiclassical discussion of quasi-energy spectra. Having given the semiclassical propagator (10.2.19), we just need to do the single x-integral in (10.3.1) and there once more employ the stationary-phase approximation. The stationary-phase condition, d S (n,α) (q, q)/dq = [(∂/∂q + ∂/∂q )S (n,α) (q, q )]q=q = p − p = 0, restricts the contributing paths to periodic orbits, for which the initial phase-space point q , p

and its nth classical iterate q, p coincide. The period n may be “primitive” or an integral multiple of a shorter primitive with period n 0 = n/r and the number r of traversals a divisor of n. All n 0 distinct points along any one period-n orbit make the same (yes, the same, see below) additive contribution to the integral which may again be found from the Gaussian approximation exp{iS (n,α) (x, x)/} ∼ exp {(i/)[S (n,α) (q, q) + 12 (S (n,α) (q, q))

(x − q)2 ]} and the Fresnel integral (10.2.12). So, we obtain Gutzwiller’s trace formula tn ∼

 α

6 n0

 2 (n,α)   ∂ S (q, q )  1 (n,α) (n,α)   e i{ S (q,q)/−μ π/2}   (n,α)



|(S (q, q)) | ∂q∂q q=q

(10.3.2)

where the sum is over all period-n orbits, q is the coordinate of any of the n points on the αth such orbit, and μ(n,α) is the Maslov index5

5 Somewhat arbitrarily but consistently, I shall speak of Morse indices in propagators and Maslov indices in trace formulae.

10.3

Gutzwiller’s Trace Formula

1 μ(n,α) = ν α (q, q) + {1 − sign((S (n,α) (q, q))

)} ≡ μαprop + μαtrace . 2

395

(10.3.3)

A word is in order on the Morse index ν α (q, q) ≡ μαprop in the integrand in (10.3.1). I already pointed out in the previous subsection that ν α (q, q ) depends on q, q and the connecting path. Upon equating q and q to the integration variable x, one must reckon with ν α (x, x) as an x-dependent quantity. On the other hand, ν α (x, x) is an integer. The n-step path under consideration will change continuously with x unless it gets lost as a classically allowed path; concurrent with continuous changes of x, its ν α (x, x) may jump from one integer value to another at some xˆ , but the asymptotic  → 0 contribution to the integral won’t take notice unless xˆ happened to be a stationary point of the action S (n,α) (x, x), i.e., unless the n-step path were a period-n orbit in phase space rather than only a closed orbit in configuration space. There is no reason to expect a point xˆ of jumping ν α (x, x) in coincidence with a point of stationary phase: Stationarity is a requirement for the first derivatives of the action while jumps of the Morse index take place when an eigenvalue of the defining the second variation of the action around the path according matrix G i(n,α) j to (10.2.11) and (10.2.15) passes through zero.6 Thus, the contribution of a point q on the αth period-n orbit involves the Morse index ν α (q, q) which characterizes a whole family of closed-in-configuration-space n-step paths near the period-n orbit in question. The question remains whether all n points on the period-n orbit have one and the same Morse index; they do, but I beg the reader’s patience for the proof of that fact until further below. An alternative form of the prefactor of the exponential in the semiclassical trace (10.3.2) explicitly displays the stability properties of the αth period-n orbit by involving its stability matrix (called monodromy matrix by many authors) ⎞ ⎡    ⎤ ⎛ Sq ,q

1 ∂q ∂q − − ⎢ ∂q p ∂ p q ⎥ ⎜ Sq,q

Sq,q ⎟ ⎟ ⎥=⎜ (10.3.4) M =⎢     ⎜ ⎟ 2 ⎦ ⎣ ∂p ∂p ⎝ (Sq,q ) − Sq ,q Sq ,q

Sq,q ⎠ − ∂q p ∂ p q

Sq,q

Sq,q

which latter defines the n-step phase-space map as linearized about the initial point q , p . The second of the above equations follows from the generating-function property (10.2.9) of the action; to save on war paint, subscripts on the action denote partial derivatives while the superscripts (n, α) have been and will from here on mostly remain dropped. Due to area conservation, det M = 1, as is indeed clear from the last member in (10.3.4). Therefore, the two eigenvalues of M are reciprocal to one another; if they are also mutual complex conjugates, we confront a stable orbit; positive eigenvalues signal hyperbolic behavior and negative ones signal inverse hyperbolicity. According to (10.3.4), the trace of M is related to derivatives

6 Should such nongeneric a disaster happen, there is a way out shown by Maslov: one changes to, say, the momentum representation; see next section.

396

10 Semiclassical Roles for Classical Orbits

√ √ of √ the action such that the prefactor takes the form · · · = 1/ | det(M − 1)| = 1/ |2 − TrM|. Hence we may write tn as [17, 18, 23]7 tn ∼



√

n0 e i{S(q,q)/−μπ/2} . | det(M − 1)|

(10.3.5)

It is quite remarkable that the trace of the n-step propagator is thus expressed in terms of canonically invariant properties of classical period-n orbits. To check on the invariance of S (n,α) (q, q), first, think of a general periodically driven Hamiltonian system with H ( p, q, t + τ ) = H ( p, q, t), and let a canonical transformation p, q → P, Q be achieved by the generating function F(Q, q) according to P = ∂ F(Q, q)/∂ Q ≡ FQ and p = −∂ F(Q, q)/∂q ≡ −Fq . The action accumulated along a classical path from q to q (equivalently, from Q to Q) during a time span t may be calculated with either set of coordinates, and the most basic $q ˜ p˜ , t˜)d t˜] = property of the generating function yields the relation q [ p˜ d q˜ − H (q, $Q



˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ Q { Pd Q − H [q( Q, P), p( Q, P), t ]d t } + F(Q, q) − F(Q , q ); but for a periodic

path with period nτ from q to q = q and similarly Q = Q , the difference F(Q, q) − F(Q , q ) vanishes such that the two actions in question turn out to be equal. A little more labor is needed to verify the canonical invariance of the trace of the stability matrix. Starting from ( ∂∂PP ) Q + ( ∂∂QQ ) P one thinks of the new phase-space coordinates P, Q as functions of the old ones and uses the chain rule to write  ∂Q  ∂ Q

P

 ∂ Q   ∂q  ∂ Q   ∂p  + ∂q p ∂ Q P

∂ p q ∂ Q P

 ∂q   ∂ p    ∂ Q   ∂q   ∂q  + = ∂q p ∂q p ∂ Q P

∂ p q ∂ Q P

  ∂ Q   ∂ p   ∂q   ∂ p   ∂ p   + + ∂ p q ∂q p ∂ Q P

∂ p q ∂ Q P

=

(10.3.6)

and similarly for the second term in the trace. Upon employing the generating function F(Q, q, t), we get d P = FQ Q d Q + FQq dq, dp = −Fq Q d Q − Fqq dq and thus the Jacobian matrices (in analogy to (10.3.4)) ⎛ ⎛ ⎞ ⎞ F F 1 − FQQqQ − FqqqQ − F1q Q ∂(q, p) ∂(Q, P) ⎝ FQq ⎠, ⎠. = F F −F F = ⎝ F F −F F Fqq FQ Q Q Q qq q Q Qq q Q Qq Q Q qq ∂(q, p) ∂(Q, P) − − FQq FQq Fq Q Fq Q With these Jacobians inserted in (10.3.6) and the corresponding ∂ P/∂ P and realizing that for a periodic point q = q , p = p as well as Q = Q , P = P we immediately conclude that 7 Remarkably, the semiclassically approximate equality in (10.3.5) becomes a rigorous equality for the cat map, as was shown by Keating in Ref. [23].

10.3

Gutzwiller’s Trace Formula

 ∂Q  ∂ Q

P

+

397

 ∂P  ∂ P

Q

=

 ∂q  ∂q

p

+

 ∂p  ∂ p

q

,

(10.3.7)

i.e., the asserted canonical invariance of the trace of the stability matrix of a periodic orbit. The more difficult proof of the canonical invariance of the Maslov index will be discussed in Sect. 10.4. At this point we can partially check, that in the trace formulae (10.3.2) and (10.3.5), the n 0 distinct points along an orbit with primitive period n 0 contribute identically such that when summing over periodic orbits rather than periodic points we encounter the primitive period as a factor: According to (10.2.8) the function S (n,α) (qi , qi ) differs for the n 0 points qi only in the irrelevant order of terms in the sum over single steps. The square root in the prefactor must also be the same for all points on the periodic orbit, simply since the stroboscopic time evolution may itself be seen as a canonical transformation and since the trace of the stability matrix is a canonical invariant, as shown right above. Similarly, once we have shown that the Maslov index is a canonical invariant, we can be sure of its independence of the periodic points along a periodic orbit. Inspection of (10.2.8) also reveals that the contribution of a period-n orbit with shorter primitive period n 0 = n/r may be rewritten as S (n,α) = r S (n 0 ,α) . As regards the prefactor for a repeated orbit of primitive period n 0 = n/r , we may invoke the meaning of M as the map linearized about the orbit to conclude the multiplicativity M (n) = (M (n 0) )r . Once again, the Maslov index is more obstinate than the other ingredients of the trace formula; its additivity for repeated traversals, μ(n,α) = r μ(n 0 ,α) ≡ r μα , which unfortunately holds only for unstable orbits, will be shown in Sect. 10.4. The Maslov index for stable orbits, as appear for systems with mixed phase spaces, need not be additive with respect to repeated traversals. Now, the quasi-energy density is accessible through the Fourier transform ∞ ∞ 4  1 3 1  tn einφ = tn einφ , N + 2 Re (10.3.8) (φ) = 2π N n=−∞ 2π N n=1 where a finite dimension N is assumed for the Hilbert space. I immediately proceed to a spectral average of the product of two densities,  2π dφ 2π 2π e)(φ) = (φ + e)(φ) (φ + N 2π N 0 2   ∞ 2π 1 |tn |2 exp{i n e} (10.3.9) = 2π N n=−∞ N  2  ∞  1 2π = |tn |2 cos (n e) . N2 + 2 2π N N n=1 This can be changed into the two-point cluster function in the usual way.8 8

Subtract and then divide by the product (1/2π )2 of two mean densities, take out the delta function provided by the self-correlation terms, and change the overall sign.

398

10 Semiclassical Roles for Classical Orbits

It may be well to emphasize that a smooth two-point function results only after one more integration since the densities provide products of two delta functions. The form factor |tn |2 is such an integral and thus free of delta spikes (but still displays wild fluctuations in its n-dependence (see Sect. 4.19)). A cheap way of smoothing the above two-point function is to truncate the sum over n at some finite n max . Already in the quasi-energy density (10.3.8), that truncation will “regularize”, roughly as δ(φ − φi ) → sin[(n max + 12 )(φ − φi )]/ sin[ 12 (φ − φi )]. To fully define the spectrum within the semiclassical treatment we are pursuing we would need all periodic orbits with periods between 1 and N /2 since the traces tn with larger n can be determined with the help of self-inversiveness, Newton’s formulae, and the Hamilton Cayley theorem, as discussed in Sect. 4.15 The most important information about the specifics of a given dynamic system with global classical chaos is expected to be encoded in short periodic orbits, since as the period n grows to infinity, typical orbits will follow the universal trend to ergodic exploration of the phase space. Such classical universality corresponds to the quantum universality of the spectral fluctuations on the (quasi)energy scale of the mean nearest neighbor spacing. The expectations just formulated constitute a strong motivation to seek semiclassical justification of the success of random-matrix theory. Now, the frame is set for a discussion of such efforts.

10.3.2 Flows Gutzwiller suggests [3] proceeding toward the energy spectrum through the energydependent propagator defined by the one-sided Fourier transform 1 G(q, q , E) = i



∞

dt eiEt/ q|e−iH t/ |q  = q|

0

1 |q . E−H

(10.3.10)

To ensure convergence, the energy variable E here must be endowed with a positive imaginary part. Indeed, the trace of the imaginary part of G(q, q , E) yields the density of levels as ImE → 0+ . After inserting the Van Vleck propagator (10.2.25), it is natural again to invoke the semiclassical limit and evaluate the time integral by a stationary-phase approximation. The new stationarity condition is E+

∂ S α (q, q , t) = E − H (q, q , t) = 0. ∂t

(10.3.11)

Note that the Hamiltonian function here does not appear with vari  its natural

, t) = ables and therefore is time-dependent through H (q, p) = H q, p(q, q  

H q , p (q, q , t) = H (q, q , t). The phase of the exponential in our time integral is stationary for the set of times solving (10.3.11). A new selection of classical paths is so taken since besides q and q , it is now the energy E rather than the time span t that is prescribed. Just as we could have labelled the paths contributing to the time-dependent propagator by their energy, we could so employ their duration

10.3

Gutzwiller’s Trace Formula

399

now. Actually, one usually does neither but rather puts some innocent looking label on all quantities characterizing a contributing path. Calling t α (q, q , E) the roots of the new stationary-phase condition (10.3.11) and confining ourselves to the usual quadratic approximation in the expansion of the phase in powers of t − t α , we get 6  π4 1   ∂ 2 S α (q, q , t α )  3 i exp [Et α + S α (q, q , t α )] − iν α G osc (q, q , E) =  

i α ∂q∂q  2  ∞  dt i ¨α S (q, q , t α )(t − t α )2 (10.3.12) × exp √ 2 i2π 0 6  1   1 ∂ 2 S α (q, q , t α )  =  S¨ α (q, q , t α )  i α ∂q∂q

3i π4 × exp [Et α + S α (q, q , t α )] − iκ α  2

where the double dot on S¨ α (q, q , t) requires two partial differentiations with respect to time at constant q and q . The “osc” attached to G osc (q, q , E) signals that the Gaussian approximation about the times t α of stationary phase cannot do justice to very short paths where t α is very close to the lower limit 0 of the time integral; their contributions will be added below under the name G such that eventually G ∼ G + G osc . The Morse index ν α counts the number of conjugate points along the αth path up to time t α , while for the successor κ α in the last member of the foregoing chain, by appeal to the Fresnel integral (10.2.12), κ α = να +

  1 1 − sign S¨ α (q, q , tα ) . 2

(10.3.13)

I shall come back to that index in the next section and refer to it as the Morse index of the energy-dependent propagator; we shall see that it is related to the num˙ ber of turning points (points where the velocity q(t) changes sign) passed along the orbit. Now it is more urgent to point to another feature of the exponential, the appearance of the time-independent action S0 (q, q , E) = S(q, q , t) + Et

(10.3.14)

which may be seen as related to the original action by a Legendre transformation. It is natural to express the preexponential factor in the last member of (10.3.12) in terms of S0 and its derivatives with respect to its “natural variables” as well. One obtains9

9 The structures of (10.2.25) and (10.3.15) are similar; one might see the difference as the result of an extension of the phase space by inclusion of the pair E, t.

400

10 Semiclassical Roles for Classical Orbits

G osc (q, q , E) =

i π 1  α A (q, q , E) exp S0α (q, q , E) − iκ α , i α  2

F ⎛ G ∂ 2 S0α G G ⎜ G ⎜ ∂q∂q

Aα (q, q , E) = G Gdet⎜ H ⎝ ∂ 2 S0α  ∂ E∂q

⎞ ∂ 2 S0α   ∂q∂ E ⎟ ⎟ ⎟  ∂ 2 S0α ⎠ ∂ E2 

(10.3.15)

by the usual prestidigitation of changing variables; here, the transformation t α → E according to a solution t α (q, q , E) of the stationary-phase condition; to avoid notational hardship, I wave good-bye to the index α, write S 0 for S0α , and denote partial derivatives of S(q, q , t), S 0 (q, q , E), t(q, q , E), and E(q, q , t) by suffixes. Starting with the stationary-phase condition (10.3.11), I note the first derivatives of the two action functions, St = −E S E0 = St t E + t + Et E = t Sq0

(10.3.16)

= Sq + St tq + Etq = Sq = p,

Sq0



= Sq = − p .

The first of the foregoing identities, together with (10.3.14), constitutes the Legendre transformation from the time-dependent to the energy-dependent action, inasmuch as it yields the time t(q, q , E) taken by a classical phase-space trajectory from q to q on the energy shell; the third line in (10.3.16) reveals that the energydependent action is the generating function for the classical transition on the energy shell, p = ∂ S 0 (q, q , E)/∂q and similarly p = −∂ S 0 (q, q , E)/∂q . Now, we 0 0 fearlessly proceed to the second derivatives, starting with Sq,q = Sq,q

+ Sq,E E q ; 0 but here E q can be expressed in terms of other second derivatives of S since 0 0 dt = tq dq + tq dq + t E d E at constant q and t yields E q = −tq /t E = −S Eq

/S E E 0 such that indeed Sqq is expressed through derivatives of S as 0 Sqq = Sqq

−

0 Sq0 E S Eq

S E0 E

.

(10.3.17)

Finally, Stt = −E t = −1/t E = −1/S E0 E , whereupon the radicand under study, Sqq

0 0 0 = Sq0 E S Eq

− S E E Sqq , Stt

(10.3.18)

assumes the asserted determinantal form. Inasmuch as dynamics with nonintegrable classical limits are to be included we must generalize to f degrees of freedom, and f ≥ 2. The change is but little for

10.3

Gutzwiller’s Trace Formula

401

G(q, q E): The determinant in the prefactor A becomes ( f + 1) × ( f + 1), with ∂ 2 S/∂q∂q an f × f matrix bordered by an f -component column ∂ 2 S/∂q∂ E to √ f −1 arises, the right and a row ∂ 2 S/∂ E∂q from below.10 An overall factor 1/i2π  as is easy to check by going back to (10.2.3) where the free-particle propagator in √ f f dimensions comes with a prefactor m/i2π τ ; of that, the part m/τ is eaten up for good by the f -dimensional generalization of (10.2.19) such that the fate of √ f 1/i2π must √ be followed; the time integral in (10.3.12) takes away one of the f factors 1/ i2π ; these remarks should also prepare the reader for the further changes in the prefactor toward the final trace formula. I shall not bother with arbitrary f but rather specialize to f = 2 from here on since this is enough for most applications known and also simplifies the work to be done. Now, we must attack the trace 

 1 0 d 2x A ei{S /−κπ/2} d x G osc (x, x, E) = √ i i2π α 2

(10.3.19)

by doing the twofold configuration-space integral, surely invoking the stationaryphase approximation. The stationary-phase condition, ( ∂∂x + ∂∂x )S 0 (x, x )|x=x = p − p = 0, again restricts the contributing paths to periodic ones. But in contrast to the period-n orbits encountered in the previous subsection, which were a sequence of n points (qi , pi ) with contributions to be summed over, now we encounter a continuous sequence of periodic points along each periodic orbit. The sum over their contributions takes the form of a single integral along the orbit, and that integral cannot be simplified by any Gaussian approximation of the integrand. It is only the integration transverse to a periodic orbit for which the stationary-phase approximation invites a meaningful Gaussian approximation. I shall specify the two configuration-space coordinates summarily denoted by x above as one along the orbit, x , and a transverse one, x⊥ ; the transverse coordinate is kin to the one and only one incurred for maps with f = 1 in the preceding subsection, cf (10.3.1). The two coordinates x , x⊥ are assumed orthogonal such that the corresponding axes necessarily change along the orbit and d 2 x = d x d x⊥ ; moreover, the x⊥ -axis is taken as centered at the orbit, i.e., x⊥ = 0 and x˙ ⊥ = 0 thereon. Such a choice helps with the twofold integral over x and x⊥ , first by giving a simple form to the 3 × 3 determinant in the prefactor A. To see this, we take various derivatives of the conservation law H (q, p) = E, H (q , p ) = E, considering the canonical momenta as well as the action S 0 (q, q , E) as functions of q, q , and E; also observing q = (x , x⊥ ), p = ( p , p⊥ ), and ∂ H/∂ p⊥ = x˙ ⊥ = 0 (and similarly for the primed quantities), we get

10 While this generalization is suggested by the very appearance of the prefactor in (10.3.15) and the previous footnote, and thus easy to guess, serious work is needed to actually verify the result.

402

10 Semiclassical Roles for Classical Orbits

  

∂ H (q, p) ∂E

∂ H (q , p ) ∂E

∂ H (q , p ) ∂ x⊥ 

 q

 q

 q ,E

= 1 = x˙

= 0 = x˙

∂ H (q , p ) ∂x 

∂p ∂ H (q, p) ∂ p = x˙ = x˙ Sx0 E , ∂p ∂E ∂E

=1 =



∂ H (q, p) ∂ x⊥

∂ p

∂E ∂ p

∂ x⊥

= −x˙ Sx0 E , = −x˙ Sx0 x⊥ ,

= 0 = x˙

q ,E



= 0 = x˙ q,E

∂ p

∂x

(10.3.20)

= −x˙ Sx0 x ,

∂p = x˙ Sx0 x , ⊥ ∂ x⊥

where on the left-hand sides the suffixes attached to the closing brackets display the variables to be held constant while taking derivatives. Thus, the prefactor A from (10.3.15) becomes F ⎛ ⎞ F G   G 0 1/x˙  G 2 S0  G ⎜ 0 G 1 ∂ 0 0 ⎟   H A(q, q , E) = G H det⎝ 0 Sx⊥ x⊥ Sx⊥ E⎠ =  x˙ x˙ ∂ x ∂ x , ⊥ ⊥ −1/x˙ S E0 x S E0 E   ⊥

(10.3.21)

whereupon the trace (10.3.19) appears in the form  d 2x G osc (x, x, E) = 7

dx |x˙ |



d x⊥ √ i2π

1  i α

6  ∂ 2 S 0 (x⊥ ,x ,x⊥ ,x   ∂ x⊥ ∂ x⊥

(10.3.22)   

,E)  x⊥ =x⊥

0 e i{S (x, x, E)/ − κπ/2} .

Integrating over x ⊥ with the usual quadratic approximation in the exponent, S 0 (x, x, E) = S(E)+[(∂/∂ x⊥ +∂/∂ x⊥ )2 S(x⊥ , x , x⊥ , x , E)]x⊥=x⊥ =0 x⊥2 /2, we get 

6   ∂ 2 S(x⊥ ,x ,x⊥ ,x ,E)  d x⊥   e iS(x, x, E)/ (10.3.23) √  ∂ x⊥ ∂ x⊥

i2π   x⊥ =x⊥

F  G G ∂ 2 S(x⊥ ,x ,x⊥ ,x ,E)/∂ x⊥ ∂ x⊥  H e i{S(E)/ − μtrace π/2} ∼  ∂   ( ∂ x⊥ + ∂ x∂ )2 S(x⊥ ,x ,x⊥ ,x ,E) 

⊥

x⊥ =x⊥ =0

(10.3.24)

10.3

Gutzwiller’s Trace Formula

403

with the index ν according to the Fresnel integral (10.2.12), μtrace =

1 (1 − sign{[(∂/∂ x⊥ + ∂/∂ x⊥ )2 S(x, x , x , x , E)]x=x =0 }). 2

(10.3.25)

Note that I have somewhat sloppily written S(E) for S 0 (0, x , 0, x , E) thus expressing the fact that the action is independent of the coordinate x along the periodic orbit and judging redundant the superscipt “0” once writing the energy as an argument of the action. The index μtrace resulting from the trace operation combines with the Maslov index κ from the energy-dependent propagator to the full Maslov index of the periodic orbit μ = κ + μtrace ≡ μprop + μtrace .

(10.3.26)

In further implementing the restriction to periodic orbits, it is well to realize that the transverse derivatives (w.r.t. x⊥ and x⊥ ) of the action taken at x⊥ = x⊥ = 0 are also independent of x . If we take for granted, subject to later Cproof, that the full Maslov index μ is x -independent as well, the final x -integral d x /x˙ = T yields the primitive period of the orbit; it is the primitive period since the original configuration-space integral “sees” the orbit as a geometrical object without noticing repeated traversals. Thus, the trace we pursue becomes  d 2x G osc (x, x, E) = F  G   G 2 

  G ∂ S x⊥ , x , x⊥ , x , E /∂ x⊥ ∂ x⊥  1 T G  2    i α H ∂  ∂ x⊥ + ∂ ∂x

S x⊥ , x , x⊥ , x , E  ⊥

(10.3.27)

e i{S(E)/ − μπ/2} . x⊥ =x⊥ =0

This trace formula looks remarkably similar to that obtained for maps with f = 1; see (10.3.2). In particular, the radicands in the prefactors are in complete correspondence such that by simply repeating the arguments of the previous subsection we express the present radicand in terms of the 2 × 2 stability matrix11 for a loop around the αth periodic orbit which should be envisaged as giving a map in a Poincar´e section spanned by x⊥ , p⊥ . The final result for our trace reads, in analogy to (10.3.5),  T 1  e iS(E)/ ≡ gosc (E) (10.3.28) d 2x G osc (x, x, E) = √ i α | det(M − 1)|

11 Note that in contrast to the preceeding chapter I here use the same symbol M for the stability matrix of maps and flows

404

10 Semiclassical Roles for Classical Orbits

where for notational convenience the “action” S(E) ≡ S(E) − μπ/2 is defined to include the Maslov phase. Upon taking the imaginary part, we arrive at the oscillatory part of the density of levels12 osc (E) =

1  T cos (S/). √ π |det(M − 1)|

(10.3.29)

It is appropriate to emphasize that orbits which are r -fold traversals of shorter primitive ones contribute with their primitive period T while the stability matrix M is the r th power of the one pertaining to the primitive orbit, just as for maps. It is well to note that the above final results for the oscillatory parts gosc (E) of the Green function and osc of the level density also hold for more than two degrees of freedom [2].

10.3.3 Weyl’s Law $∞ An integral of the form τ dt A(t)eiνΦ(t) with large ν draws its leading contributions not only from the “points” tα of stationary phase but also from near the boundary τ of the integration range. The pertinent asymptotic result is [24] 

∞

τ

dt A(t)eiνΦ(t) ∼

i A(τ )eiνΦ(τ ) + {stationary-phase contributions}. ˙ ) ν Φ(τ

(10.3.30)

The one-sided Fourier transform defining the energy-dependent propagator in (10.3.10) is such an integral with 1/ as ν and τ → 0. While the stationaryphase contributions were evaluated in the preceding subsection as G osc (E), now we turn to the boundary contribution G(E). Employing the Van Vleck form of the time-dependent propagator, we run into the phase Φ(t) = Et + S(q, q , t) and its ˙ derivative Φ(t) = E − H (q, q , t). Thus, the boundary term reads 

 d fq G(q, q, E) ∼ lim

τ →0

d fqd fq

δ(q − q ) ˆ q|e−i H τ/ |q  (10.3.31)

E − H (q, q , τ )

where for brevity q|e−i H τ/ |q  stands for the Van Vleck propagator; the reader ˆ as the Hamiltonian operator and distinguish it from the Hamilwill recognize H tonian function H (q, q , τ ) appearing in the denominator. To see that the foregoing (τ → 0)-limit is well defined (rather than a bewildering$ 0 × ∞), we repre∞ sent the delta function by a Fourier integral, δ(q) = (2π )− f −∞ d f p e−i pq/ , and think of the q -integral as done by stationary phase. The stationary-phase condition ˆ

12 Departing from the convention mostly adhered to in this book, here I do not restrict myself to Hilbert spaces with the finite dimension N and thus do not write a factor 1/N into the density of levels; the reader is always well advised to check from the context on the normalization of (E).

10.3

Gutzwiller’s Trace Formula

405

∂ (S(q, q , τ ) ∂q

+ pq ) = p − p (q, q , τ ) = 0 restores the natural variables q, p to the Hamiltonian function in the denominator, such that we may write 





e−i p(q−q )/ ˆ d qd q d p q|e−i H τ/ |q . E − H (q, p) (10.3.32) Note that while taking advantage of the benefit H (q, q , t) → H (q, p) of the aforementioned stationary-phase approximation, I have otherwise still refrained from integrating over q . Such momentary hesitation is not necessary but conveˆ nient since after first taking the limit τ → 0 to get q|e−i H τ/ |q  → δ(q − q ), the

q -integral becomes trivial. Thus, the desired result 1 d q G(q, q, E) ∼ lim f τ →0 h f

 d fq G(q, q, E) ∼

f

1 hf

f

f

 d fqd f p

1 E − H (q, p)

(10.3.33)

is reached. Providing the energy E with a vanishingly small imaginary part and using the familiar identity Im{1/(x − i0+ )} = π δ(x), we arrive at Weyl’s law, (E) ∼

1 hf

 d fqd f p δ(E − H ),

(10.3.34)

which expresses the nonoscillatory alias average density of states as the number of Planck cells contained in the classical energy shell. Following common usage, I refer to (E) also as the Thomas–Fermi density. Variants of and corrections to Weyl’s law can be found in Refs. [25–28]. Such corrections are not only interesting in their own right but become indispensable ingredients for semiclassical determinations of spectra for systems with more than two freedoms [29].

10.3.4 Limits of Validity and Outlook Most obvious is the limitation to the lowest order in , due to the neglect of higher orders when approximating sums and integrals by stationary phase. Readers interested in corrections of higher order in  may consult Gaspard’s work [30] as well as the treatments of three-dimensional billiards by Primack and Smilansky [29] and Prosen [31]. Interestingly, there are dynamical systems for which the trace formulae become rigorous rather than lowest-order approximations. Prominent among these are the cat map [23], billiards on surfaces of constant negative curvature [32], and graphs [33]. Inasmuch as all periodic orbits are treated to make independent additive contributions to the trace, one implicitly assumes that the orbits are isolated. Strictly speaking, one is thus reduced to hyperbolic dynamics with only unstable orbits. In practice, systems with stable orbits and mixed phase spaces can be admitted, provided one stays clear of bifurcations. Near a bifurcation, the periodic orbits involved, about to disappear or just arisen, are nearly degenerate in action and must be treated

406

10 Semiclassical Roles for Classical Orbits

as clusters; that necessity is signalled by the divergence of the prefactor A in the single-orbit contribution AeiS/ , due to the appearance of eigenvalues unity in the stability matrix. Some more remarks on mixed-phase spaces will be presented in Sect. 10.12. The greatest worry is caused by phase-space structures finer than a Planck cell, dragged into the trace formulae by orbits with long periods. Imagine a stroboscopic map for a system with a compact phase space with total volume Ω, a Hilbert space of finite dimension N , and a Planck cell of size h f ≈ Ω/N . Since the number of periodic orbits with periods up to n grows roughly like eλn , a Planck cell, on average, begins to contain more than one periodic point once the period grows larger than the Ehrenfest time, n E = λ−1 ln (Ωh − f ), and is crowded by exponentially many periodic points for longer periods. For maps with finite N , one escapes the worst consequences of that exponential proliferation by taking the periodic-orbit version of the tn only for 1 ≤ n ≤ N /2 and using unitarity, Newton’s formulae, and the Hamilton–Cayley theorem to express all other traces in terms of the first N /2 traces, as explained in Sect. 4.17.2. In particular, the N eigenphases can be obtained once the first N /2 traces tn are known. The proliferation seems more serious for the trace formulae (10.3.28) and (10.3.29) for flows since orbits of all periods, even infinitely long ones, are involved. Much relief was afforded by a series of papers by Berry and Keating that culminated [34–36] in a resummation, relating the contributions of long orbits to those of short ones in a certain functional equation to be presented in Sect. 10.5. It is interesting to see that convergence may be enforced for the trace formula (10.3.28) by letting the energy E become complex, E = E + i with [37],  = Im E > c ≡ λ/2,

(10.3.35)

where λ is the Lyapunov exponent that roughly gives the eigenvalues of the stability matrix M as e±λT . Indeed, assuming all orbits unstable and recalling that exponential proliferation implies that the number of orbits with periods near T is roughly eλT , we conclude that √ the contributions of long orbits run away as {# of p.o. s with periods near T}/ det M ≈ eλT /2 ; but the addition of a classically small imaginary part to the energy ( of order ) provides an exponential attenuation exp ImS(E + i)/ = e−T / sufficient to overwhelm the exponential runaway if  > c . Deplorably, an imaginary part Im E = O() wipes out all structure in the density of levels on the scale of a mean level spacing since the latter is, according to Weyl’s law, of the order  f , i.e., already smaller than the convergence-enforcing c for the smallest dimension of interest, f = 2 for autonomous systems; for periodically driven single-freedom systems whose quasi-energy spacing is O(), complexifying the quasi-energies might be more helpful but has not, to my knowledge, been put to test yet. An interesting attempt at unifying the treatment of maps and autonomous flows and at fighting the divergence due to long orbits was initiated by Bogomolny [38]. It is based on semiclassically quantizing the Poincar´e map for flows. Similar in spirit and as fruitful is the scattering approach of Smilansky and coworkers [39]; these authors exploit a duality between bound states inside and scattering states outside

10.4

Lagrangian Manifolds and Maslov Theory

407

billiards. Respectful reference to the original papers must suffice here, to keep the promise of steering a short course through semiclassical terrain. As yet insufficient knowledge prevents expounding yet another promising strategy of avoiding divergences for flows in compact phase spaces which would describe flows through stroboscopic maps. One would choose some strobe period T and aim at the traces tn = Tre−inT H/ with integer n up to half the Heisenberg time. Thus, the unimodular eigenvalues of e−iH T / are accessible, and the periods of the classical orbits involved not exceeding half the Heisenberg time. The strobe period T would have to be chosen sufficiently small such that the energy levels yield phases E i T / fitting into a single (2π )-interval. To efficiently determine spectra from semiclassical theory, we must learn how to make do with periodic orbits much shorter than half the Heisenberg time. Interesting steps in that direction have been taken by Vergini [40].

10.4 Lagrangian Manifolds and Maslov Theory 10.4.1 Lagrangian Manifolds To bridge some of the gaps left in the proceeding two sections and to get a fuller understanding of semiclassical approximations, I pick up a new geometrical tool, Lagrangian manifolds [22]. These are f -dimensional submanifolds of a (2f)-dimensional phase space to whose definition I propose to gently lead the willing here. We shall eventually see that these hypersurfaces have “generating functions” that are just the actions appearing in semiclassical wave functions or propagators of the structure AeiS/ . Therefore, Lagrangian manifolds provide the caustics at which semiclassical wavefunctions diverge. Above its q-space caustics, one may climb up or down a Lagrangian manifold while appearing to stay put in q-space. A simple such manifold was encountered in Fig. 10.2 for f = 1 as the set of all points with a single fixed value of the coordinate and arbitrary momentum; it naturally arose as the manifold specified by the initial condition of the propagator, q|e−iH t |q t=0 = δ(q − q ). Inasmuch as all points on that manifold project onto a single point in q-space, the latter point is a highly degenerate caustic. The image of that most elementary Lagrangian manifold under classical Hamiltonian evolution at some later time t also qualifies as Lagrangian. For f ≥ 1, the foregoing examples are immediately generalized to all of momentum space above a single point in configuration space and the images thereof under time evolution. These may be seen as f -component vector fields p(q) defined on the f -dimensional configuration space, but not all vector fields qualify as momentum fields for classical Hamiltonian dynamics. Inasmuch as phase space allows for reparametrization by canonical transformations, we may think of the admissible momentum fields as gradients p = ∂ S/∂q of some scalar generating function S, and such vector fields are curl-free, ∂ pi /∂q j = ∂ p j /∂qi . Curl-free vector fields are Lagrangian manifolds albeit not the most general ones. It has already become clear in the preceding section that the initial manifold specified by the propagator may and

408

10 Semiclassical Roles for Classical Orbits

in general does develop caustics under time evolution, i.e., points in configuration space where some of the derivatives ∂ pi /∂q j become infinite and in some neighborhood of which the momentum field is necessarily multivalued (see Fig. 10.2 again). Therefore, the proper definition of Lagrangian manifolds avoids derivatives: An f -dimensional manifold in phase space is called Lagrangian if in any of its points any two of its (2 f )-component tangent vectors, δz 1 = (δq 1 , δp 1 ) and δz 2 = (δq 2 , δp 2 ), have an antisymmetric product ω(δz 1 , δz 2 ) ≡ δp 1 · δq 2 − δp 2 · δq 1 which vanishes, ω(δz 1 , δz 2 ) ≡ δp 1 · δq 2 − δp 2 · δq 1 = 0 ;

(10.4.1)

here the dot between  f two f -component vectors means the usual scalar product, like δp 1 · δq 2 = i=1 δpi1 δqi2 . I leave to the reader as Problem 10.7 to show that the antisymmetric product ω(δz 1 , δz 2 ) is invariant under canonical transformations, i.e., independent of the phase-space coordinates used to compute it. We can easily check that the definition (10.4.1) comprises all of the aforementioned examples. For f = 1, the antisymmetric product ω(δz 1 , δz 2 ) measures the area of the parallelogram spanned by the two vectors in phase space; requiring that area to vanish means restricting the two two-component vectors in question so as to be parallel to one another; indeed, then, we realize that all curves p(q) are Lagrangian manifolds for single-freedom systems since all tangent vectors in a given point are parallel. For f > 1, one may use the antisymmetric product ω(δz 1 , δz 2 ) to define phase-space area in two-dimensional subspaces of the (2 f )dimensional phase space. Moreover, we can consider two tangent vectors such that δz 1 has only a single nonvanishing q-ish component, say δqi , and likewise only a single nonvanishing p-ish one, δp j , while, similarly, δz 2 has only δq j and δpi as nonvanishing entries. For the f -dimensional manifold in question to be Lagrangian, we must have δp j δq j = δpi δqi ; now we may divide that equation by the nonvanishing product δqi δq j and conclude that curl-free vector fields yield Lagrangian manifolds. Finally, we check that {all of p-space above a fixed point in q-space} (as distinguished by the initial condition of the propagator in the q-representation) as well as {all of q-space above a fixed point in p-space} (distinguished by the initial condition of the propagator in the p-representation) fit the definition (10.4.1). Then, the image of the latter manifold under time evolution is Lagrangian as well since time evolution is a canonical transformation. It may be well to furnish examples for concrete dynamical systems. The simplest is the harmonic oscillator with f = 1 where the propagator in the coordinate representation distinguishes the straight line q = q in the phase plane as the initial manifold L . The time-evolved image L arises through rotation about the origin by the angle ωt; once every period, at the times nπ/ω with n = 2, 4, 6, . . ., L coincides with L . Clearly, that initial Lagrangian manifold is a caustic and even highly degenerate. One more such caustic per period arises at q = −q , at the times nπ/ω with n = 1, 3, 5, . . .. None of these caustics corresponds to turning points unless the initial momentum is chosen as p = 0. The caustics at q = ±q

define mutual conjugate points in coordinate space since all trajectories fanning

10.4

Lagrangian Manifolds and Maslov Theory

409

1 0.5

p –0.5 –1 –3

–2

–1

q

1

2

3

Fig. 10.3 Temporal successors of the line p = 0 (a Lagrangian manifold, see Sect. 10.4) for a kicked top under conditions of near integrability (left column) and global chaos after, from top to bottom, n = 1, 2, 3, 4 iterations of the classical map. The intersections with the line p = 0 define period-n orbits. The beginning of the exponential proliferation of such orbits is illustrated by the right column. Points qc with dp/dq = ∞ give configuration-space caustics and, correspondingly, points pc with dq/dp = ∞ give momentum-space caustics

out of them refocus there again (and again). A less trivial example is presented in Fig. 10.3 which refers to a kicked top with a two-dimensional spherical phase space and depicts an initial Lagrangian manifold spanned by all of configuration space at fixed momentum.

410

10 Semiclassical Roles for Classical Orbits

A type of Lagrangian manifold not contained in the set of examples given above arises when dealing with the energy-dependent propagator (10.3.15); I shall denote such manifolds by L E . Within the (2 f )-dimensional phase space is the (2 f − 1)dimensional energy shell, wherein upon picking a point q in coordinate space, we consider a sub-manifold of f − 1 dimensions as well as its time-evolved images for arbitrary positive times. In the course of time, these time-evolved images sweep out an f -dimensional manifold that we recognize as Lagrangian as follows. For the sake of simplicity, I spell out the reasoning for f = 2 for which Fig. 10.4 provides a visual aid; moreover, I assume a Hamiltonian of the form H = p 2 /2m +V (q). The manifold L E to be revealed as Lagrangian originates by time evolution from a set of above the picked point q in the (q1 q2 )-plane; in points along, say, the p1 -direction %

the p2 -direction, p2 = ± 2m(E − V (q)) − p12 determined by the energy through H√= E; the latter also restricts √ the set of admitted values of p1 to the interval − 2m(E − V (q)) ≤ p1 ≤ + 2m(E − V (q)). The initial one-dimensional manifold so characterized sweeps out L E as every initial point is dispatched along the trajectory generated by the Hamiltonian H . Within the three-dimensional q1 q2 p1 space illustrated in Fig. 10.4, L E appears as a sheet which starts out flat but may eventually develop ripples and thus caustics. By complementing the three coordinates with the uniquely determined p2 attached to each point on the sheet, one gets L E as a two-dimensional submanifold of the four-dimensional phase space. Two independent tangent vectors offer themselves naturally at each point on ˙ q , p1 ), p˙ (t; q , p1 )}δt, L E : One along the trajectory through the point, δz 1 = {q(t;



2



and the other, δz = {(∂q(t; q , p1 )/∂ p1 ), ∂ p(t; q , p1 )/∂ p1 )}δp1 , pointing toward a neighboring trajectory. Once more, by expressing the fourth components of these vectors in terms of the first three and invoking Hamilton’s equations, we immediately check that their antisymmetric product vanishes, ω(δz 1 , δz 2 ) = 0, and thus reveals L E as Lagrangian. To acquire familiarity with the technical background of

Fig. 10.4 Lagrangian manifold L E associated with the energy-dependent propagator for f = 2, depicted as a surface in the three-dimensional energy shell. L E is swept out by the time-evolved images of the initial line of fixed q1 , q2 ; it starts out flat but in general tends to develop ripples and thus caustics. Note that the trajectories lie within L E . Courtesy of Creagh, Robbins, and Littlejohn [42]

10.4

Lagrangian Manifolds and Maslov Theory

411

this reasoning, the interested reader will want to write out the pertinent few lines of calculation. The two tangent vectors just mentioned may be seen as elements of a texture covering the Lagrangian manifold L E associated with the energy shell. In particular, the trajectories running along L E climb “vertically” in some momentum direction as they pass through a caustic of L E . It follows that a caustic arising in the energydependent propagator corresponds to turning points of the trajectory involved (see Fig. 10.5). This is in contrast to the contributions of classical trajectories to the timedependent Van Vleck propagator for which we have seen that caustics are related to points conjugate to the initial one, rather than to turning points. The statement that the stable manifold L s of any periodic orbit is Lagrangian is useful for our subsequent discussion of Maslov indices13 . Imagine a point (q0 , p0 ) on L s and a tangent vector δz = (δq, δp) attached to it. As this tangent vector changes into δz(t) by transport along the trajectory through q0 , p0 , eventually, by the definition of a stable manifold, it must become parallel to the flow vector δz p.o. (t) = (∂ H/∂ p, −∂ H/∂q)p.o. along the periodic orbit whose stable manifold is under consideration. Thus, the antisymmetric product ω(δz(t), δz p.o. (t)) vanishes for t → ∞; but since it is also unchanged in time by Hamiltonian evolution, it must vanish at all times. So L s is indeed Lagrangian. This statement will turn out to be useful for the discussion of Maslov indices in Sect. (10.4.3) below.

Fig. 10.5 While momentarily climbing vertically in L E , a trajectory goes through a caustic that is associated with a turning point. Courtesy of Creagh, Robbins, and Littlejohn [42]

13 The stable and unstable manifolds of a periodic orbit are defined as the sets of points which the dynamics asymptotically carries toward the orbit as, respectively, t → +∞ and t → −∞; the reader is kindly asked to think for a moment about why L s , as well as the unstable manifold L u , are f -dimensional.

412

10 Semiclassical Roles for Classical Orbits

Fig. 10.6 A configurationspace caustic (at qc ) looks level from momentum space, as does a momentum-space caustic (at pc ) from configuration space. Near a caustic a Lagrangian manifold is double-valued

While the notion of a Lagrangian manifold is defined without reference to a particular set of coordinates in phase space, their caustics are very much coordinatedependent phenomena, as is clear already for f = 1 from Fig. 10.6. There we see a “configuration-space” caustic, i.e., a point with diverging ∂ p/∂q; the canonical transformation q → P, p → −Q turns the curve p(q) into one P(Q) for which the previous caustic becomes a point with ∂ P/∂ Q = 0 such that no configurationspace caustic appears; instead, we incur what should be called a “momentum-space” caustic inasmuch as the point in question comes with ∂ Q/∂ P = ∞. Conversely, we could say that in the original coordinates the coordinate-space caustic is not a momentum-space caustic there since ∂q/∂ p = 0. For f > 1, a little more care is indicated in explaining caustics. One may uniquely label points on a Lagrangian manifold L by f suitable coordinates ξ1 , ξ2 , . . . , ξ f such that q and p become functions q(ξ ) and p ) on L. Whereever the q(ξ ) are locally invertible functions, ξ = ξ (q), and we can deal with the momenta as a vector field p[ξ (q)]; such invertibility requires that the Jacobian det(∂q/∂ξ ) not vanish. Conversely, (configuration-space) caustics arise where det(∂q/∂ξ ) = 0 since in such points one finds tangent vectors δz = (δq, δp) to L such that δq = (∂q/∂ξ )δξ = 0, even though δξ = 0. For every independent such δξ , the Lagrangian manifold L appears perpendicular to coordinate space, unless it happens that δξ is also annihilated by ∂ p/∂ξ . The number of such independent δξ is called the order of the caustic; the most common value of that order is 1, but orders up to f may arise, and the latter extreme case indeed arises for the L distinguished by the initial form of the propagator q|e−iH t/ |q . Again, caustics are phenomena depending on the coordinates used; if one encounters a coordinatespace caustic, one can canonically transform, as qi → Pi , pi → −Q i , for one or more of the f components so as to ban the caustic from coordinate space to momentum space. When dealing with the semiclassical approximation of a prop-

10.4

Lagrangian Manifolds and Maslov Theory

413

agator, such a change corresponds to switching from the qi -representation to the pi -representation. In a region devoid of configuration-space caustics, a Lagrangian manifold L is characterized by a curl-free momentum field p(q) which in turn can be regarded as the gradient ∂ S/∂q of a scalar field S. Taking up Littlejohn’s charming manner of speaking [22], I shall call S the generating function of L; this name reminds us of the other, already familiar role S plays in semiclassical games, that of the generating function of classical time evolution as a canonical transformation. Indeed, in the short-time version of the Van Vleck propagator (10.2.25), the action S(q, q , t) is a generating function in both meanings. As the generating function of L, we may get S from a path-independent integral in configuration space  q d x p(x). (10.4.2) S(q) = The starting point for that integral is arbitrary in accordance with the fact that the generating function of L is determined only up to an additive constant. Even within the semiclassical form AeiS/ of a wave function such as the short-time version of the Van Vleck propagator, the additive constant in question is of no interest inasmuch as a constant overall phase factor is an unobservable attribute of a wave function. However, if L has caustics such as shown in Figs. 10.6 and 10.7, one may divide it into caustic-free regions separated by the caustics, here each region possesses a different generating function S α (q) and momentum field p α (q) = ∂ S α (q)/∂q. In other words, the momentum field is no longer single-valued when caustics are around. Then,  semiclassical wave functions take the form of sums over several branches, α ψ = α Aα eiS / . From a classical point of view, one might be content for each S α to have an undetermined additive constant. For a wave function only, an overall phase factor is acceptable as undetermined, however; relative phases between the

p

p

2

1

L B

A

Fig. 10.7 Two branches (sheets, rather) of a Lagrangian manifold which may be seen as divided by a caustic (a configuration-space caustic here, depicted as the dashed line). Courtesy of Littlejohn [22]

q

q q

1

2

414

10 Semiclassical Roles for Classical Orbits α

various Aα eiS / determine interference between them.Therefore, we can leave open only one additive constant for one of the S α ’s and must uniquely determine all the other S α ’s relative to an arbitrarily picked “first” one. With this remark, we are back to Maslov indices.

10.4.2 Elements of Maslov Theory In Sect. 5.3, I already mentioned Maslov’s idea to temporarily switch to the momentum representation when a configuration-space caustic is encountered. It is now about time at least to sketch the implementation of that idea. Let us first recall the time-dependent Van Vleck propagator (10.2.25) which in α α general consists of several “branches”, i.e., terms of the structure Aα ei(S /−ν π/2) . Each of these corresponds to a classical trajectory leading from some point on the initial Lagrangian manifold L at q to a point with coordinate q on the final manifold (See Fig. 10.8). The various such trajectories have different initial and final momenta, p α = −∂ S α /∂q and p α = ∂ S α /∂q, respectively. But as explained above, for Hamiltonians of the structure H = T + V and sufficiently small times t, only a single term of the indicated structure arises, corresponding to a Lagrangian manifold L still without configuration-space caustics. Now assuming t so large that L has developed caustics, I consider a region in configuration space around one of them. If dealing with f = 1, it suffices to imagine that there are only two branches of the curve p(q) making up L “above” the q-interval in question; these branches are joined together at the phase-space point whose projection onto configuration space is the caustic. As pointed out in the previous subsection, there is no momentum-space caustic around then. For f > 1, I analogously consider a part of L with a single configuration-space caustic and no momentum-space caustic. Then, it is advisable to follow Maslov into the momentum representation where Schr¨odinger’s equation reads L' Momentum

Fig. 10.8 Trajectory c has reached a caustic at the final time t, while that incident is imminent for orbit 1 and has already passed for orbit 2. Note again that the Lagrangian manifold associated with the time-dependent propagator is transverse to the trajectories. Courtesy of Littlejohn [22]

L 2 c 1

q'

Coordinate

10.4

Lagrangian Manifolds and Maslov Theory

˙˜ p, t) = iψ(



  p2 ∂ ˜ p, t). ψ( + V i 2m ∂p

415

(10.4.3)

A semiclassical solution may here be sought through the WKB ansatz ˜ p,t)/ ˜ p, t)ei S( ˜ p, t) = A( ψ( .

(10.4.4)

Inasmuch as this is a single-branch function due to the assumed absence of momentum-space caustics, no problem with Maslov or Morse phases arises. In full analogy to the usual procedure in the q-representation (see Problem 10.1), ˜ p, t)/ obeys the Hamilton–Jacobi we find that to leading order in  the phase S( equation ˙˜ p, t) + S(



 ˜ p, t)  ∂ S( p2 +V − = 0, 2m ∂p

(10.4.5)

and thus may be interpreted as a momentum-space action. The next-to-leading order yields a continuity equation of the form  ∂  ˜ ∂ ˜ ˜ p, t)] = 0 ; | A( p, t)|2 − | A( p, t)|2 V [− S( ∂t ∂p

(10.4.6)

˜ p, t)|2 here, the force −V provides the velocity in momentum space that makes −| A(

˜ V (− S( p, t)) a probability current density in momentum space. Coordinate-space and momentum-space wave functions are of course related by a Fourier transform. In our present semiclassical context, that Fourier transform is to be evaluated by stationary phase, in keeping with the leading order in  to which we are already committed. Therefore, I may write  ψ(q, t) =

df p ˜ p, t) ei pq/ ψ( (2π) f /2

(10.4.7)

with the single-branch momentum-space wave function (10.4.4). Momentarily setting f = 1, we arrive at the stationary-phase condition ˜ p q = −∂ S/∂

(10.4.8)

which for the envisaged situation has two solutions pα (q) with α = 1, 2. Thus, ˜ p) gives rise to a two-branch ψ(q) which with the help of the the single-branch ψ( Fresnel integral (10.2.12) we find as

416

10 Semiclassical Roles for Classical Orbits

ψ(q) = α

2 

Aα (q)eiS

α

(q)−iν α π/2

α=1 α

˜ pα (q)) S (q) = p (q)q + S( F 6 G  −1 G d 2 S( ˜   dp α (q)    p) H α α α     ˜ ˜ A (q) = A p (q)  = A( p (q))  dp 2  p= pα dq 

(10.4.9)

  2˜   α   1 d S( p) dp (q) 1 1 − sign 1 + sign . = ν = 2 2 dp 2 dq α

The two branches of the coordinate-space wave function differ not only by having their own action and prefactor each but also by the relative Maslov alias Morse phase π/2.14 A word and one more inspection of Fig. 10.8 are in order about the classical trajectories associated with the two branches of the wave function ψ(q, t) in (10.4.9); each goes from a point on the initial Lagrangian manifold L to a point on the image L of L at time t. One of them, that labelled “1”, has at some previous moment passed the caustic shown, while the trajectory labelled “2” has not yet gone through the “cliff”; the critical moment tc has arrived for the trajectory designated “c”, when the final point q is conjugate to the initial q . If, as shown in the figure, the slope dp/dq of the Lagrangian manifold at the caustic changes from negative to positive (while passing through infinity), the Maslov alias Morse index of branch “1” is larger by one than that of branch “2”, according to the above expression for ν α . This is in accord with our previous interpretation of the Morse index: The slope dp/dq of L should be written as the partial derivative ∂ p(q, q )/∂q when the initialvalue problem pertaining to the propagator is at issue; but when ∂ p/∂q changes from negative to positive so does ∂q/∂ p and also, since points on L may still be uniquely labelled by the initial momentum p , the derivative ∂q/∂ p ; we had seen that ∂q/∂ p = 0 is the conjugate-point condition and that the Morse index increases by one when a conjugate point is passed. A similar picture arises for the energy dependent-propagator. The only difference is that the relevant Lagrangian manifold L E has the trajectories lying within itself (rather than piercing, as is the case for the time-dependent propagator and its L). Even more intuitively, then, the trajectory which has climbed through the vertical cliff in L E with the slope changing from negative to positive yields a Maslov/Morse index larger by one than that of the trajectory not yet through the cliff. When attempting to solve the initial-value problem for ψ(q, t) by a WKB ansatz in the coordinate representation, one finds a single-branch solution as long as the 14 In this subsection, it would seem overly pedantic to insist on the name Morse phase or index for wave functions or propagators, reserving the name Maslov index for what appears in traces of propagators; as a compromise I speak here of Morse alias Maslov phases for multibranch wave functions or propagators.

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Lagrangian Manifolds and Maslov Theory

417

Lagrangian manifold L originating from the initial L is free of caustics. The singlebranch ψ(q) diverges at precisely that moment when a caustic of L arrives at the configuration-space point q. A little later, a WKB form of ψ(q, t) arises again but now as a two-branch function. That sequence of events might leave a spectator puzzled who is born to the coordinate representation. Maslov’s excursion into momentum space opens a caustic-free perspective and thus avoids the catastrophic but only momentary breakdown of WKB. Conversely, when in the momentum representation a momentum-space caustic threatens failure of WKB, one may take refuge in coordinate space (representation). At any rate, Maslov’s reasoning explains quite naturally why the breakdown of the coordinate-space Van Vleck propagator at a coordinatespace caustic is only a momentary catastrophe. Incidentally, the divergence of the time-dependent propagator upon arrival of a caustic at q may but does not necessarily imply breakdown of the semiclassical approximation. The harmonic oscillator treated in Problem 10.5 provides an example: The divergence of the semiclassical Van Vleck propagator takes the form of a delta function, G(q, q , tc ) = δ(q + q ) for times tc = (2n + 1)T /2, n = 1, 2, . . . and G(q, q , tc ) = δ(q − q ) for times tc = nT, n = 1, 2, . . ., i.e., whenever a point conjugate to the initial q

is reached; but this behavior faithfully reproduces that of the exact propagator. Conjugacy means simultaneous reunification at q of a one-parameter family of orbits which originated from the common q with different initial momenta and that family of orbits interferes constructively in building the “critical” propagator G(q, q , tc ) = q|e−iH tc / |q . The occurrence of multibranch WKB functions may be understood as due to several effective caustics. Maslov’s procedure then naturally assigns to each branch its own action branch S α (q) and Maslov alias Morse phase −ν α π/2, leaving open only a single physically irrelevant overall phase factor. It is well to keep in mind that each branch is provided by a separate classical trajectory and that the Maslov index equals the net number of passages of the trajectory through vertical cliffs in the relevant Lagrangian manifold where the slope dp/dq changes from negative to positive; “net” means that a passage in the inverse direction decreases the index by one. To liven up my words a bit, I shall henceforth speak of clockwise rotation of the tangent to L or L E at the caustic when the slope dp/dq changes from negative to positive and of anticlockwise rotation when the slope becomes negative. Thus, the net number of clockwise passages of that tangent through the momentum axis along a single traversal of a periodic orbit gives the Morse alias Maslov index in the contribution of that orbit to the propagator. The above (f =1) result (10.4.9) can be generalized to more degrees of freedom [22, 41]. The essence of the change is to replace |dp α (q)/dq| by | det dp α (q)/dq| ˜ p)/∂ p∂ p in the amplitude Aα (q). If only a single eigenvalue of the matrix ∂ 2 S( vanishes at the coordinate space caustic, one may imagine axes chosen such that in the above expression for the Maslov index dp(q)/dq → ∂ pl (q)/∂qm where the subscripts indicate the components of p and q with respect to which the coordinatespace caustic appears with ∂ pl (q)/∂qm = ∞.

418

10 Semiclassical Roles for Classical Orbits

10.4.3 Maslov Indices as Winding Numbers We can finally proceed to scrutinizing an unstable periodic orbit for its Maslov index μ, staying with autonomous systems of two degrees of freedom. The presentation will closely follow Creagh, Robbins, and Littlejohn [42] but Mather’s pioneering contribution [43] should at least be acknowledged. The generalization to arbitrary f can be found in Robbins’ paper [44]. Intuition will be furthered by focusing on the three-dimensional energy shell. Therein the stable (and the unstable) manifold of the periodic orbit is a two-dimensional surface. Now imagine a surface of section Σ, also two-dimensional, transverse to the orbit, and let that Σ move around the periodic orbit. As in deriving the trace formula in Sect. 10.3.2, I shall specify the location of Σ by the coordinate x along the orbit and parametrize Σ(x ) by the transverse coordinate x⊥ and its conjugate momentum p⊥ . The stable manifold of the periodic orbit intersects Σ(x ) in a line σ (x ) through the origin where the orbit pierces through Σ(x ); near the origin, σ (x ) will appear straight. As Σ(x ) is pushed along the orbit, the stable manifold σ (x ) will rotate, possibly clockwise at times and then anticlockwise, as may happen. But σ (x ) must return to itself when Σ(x ) has gone around the periodic orbit. The net number of clockwise windings of σ (x ) about the periodic orbit during a single round trip of Σ(x ) is obviously an intrinsic property of that orbit, independent of the phase-space coordinates employed. We shall see presently that twice the net number of clockwise windings of the stable manifold for one traversal of the periodic orbit in the sense of growing time is the Maslov index μ appearing in the trace formula (10.3.33). With that fact established, the constancy of μ along the periodic orbit (i.e., the independence of x ) is obvious, as is the additivity with respect to repeated traversals: If μ refers to a primitive orbit, its r -fold repetition has the Maslov index r μ. Thus, the trace formula (10.3.32) for the density of levels may be rewritten by explicitly accounting for repetitions of primitive orbits,

osc (E) =

∞ T 1   cos r (S/ − μπ/2), √ π  r=1 prim.orb. |2 − TrM r |

(10.4.10)

where T, M, S, and μ all refer to primitive orbits and complete “hyperbolicity” is assumed, i.e., absence of stable orbits. To work, then! We shall have to deal with mapping the “initial” surface of section Σ(x ) to the “running” one, Σ(x ), as linearized about the origin (i.e., about the orbit in question), 

δx⊥ δp⊥



⎡

 

⎢ =⎢ ⎣

 

 ⎤

∂ x⊥ ∂ x⊥ p

⊥

∂ x⊥

∂ p⊥ x⊥

∂ p⊥ ∂ x⊥ p

⊥

∂ p⊥ x⊥

       ⎥ δx

δx⊥ ab δx⊥

⊥ , x ) = = M(x .  ⎥



⎦ δp⊥ δp⊥ δp⊥ cd ∂ p⊥

10.4

Lagrangian Manifolds and Maslov Theory

419

The special case of the matrix M(x , x ) pertaining to a full traversal of the periodic orbit is the stability matrix M entering the trace formula (10.3.33); kin to the present M(x , x ) is that found for area-preserving maps in (10.3.4); indeed, the close analogy of area preserving stroboscopic maps to mappings between surfaces of section for Hamiltonian flows permeates all of this chapter. Now instead of immediately concentrating on the (section with Σ(x ) of the) stable manifold σ (x ), it is convenient first to look at the section λ(x ) of Σ(x ) with the Lagrangian manifold L E associated with the energy-dependent propagator. The reader will recall that L E is the two-dimensional sheet swept out by the set of all trajectories emanating from the set of all points within the energy shell projecting to one fixed initial point in configuration space. The line λ(x ) arises as the image under the mapping M(x , x ) of the momentum axis in Σ(x ) or rather that interval of the momentum axis accommodated in the energy shell. Like σ (x ), the line λ(x ) goes through the origin of the axes used in Σ(x ); close to that origin λ(x ) looks straight and can thus be characterized by the vector (b, d) in Σ(x ); that vector even provides an orientation to λ when attached to the origin like a clock hand. I shall refer to the part of λ extending from the origin along the vector b, d as the half line λ+ . Now, recall from (10.3.31) that the full Maslov index of a periodic orbit, μ = μprop + μtrace , gets the contribution μprop from the energy-dependent propagator while the trace operation furnishes μtrace . The first of these has already been given a geometric interpretation in Sect. 10.4.2. The consideration there was worded for a two dimensional phase space and thus fits our present Σ(x ), and the slope of the Lagrangian manifold here is given by d/b. The net number of clockwise rotations of the vector (b, d) through the momentum axis while one goes around the periodic orbit gives μprop . Only a subtlety regarding the initial Lagrangian manifold, i.e., the momentum axis (rather, the part thereof fitting in the energy shell), needs extra comment. That initial λ(x ) is as a whole one caustic and one that is not climbed through. Whether or not the index μprop gets a contribution here depends on the sign of ∂ p⊥ /∂ x⊥ at infinitesimally small positive times, t = 0+ ; if positive, no contribution to the index arises, whereas for negative slope, the index suffers a decrement of 1; this simply follows from our discussions of the Van Vleck propagator. In fact, we already saw that Hamiltonians of the structure H = T + V always give rise to minimal (rather than only extremal) actions for small times, such that the initial caustic is immaterial for μprop there. To proceed toward μtrace , we must determine where the vector (b, d) ends up after one round trip through the periodic orbit. Since we are concerned with an unstable orbit, we can narrow down the final direction with the help of the stable and unstable manifolds whose tangent vectors at the origin are given by the two real eigenvectors es and eu of the 2 × 2 matrix M pertaining to the completed round trip; the respective eigenvalues will be called 1/τ and τ with |τ | > 1. As shown in Fig. 10.9, these directions divide Σ(x ) into four quadrants, labelled clockwise H, I, J, and K . H is distinguished by containing the upper momentum axis, so the positive- px part of the initial λ(x ) lies therein. The final vector (b, d) must lie in that same quadrant if the orbit is hyperbolic (τ > 2); conversely, for hyperbolicity with reflection (τ < −2), the vector ends up in the opposite quadrant, J .

420

10 Semiclassical Roles for Classical Orbits

Fig. 10.9 The surface of section Σ transverse to a periodic orbit is divided into four sectors H, I, J, and K by the (sections with Σ of the) stable and unstable manifolds which look locally straight. At some arbitrarily chosen initial moment, the section λ of the Lagrangian manifold with Σ starts out as the momentum axis; its positive part, the half line λ+ , rotates as the section Σ(x ) is carried around the periodic orbit and ends up, after one full traversal, either in sector H (hyperbolic case) or in sector J (inverse hyperbolic case). If λ+ ends up in H+ or J+ , μtrace = 0 while arrival in H− or J− implies μtrace = 1. Courtesy of Creagh, Robbins, and Littlejohn [42]

The index μtrace depends on where the vector (b, d) and thus the half line λ+ end up within the quadrants H or J . The definition (10.3.30) implies that μtrace = 0 if the quantity w ≡ {[(∂/∂ x⊥ + ∂/∂ x⊥ )2 S(x⊥ , x , x⊥ , x , E)]x⊥ =x⊥ =0 } TrM − 2 τ + 1/τ − 2 a+d −2 = = = b b b

(10.4.11)

is positive while μtrace = 1 if w < 0. But since |τ + 1/τ | > 2, the signs of τ + 1/τ − 2, τ + 1/τ , and τ all coincide. It follows that the numerator of w is positive when M is hyperbolic and negative if M is hyperbolic with reflection. The denominator b, on the other hand, is positive or negative when the vector (b, d) ends up, respectively, to the right or the left of the momentum axis. Therefore, it is indicated to subdivide the quadrants H and J into the sectors H+ , H− , J+ , and J− , as shown in Fig. 10.9. The negative subscripts denote sectors counterclockwise to the momentum axis while the sectors clockwise of it have positive subscripts. It follows that w > 0 if the vector (b, d) ends up in either H+ or J+ since in that case, τ and b have the same sign; conversely, w < 0 if the said vector ends up in either H− or J− , τ and b differs in sign then. What finally matters is that μtrace = 0 if the half line λ+ ends up running through the sectors H+ and J+ while μtrace = 1 if it ends up running through H− and J− . With both μprop and μtrace geometrically interpreted, now we can tackle their sum, the full Maslov index. To that we extend the definitions of the quadrants H, I, J, and K to the intermediate surfaces of section Σ(x ). We do that by introducing the vectors es∗ (x ) and eu∗ (x ) such that their components along the x-axis and the px -axis in Σ(x ) remain the same as those of es and eu in the initial surface of

10.4

Lagrangian Manifolds and Maslov Theory

421

section Σ(x ). Note that these starred vectors are not the ones specifying the stable ∗ of M(x , x ), i.e., and unstable manifolds in Σ(x ) through the eigenvectors es,u ∗ es,u (x ) = es,u (x ). Rather, the starred vectors carry the quadrants H, I, J, and K along in rigid connection with the x- and px -axes, as we go through the sequence of sections Σ(x ) around the orbit. During that round trip, we may define the current value of μprop as the net number of clockwise passages of the vector (b, d)(x ) through the momentum axis along the part of the voyage done. A little more creativity is called for in defining intermediate values of μtrace and thus the sum μ since the connection with the trace must be severed. It proves useful to define μtrace = 0 if λ(x ) and thus the vector (b, d)(x ) is clockwise of the momentum axis, i.e., in the sectors H+ or J+ and μtrace = 1 if λ(x ) is anticlockwise of it in the sectors H− or J− . There is no need to define μtrace for the quadrants I and K since, as seen above, λ cannot end up therein. Now the full Maslov index μ behaves quite simply as we go through the orbit. When λ(x ) sweeps through the quadrants H and J , no change of μ occurs, since whenever λ(x ) rotates through the momentum axis, the changes of μprop and μtrace cancel in their sum. But μ does increase by one for every completed clockwise passage of the vector (b, d)(x ) through either quadrant I or K . Finally, I hurry to formulate the result just obtained in terms of the stable manifold σ (x ). This is necessary since in contrast to the Lagrangian manifold λ(x ), the stable manifold σ (x ) is obliged to return to itself after one round trip through the periodic orbit. Incidentally, if λ(x ) does not coincide with σ (x ) initially, it cannot do so later either since the map M(x , x ) is area-preserving and cannot bring about coincidence of originally distinct directions. This means that λ(x ) shoves along σ (x ), though not with a constant angle in between. But both lines must have the same net number of clockwise passages through the quadrants I and K . When σ (x ) returns to itself after one traversal of the orbit it must have undergone an integer number of “half-rotations” in the surfaces of section Σ(x ) ; half-rotation means a rotation by π about the origin. During a half-rotation σ (x ) sweeps through I and K . The number of half-rotations during one round trip through the orbit is the full Maslov index; we might also say, that μ is twice the number of full 2π -rotations if we keep in mind that the number of full rotations is half-integer if the orbit is hyperbolic with reflection. Now the previous claim that the Maslov index is a winding number are shown true. In addition, we conclude that μ is even for hyperbolic orbits and odd for hyperbolicity with reflection. It may be well to remark that odd need not, but may be taken to mean either 1 or 3, and even either 0 or 2, since in the trace formula, μ appears only in a phase factor e−iμπ/2 . Before concluding, a sin must be confessed, committed in employing the specific coordinates x , p , x⊥ , p⊥ . Helpful as they indeed are, they may be ill defined at certain moments during the round trip along the periodic orbit. Such instants occur if when arriving at a caustic, the orbit momentarily comes to rest (see Fig. 10.10), such that (not only x˙ ⊥ = 0 which, by definition, is always the case but also) x˙ = 0. In coordinate space, the orbit then traces out a cusp. In the tip of the cusp, the coordinates x , x⊥ are indeed ill defined since both axes become inverted there. Our final result for μ holds true, even when such cusps arise as one may check

422

10 Semiclassical Roles for Classical Orbits

Fig. 10.10 An orbit momentarily coming to rest at a caustic. In this nongeneric case, the coordinates x , p are momentarily ill defined. Then, the projection of the orbit onto configuration space displays a cusp. Courtesy of Creagh, Robbins, and Littlejohn [42]

by temporarily switching to the momentum representation a` la Maslov where the caustic is absent entirely. While the considerations of the present section establish the canonical invariance of all ingredients of the Gutzwiller trace formula (10.4.10) for Hamiltonian flows of autonomous systems with two degrees of freedom, the question is still open for the case of maps with f = 1. To settle that question, I proceed to show the equivalence, both classical and quantum mechanical, of autonomous dynamics with f = 2 and periodically driven ones with f = 1.

10.5 Riemann–Siegel Look-Alike Inspired by the famous Riemann–Siegel formula for Riemann’s zeta function, Berry and Keating [34–36] devised a semiclassical representation of the spectral determinant det (E − H ) with Im E ≥ 0 which is convergent and becomes real when the energy argument does, Im E → 0. The essence of the argument lies in the restriction to periodic orbits whose periods are bounded from above by half the Heisenberg time. The resulting Riemann-Siegel look-alike will turn out useful for the demonstration of spectral universality in the later sections. Readers with the strong nerves of physicists are invited to adventure in matheˆ matically unsafe territory, starting with the definition of a zeta function, H ˆ )(E − H ˆ )} = ζ (E) = det{ A(E, H

, { A(E, E j )(E − E j )} j

ˆ ) exp Tr ln(E − H ˆ ). = det A(E, H

(10.5.1)

10.5

Riemann–Siegel Look-Alike

423

To provide a notational distinction between definitely real and possibly complex energies, I shall denote the former by E and the latter by E. The function A is arbitrary, save for having no real zeros, being real for real E, and ensuring convergence should the Hilbert space dimension N not be finite; elementary examples for such “regularizers” are found in [34] and Problems 10.9 and 10.10. Then, the zeta function has the eigenvalues E j as the only real zeros. The logarithm appearing above can be expressed as the integral of the Green function g(E) = Tr

1 ˆ E−H

(10.5.2)

1 which yields the level density for E = E + i0+ as (E) = − Im g(E + i0+ ) and π thus the level staircase as N (E) = −

1 π



E

d E Im g(E + i0+ ) ;

(10.5.3)

0

ˆ begins immediately to the right of for convenience I imagine that the spectrum of H E = 0. Now, I replace the logarithm, 

E

ˆ ) − Tr ln (− H ˆ)= Tr ln (E − H 

dE g(E )

0 E

= 0

dE gosc (E ) +

(10.5.4) 

E

dE g(E ),

0

splitting the Green function into an “oscillatory” and a smooth part; the latter is defined as usual by a local spectral average. The smooth part is again split into $E $E two pieces such that 0 dE g(E ) = −iπ N (E) + { 0 dE g(E ) + iπ N (E)}; as indicated by a somewhat cavalier notation, the first of these is defined so as to become −iπN (E) for E = E + i0+ while the second becomes real in that limit; the local spectral average N (E) is the Weyl staircase, i.e., the number of Planck cells contained in the part of phase space with energy up to E, 

N (E) = (2π )− f

d fqd f p Θ[E − H (q, p)].

(10.5.5)

Thus, the zeta function assumes the form  E 3 4 ζ (E) = B(E) exp −iπ N (E) + dE {g(E ) − g(E )} ,

(10.5.6)

0

wherein the prefactor ˆ ) exp B(E) = det(−A H

3 0

E

4

dE g(E ) + iπ N (E)

(10.5.7)

424

10 Semiclassical Roles for Classical Orbits

becomes real and free of zeros for real E = E. All is now prepared for the intended jump into semiclassical terrain which amounts to invoking the periodic-orbit expansion (10.3.29) of the oscillatory part of the level density, gosc (E) =

∞ e ir S p (E)/ 1  , Tp % i p r=1 | det(M rp − 1|

(10.5.8)

where r -fold repetitions of primitive periodic orbits p are taken care of explicitly. Since we actually need the integral of gosc we should recognize that periodic-orbit sum as nearly a total energy derivative due to ∂S p /∂E = T p (E), i e ir S p (E)/ ∂ e ir S p (E)/ % = r Tp % + ... ; ∂E | det(M r − 1|  | det(M rp − 1| p

(10.5.9)

the dots stand for a term involving the derivative of the denominator which is smaller by one order in  than the first term and may be dropped. Upon inserting all of this periodic-orbit stuff into our zeta function from (10.5.6), we get the periodic-orbit approximation ∞ 3    e ir S p (E)/ 4 % . (10.5.10) ζ (E) ∼ B(E) exp − iπ N (E) exp − r p r=1 r | det(M p − 1)|

Of the three factors in the zeta function, the last two are of prime importance: the second one, e−iπ N (E) , becomes a “Weyl type” phase factor for real energies E; the third one, the exponentiated periodic-orbit sum, contains all semiclassical information about spectral fluctuations. For convergence, the periodic-orbit sum needs to be restricted to complex energies further away from the real energy axis than the limit established at the end of Sect. 10.3.4, Im E > λ/2. It is helpful to write the infinite product over periodic orbits and their repetitions in (10.5.10) as a series. To that end, I first do the sum over repetitions. Barring dynamics with stable periodic orbits, I must reckon with eigenvalues of the stability matrix M p that read15 exp(±λ p T p ) − exp(±λ p T p )

Poincar´e map hyperbolic

(10.5.11)

Poincar´e map hyperbolic with reflection.

Focussing on the first case I write the determinant det (M rp − 1) = 2 − Tr M rp as

15 For simplicity, I specialize to two-freedom systems from hereon; there is only a single Lyapunov exponent λ p for the pth orbit then.

10.5

Riemann–Siegel Look-Alike

425

 2  2 | det(M rp − 1)| = erλ p Tp 1 − e−r λ p Tp = erλ p Tp /2 − e−r λ p Tp /2

(10.5.12)

and expand the stability prefactor in a geometric series, %

1 | det(M rp

− 1)|

= e−r λ p Tp /2

∞ 

e−kr λ p Tp .

(10.5.13)

k=0

∞ 1 The sum over repetitions then yields − r=1 exp r [iS p / − (k + 12 )λ p T p ] = r 1 ln (1 − exp [iS p / − (k + 2 )λ p T p ]). In the zeta function we have thus bargained one product against another, ζ (E) = B(E)e −iπN (E)

4  1 − exp iS p / − (k + 12 )λ p T p . (10.5.14)

∞ 3 ,, p k=0

A sum is to come through Euler’s identity ∞ ,

(1 − ax k ) = 1 +

k=0

∞  r=1

(−a)r x r(r −3)/4 (x −1/2 − x 1/2 )(x −1 − x) . . . (x −r/2 − x r/2 )

(10.5.15)

and reads ζ (E) = B(E)e −iπ N (E) (10.5.16)  5  ∞ 1  , exp ir S p / − 4 r (r − 1)λ p T p (−1)r 1+ . × +r j 1/2 p j=1 | det(M p − 1)| r=1 The phase factors suggest interpreting the summation variable r as a new repetition number. Upon expanding the product over primitive orbits, we finally arrive at the desired series  FP (E) e i S P (E)/ . (10.5.17) ζ (E) = B(E)e −iπ N (E) P

Now, the summation is over “pseudo-orbits” in which each primitive orbit is repeated rp times (possibly rp = 0) such that P is a multiple summation variable, P = {rp },

rp = 0, 1, 2, . . . .

(10.5.18)

A pseudo-orbit has as its action and pseudo-period the pertinent sums over the contributing primitive orbits, SP =

 p

rp S p ,

TP =

 ∂S P rp T p . = ∂E p

(10.5.19)

426

10 Semiclassical Roles for Classical Orbits

The pseudo-orbit series (10.5.17) can be thought ordered by increasing pseudoperiod TP . The weight FP (E) of the Pth pseudo-orbit is FP (E) =

, p



5  exp − 14 rp (rp − 1)λ p T p (−1) +rp . j 1/2 j=1 | det(M p − 1)| rp

(10.5.20)

I leave to the reader to show that primitive orbits that are hyperbolic with reflection show up like hyperbolic ones in the pseudo-orbit sum (10.5.17), except for the replacement (−1)rp → (−1)Int[(rp +1)/2] in the weight FP (E). Incidentally, multiple repetitions of a primitive orbit within a pseudo-orbit have rapidly decreasing weights as rp grows, due to the factor  exp − 14 rp (rp − 1)λ p T p −→ exp{− 12 rp2 λ p T p }. +rp j 1/2 | det(M − 1)| p j=1

(10.5.21)

Thus, the most important long pseudo-orbits are those composed of singly traversed primitive orbits. The predominance of singly traversed primitive orbits in the pseudo-orbit sum (10.5.17) is also due to the exponential proliferation of orbits with growing period that we have seen to follow from the sum rule of Hannay and Ozorio de Almeida in Sect. 9.12. Mustering a lot of courage I follow Berry and Keating to real energy in the pseudo-orbit sum for the zeta function. Convergence is surely lost; reality, while required by the definition (10.5.1), is no longer manifest and definitely destroyed by any finite truncation of the infinite sum. To make the best of the seemingly desperate situation we may impose reality,

e

−iπ N (E)

∞ 

FP (E) e

i S (E)  P

= e iπN (E)

P

∞ 

FP (E) e−  S P (E) , i

P

(10.5.22) in the hope of thus forbidding the divergence to play the most evil of games with us. But no amount of conjuration will retrieve convergence. Not even refuge to complex energies is possible since the l.h.s. would require Im E > λ/2 (see Sect. 10.4.3) and the r.h.s. Im E < −λ/2, leaving no overlap of the domains of convergence. Keating modestly calls the foregoing reality condition the “formal functional equation”[35], before cunningly drawing most useful consequences. $ E+α() A Fourier transform E−α() d E exp (iτ E /) (. . .) with respect to a classically small but semiclassically large energy interval, α() → 0,

α()/ → ∞

for

 → 0,

(10.5.23)

is now done on the functional equation. Neither the Weyl staircase N (E) nor the action S P vary much over the small integration range and may thus both be expanded around E to first order such that the l.h.s. of (10.5.22) yields

10.5

Riemann–Siegel Look-Alike

 i[−πN (E)+τ E/]

+α()

427

d E eiE [−π (E)+τ/]

e



−α()



FP (E) ei(S P (E)+E TP )/ (10.5.24)

P

or, after rescaling the integration variable as E / → ω and boldly interchanging the order of integration and summation,  +α()/   ei[−πN (E)+τ E/] FP (E) eiS P / dω eiω[−π (E)+τ +TP ] . (10.5.25) −α()/

P

But now the integration range is large, and the η-integral gives a fattened delta function 2π δ/α() [−π (E) + τ + TP ] whose width /α() vanishes as  → 0. Exactly the same procedure brings the r.h.s. of the functional equation into to a form identical to (10.5.26) save for sign changes in front of N , , S P , and TP . Hence, the functional equation reappears as    FP (E) eiS P (E)/ δ/α() − π(E) + τ + TP (E) (10.5.26) e−iπ N (E) P

=e

iπN (E)



  FP (E) e−iS P (E)/ δ/α() π (E) + τ − TP (E) .

P

As a final step we integrate on both sides over τ from 0 to ∞ and recognize π(E) = TH (E)/2 as half the Heisenberg time, TP ≤

TH

2

TP >

FP exp i(S P / − πN ) =

P

TH

2

FP exp i(−S P / + π N ).

(10.5.27)

P

We have arrived at a sum rule connecting the manifestly finite contribution from pseudo-orbits with pseudo-periods below half the Heisenberg time to the ill-defined contribution of the infinitely many longer pseudo-orbits; the two contributions are mutual complex conjugates. If we import that sum rule into the zeta function (10.5.17), we get a finite semiclassical approximation, the celebrated Riemann– Siegel look-alike 

TP (E)≤TH (E)/2

ζs.cl. (E) = 2B(E)

 FP (E) cos S P (E)/ − π N (E) .

(10.5.28)

P

What a relief indeed, after all the chagrin about lost convergence! The zeros of this finite semiclassical zeta function [45, 46] give semiclassically accurate energy eigenvalues up to the energy E which one is free to choose, provided that the periodic orbits with periods up to TH (E)/2 are available. It is immaterial now whether the Hilbert space is finite- or infinite-dimensional. Some further remarks are in order in view of later applications. First, even though the special charm of the Riemann–Siegel lookalike lies in the reality of ζ (E) for real E, it is allowable to wander to E ± = E ± iγ and

428

10 Semiclassical Roles for Classical Orbits

  + − 2 cos S P (E)/ − iπN (E) → ei(S P (E )/−π N (E)) + e−i(S P (E )/−π N (E)) , (10.5.29) with the aim to enhance convergence in the limit  → 0 ⇔ TH ∝ − f +1 → ∞; of course, the quantities N (E) and B(E) can be kept at real E since they are the same for all pseudo-orbits; the stability factor FP (E) also does not need to be complexified since the difference F(E) − F(E ± ) is not referred to a quantum scale in the above sums. Second, if one wants to forsake orbit repetitions to begin with, the Riemann– Siegel lookalike can be obtained somewhat faster, by dropping all terms with r > 1 from the exponentiated orbit sum (10.5.10) and expanding the exponential in powers of the sum over primitive orbits; see Problem 10.11. Finally, I kindly invite the reader to think about the generalization of the reasoning of the present section to more than two degrees of freedom.

10.6 Spectral Two-Point Correlator The universality of spectral fluctuations under conditions of classical hyperbolicity and in the absence of any symmetries (beyond the ones distinguishing the WignerDyson universality classes) will be illustrated here, by a semiclassical determination of the two-point correlator of the density of levels. Due to a suitable energy average within a single energy spectrum, that correlator takes the form predicted by randommatrix theory. The four essential ingredients to be used are (i) a generating function involving four spectral determinants, (ii) Gutzwiller sums for the spectral determinants, (iii) the Riemann–Siegel lookalike, and (iv) bunches of orbits differing in close self-encounters giving constructively interfering contributions to the fourfold Gutzwiller sum. Throughout the present section I shall closely follow Ref. [6].

10.6.1 Real and Complex Correlator The simplest indicator of spectral correlations is the connected correlator

R(e) =

  A 1 @  e e  E − −1  E + (E)2 2π(E) 2π (E)

(10.6.1)

where the angular brackets (or an overbar) again denote a local average over the center energy E. The dimensionless variable e gives the energy offset in units of the mean level spacing 1/ ≡ 1/¯ (divided by 2π , for convenience). Dyson’s cluster function which was studied in the framework of random-matrix theory in Chapt. 4 is related to the above correlator as Y (e) = δ(e) − R(e). RMT yields

10.6

Spectral Two-Point Correlator

429

⎧ ⎨−s(e)2 = − 1 − cos 2e 2e2 R(e) − δ(e) = 1  ⎩ 2

−s(e) + s (e) π Si(e) − 12 sgn(e)

unitary class

(10.6.2) orthogonal class

$e sin e and Si(e) = 0 de s(e ). e The correlation function R(e) is closely connected to the complex correlator of Green functions of the Hamiltonian H

with s(e) =

C(e+ ) =

A 1 1 @ 1 1 Tr Tr − + + e e 2π 2 ¯ 2 2 E + 2π −H E − 2π −H ¯ ¯

(10.6.3)

where the superscript + denotes a positive imaginary part +iη. The first Green func¯ > 0 while the second is tion in (10.6.3) is a retarded one due to Im (E + e+ /2π ) an advanced one/. The real part of C(e+ ) at real energy offset yields R(e) = Re lim C(e+ ).

(10.6.4)

η→0

The RMT prediction for the complex correlator can be written as ⎧ + ⎪ e2ie 1 ⎪ ⎪ − ⎪ + 2 ⎪ 2(ie+ )2 ⎪ ⎪ 2(ie ) ⎨  1 (n − 3)!(n − 1) C( + ) ∼ + ∞ n=3 + )2 ⎪ (ie 2(ie+ )n ⎪ ⎪ ⎪ ⎪ (n − 3)!(n − 3) + ∞ ⎪ ⎪ ⎩+ e2ie n=4 2(ie+ )n

unitary class (10.6.5) orthogonal class,

and the semiclassical confirmation of that prediction is the principal goal of the present section. For the unitary symmetry class, C(e+ ) is the sum of a monotonous term ∝ e12 and an oscillatory term ∝ e12 e2ie . That structure of C also arises for the orthogonal class, except that both the monotonous term and the cofactor of the oscillating exponential e2ie are asymptotic series in 1e ; the series can be Borel summed and a closed from analogous to (10.6.2) be given. Before launching the semiclassical calculation of C(e+ ) it is well to pause for an interlude and prove the relation (10.6.4) between complex and real correlator.

10.6.2 Local Energy Average I start with N consecutive levels E 1 < E 2 < . . . E N , disregarding levels below E 1 and above E N . The size ΔE = E N − E 1 of the energy window so filled should be much larger than (i) the range allowing to define a smooth local mean density of levels and (ii) the range over which level correlations extend. On the other hand,

430

10 Semiclassical Roles for Classical Orbits

the window must not be too large; the local mean density must be constant therein and thus equal ¯ = E N N−E1 ; for the sake of concreteness I may imagine the window size ΔE to be the maximal one compatible with the internal constancy of . ¯ Any location of the window in the complete spectrum is possible. The included levels allow definition of “local” + N Green function and spectral determinant as  N for the 1 and Δ(E) = g(E) = i=1 i=1 (E − E i ). E−E i With the goal of picking up spectral correlations somewhere in the spectrum, say at E, I place a window of the above kind symmetrically around E. I can now $ E+δ E/2 specify a local energy average as . . . = δ1E E−δ E/2 d E (. . .), with the length δ E of the averaging interval small compared to ΔE but larger than the correlation range and thus containing many levels. The symmetry of both the averaging interval and the larger energy window is a matter of convenience. In the semiclassical limit I have the hierarchy of energy scales 1¯  δ E  ΔE and can even afford the limit δE → 0. ΔE The “local” complex correlator can now be written as16 N  1 1 C(e ) + = 2 2π 2 ¯ 2 δ E i,k=1 +

1 e+ π ¯

+ Ei − Ek

 ln

E − E +

e+ 2π ¯ e+ 2π ¯

− E i E =+δ E/2 − Ek

E =−δ E/2

where for notational convenience I have chosen the reference energy E as the origin for the energy axis. With (x + iη)−1 ↔ P x −1 − iπ δ (x) for η → 0+ and ln x = ln |x| + i arg x, the real part of C can be written as Re C(e+ ) +

1 = RI I + RR R 2

where R I I (R R R ) stems from the product of the imaginary (real) parts of the fraction  and the square bracket in the above double sum ik . Starting with R I I , I take Im e+ /¯ = η/¯ → 0+ . In the (i, k)-th summand in the above double sum for C(e+ ), the phase of the expression in the square brackets can be (i) 2π (both E i and E k lie within the averaging interval [−δ E/2, δ E/2]); (ii) π (only one eigenvalue within [−δ E/2, δ E/2]); (iii) zero (both eigenvalues outside [−δ E/2, δ E/2]). Of exclusive interest are values of the energy offset e small compared with the size δ E of the averaging interval and for these, in view of the factor δ( πe +E i −E k ), the possibility (ii) can be discarded. Up to an additive constant, R I I must then coincide with the real correlation function,    1 e δ + E i − E k = R (e) + 1. RI I = δ E −δ E/2 2 in one is split into an (l − 1)-encounter and a 2-encounter in the other, and the respective contributions cancel. In the absence of time reversal invariance only the diagonal contribution thus survives. For time reversal invariant dynamics from the orthogonal symmetry class, the foregoing cancellation mechanisms still “wipe out” all quadruplets with exclusively parallel encounters.

10.9 Semiclassical Construction of a Sigma Model, Unitary Symmetry Class Even though the nullity of off-diagonal corrections for the unitary symmetry class is now established there is more to learn about that class. I propose to encapsulate the whole pseudo-orbit series for Z (1) in a matrix integral which will turn out to coincide with the one known from the so-called sigma model in RMT.

450

10 Semiclassical Roles for Classical Orbits

10.9.1 Matrix Elements for Ports and Contraction Lines for Links Structures of pseudo-orbit quadruplets were defined by (i) fixing the set of encounters and numbering the encounter ports, (ii) connecting the encounter ports by links and thereby defining the pre- and post-reconnection periodic orbits, and (iii) dividing the pre-reconnection orbits between the pseudo-orbits A, C and the postreconnection orbits between B and D. Now I encode each structure in (10.8.14) by a sequence of alternating symbols19 Bk j and B˜ jk respectively standing for the entrance and exit encounter ports. For each encounter starting with the first entrance and exit ports and following the port labels I pairwise list the symbols for all further ports. The indices of B, B˜ indicate the ˜ affiliation with the pseudo-orbits: the index j, the second on B and the first on B, determines whether before reconnection the port belongs to A (then j = 1) or to C ( j = 2). Similarly, k shows whether the port belongs after reconnection to B (if k = 1) or to D (k = 2). Due to the two possible values for both k and j, altogether four Bk j ’s and four B˜ jk ’s are brought into play. The port numbering introduced before will now prove well suited to the goal posed (Recall that each entrance port is connected by an encounter stretch to the exit port with the same number before reconnection, and the number obtained by backward cyclic shift, within each encounter, after reconnection). The ports connected by an encounter stretch necessarily enter the same orbit and pseudo-orbit. Consequently, neighboring indices of successive port symbols must coincide: Bk j is followed by B˜ jk and preceded by B˜ j k . In particular, a 2-encounter with two entrances and exits as in Fig. 10.13a is assigned the four-symbol sequence Bk1 , j1 B˜ j1 ,k2 Bk2 , j2 B˜ j2 ,k1 ,

(10.9.1)

such that the indices follow one another just like in a trace of a matrix product; however, until further notice no summations are implied; note that the sub-indices on the indices j, k signal encounter stretch labels within the encounter. For a structure with arbitrarily many encounters indexed by σ = 1, 2, . . . V and stretches inside each encounter indexed by a = 1, 2, . . . l(σ ) the sequence of ports gives rise to the product V , 

 Bkσ 1 , jσ 1 B˜ jσ 1 ,kσ 2 Bkσ 2 , jσ 2 . . . B˜ jσ l(σ ) ,kσ 1 ;

(10.9.2)

σ =1

here, the indices j and k carry two subindices each, σ = 1, 2, . . . , V to label the encounter and i = 1, 2, . . . , l(σ ) to label the stretches within an encounter; such proliferation of address labels notwithstanding there are still only four different Bk j ’s and four B˜ jk ’s; whenever possible the subindices on k and j will be sup19 I must apologize for using the tilde on one of these two symbols which in contrast to everywhere else in this book does not mean transposition.

10.9

Semiclassical Construction of a Sigma Model, Unitary Symmetry Class

451

  B12 pressed. Taking Bk j , B˜ jk as elements of 2 × 2 matrices B = BB11 and similarly B 21 22 ˜ for B I can interprete the foregoing symbol sequence could be as one of the sum l(σ ) + . mands in the expansion of the trace product σV=1 tr B B˜ The links of a structure can be indicated on the symbol sequence by drawing “contraction lines” between the respective exit and entrance ports. For instance, in the four-symbol sequence (10.9.1) contraction lines can be drawn as

Bk1 j1 B˜ j1 k2 Bk2 j2 B˜ j2 k1

Bk1 j1 B˜ j1 k1 Bk1 j2 B˜ j2 k1 ,

or

(10.9.3)

in correspondence to the link configurations in Fig. 10.13b, c, respectively. The periodic orbit sets before and after reconnection can be read off from the ˜ B B-sequence with the link lines drawn. Namely, complementing the plot by the encounter stretches connecting each Bk j with the cyclically following (within each encounter) exit port B˜ jk we obtain a graph of pre-reconnection periodic orbits. Similarly drawing the encounter stretches in the opposite direction from each Bk j to the cyclically preceding B˜ j k we obtain the post-reconnection orbits. In our above example of a four-symbol sequence all subscripts might be chosen as 1 (thereby ascribing all pre-reconnection orbits to A and all post-reconnection ones to B) we obtain two structures each depictable by two plots,

B11 B˜ 11 B11 B˜ 11

and

B11 B˜ 11 B11 B˜ 11

(10.9.4)

B11 B˜ 11 B11 B˜ 11

and

B11 B˜ 11 B11 B˜ 11 ,

(10.9.5)

or

where lower lines indicate encounter stretches. The upper pair represents the first structure in the list (10.8.3) and pertains to Fig. 10.13b while the lower pair is one of the eight structures pertaining to Fig. 10.13c. The previous diagrammatic rules for the contribution of a structure in (10.8.14) can be translated to the new language of contracted sequences of matrix elements and start with the link factors. The two ports connected by a link belong to the same pseudo-orbit, both before and after reconnection. Consequently, a contraction of Bk j to B˜ j k may exist only if k = k and j = j . Recalling the link factor in (10.8.14), [−i( A or C +  B or D )]−1 , we can write the factor provided by a contraction between Bk j , B˜ j k as [−i( j + k )]−1 δkk δ j j . Here  j stands for  A ( j = 1) or C ( j = 2); similarly k equals  B for k = 1 and  D for k = 2. The primed energy will always refer to the post-reconnection pseudo-orbits B or D. The indices in an encounter factor i ( A or C +  B or D ) in (10.8.14) are determined by the pseudo-orbits containing the first entrance port of the encounter. In (10.9.2)

452

10 Semiclassical Roles for Classical Orbits

the first entrance port of the σ th encounter is denoted by Bkσ 1 , jσ 1 such that the respective encounter factor can be rewritten as i( jσ 1 + k σ 1 ).

10.9.2 Wick’s Theorem and Link Summation Summation over all structures involves, in particular, summation over all possible links alias contractions. A useful tool for that summation is provided by Wick’s theorem. Consider the Gaussian integral 

∗

dbdb∗ eibb f (b, b∗ )

(10.9.6)

over the complex plane with dbdb∗ ≡ dReb dImb/π , f (b, b∗ ) a product involving an equal number of b’s and b∗ ’s, and Im  > 0 to ensure convergence. That integral can be written as a sum over diagrams where each b in f (b, b∗ ) is connected to one b∗ by a “contraction line”. These diagrams can be evaluated using the rule 

dbdb∗ b b∗ g(b, b∗ ) =

1 −i



d 2 b g(b, b∗ ) ;

(10.9.7)

step by step one can remove contraction lines and the associated pairs b, b∗ and in each step one obtains a factor − i1 . For instance, if g(b, b∗ ) = (bb∗ )n one so obtains $ ∗ n! dbdb∗ eibb (bb∗ )n = (−i) n+1 with the factorial giving the number of possibilities to draw contraction lines connecting each b with some b∗ . The same result holds if the integration variables are replaced by Grassmann variables20 η and η∗ . Imagining exp (iηη∗ ) Taylor-expanded we obtain 

∗

∗ iηη∗

dηdη η η e

 ≡

∗

∗ iηη∗

dηdη η η e

1 = −1 = i



∗

dηdη∗ eiηη . (10.9.8)

This is the analogue of (10.9.7) with g = 1 which is the only case of interest since all powers of the Grassmann variables higher than 1 vanish. I shall now employ Wick’s theorem to represent sums of all possibilities of contracting Bk j ’s and B˜ jk ’s as in (10.9.3), for general products of the form (10.9.2). To that end I declare the Bk j to be either complex Bosonic or Fermionic variables and stipulate the variables Bk j and B˜ jk to be connected by contraction lines to be mutual complex conjugates up to a sign, B˜ jk ≡ ±Bk∗j ; the choice of the Bosonic or Fermionic character of the variables and of the sign is reserved for later. Taking the general symbol sequence, I integrate with a Gaussian weight that has a term proportional to Bk j Bk∗j in the exponent. Wick’s theorem, written with an arbitrary

20 Grassmann algebra and Grassmann integrals will accompany us in the remainder of this chapter. The reader may want to refresh versalitlity with these techniques by a glance at Sect. 4.13.

10.9

Semiclassical Construction of a Sigma Model, Unitary Symmetry Class

453

multiplier i in the exponent, yields a factor − i1 for each contraction line. The desired link factor [−i(e j + ek )]−1 arises if the multiplier of Bk j Bk∗j in the Gaussian exponent is chosen as i(e j + ek ). By simply including the encounter factor i(e jσ 1 + ek σ 1 ) I am led to the integral  ±  ×

˜ e± d[B, B]

 j,k=1,2

i(e j +ek )Bk j Bk∗j

(10.9.9) 5

V ,  σ =1

   ±i e jσ 1 + ek σ 1 Bkσ 1 , jσ 1 B˜ jσ 1 ,kσ 2 Bkσ 2 , jσ 2 . . . B˜ jσ l(σ ) ,kσ 1

where the integration the product of all independent differen+ measure involves ∗ ˜ ∝ 2 d B d B . All (convergent) expressions of this type will tials, d[B, B] k j k, j=1 kj presently be seen to generate the right contributions for each structure.

10.9.3 Signs The most delicate task is to get the sign factor (−1)nC +n D in the sum over structures (10.8.14) by appropriately choosing the Bosonic vs. Fermionic character of the integration variables and the signs in (10.9.9). The choices to be made are inspired the supersymmetric sigma model of random-matrix theory (see Chap. 11) but will here be justified on purely semiclassical grounds. I choose B11 , B22 Bosonic and B12 , B21 Fermionic, define the relation between the “supermatrices” B and B˜ as  B˜ =

∗ ∗ B11 B21 ∗ ∗ B12 −B22

 ,

(10.9.10)

and pick the signs in the exponent of the Gaussian in (10.9.9) as ±



i(e j + ek )Bk j Bk∗j −→

(10.9.11)

j,k=1,2 ∗ ∗ ∗ ∗ i(e1 + e1 )B11 B11 − i(e1 + e2 )B21 B21 + i(e2 + e1 )B12 B12 + i(e2 + e2 )B22 B22 .

To compact the notation I employ the diagonal 2 × 2 matrices eˆ = diag(e1 , e2 ),

eˆ = diag(e1 , e2 )

(10.9.12)

and introduce the “supertrace” of 2 × 2 supermatrices with Bosonic diagonal entries and Fermionic off-diagonal entries as   M11 M12 = M11 − M22 . Str M21 M22

(10.9.13)

454

10 Semiclassical Roles for Classical Orbits

Readers not familiar with superalgebra are kindly requested to spend a minute with checking that the foregoing definition of a supertrace entails the property of cyclic invariance, Str X Y = Str Y X , precisely due to the Fermionic character of the off-diagonal elements M12 , M21 of our supermatrices. Profiting from such notational artifices we may now enjoy the slimmed version of the above quadratic form (10.9.11), 

±

  i(e j + ek )Bk j Bk∗j −→ Str eˆ B B˜ + eˆ B˜ B .

(10.9.14)

j,k=1,2

The choices for the signs and the “superjargon” just established yield the Gaussian integral in (10.9.9) correspondingly compacted to  −

   ˜ exp i Str eˆ B B˜ + eˆ B˜ B . . . ≡ {. . .} ; d[B, B]

(10.9.15)

here the curly brackets {. . .} have the same content as in (10.9.9), namely the sequence of 2V alternating factors B and B˜ involved in the representation of a general structure; for future convenience I denote the integral with the (negative and non-normalized) Gaussian weight by double angular brackets . . .; the integration measure is finally fixed as ∗ ∗ ˜ = dRe B11 dIm B11 dRe B22 dIm B22 d B12 d B12 d B21 d B21 . d[B, B] π π

(10.9.16)

Repeated application of the contraction rule to all integrals of the type (10.9.15) ˜ until finally only the elementary integral successively removes all B’s and B’s  1 = − =

  d[B] exp i Str eˆ B B˜ + eˆ B˜ B

(e1 + e2 )(e2 + e1 ) (e A + e D )(eC + e B )

= (e1 + e1 )(e2 + e2 ) (e A + e B )(eC + e D )

(10.9.17)

(1) is left. Remarkably, the diagonal part Z diag of the generating function, up to the Weyl factor ei(e A +e B −eC −e D )/2 thus arises. A final specification of signs will presently prove beneficial: For each encounter, I insert a factor sk1 which equals 1 if k1 = 1 (after the reconnection the first exit port of the encounter belongs to B), and sk1 = −1 if k1 = 2 (the port belongs to D). Moreover, I install the weight 1/V ! of a structure in (10.8.14) and the Weyl factor. The resulting expression

10.9

Semiclassical Construction of a Sigma Model, Unitary Symmetry Class

455

ei(e A +e B −eC −e D )/2 Z (1) (V, {l(σ ), jσ i , kσ i }) = (10.9.18) V! 99 88 V , 

˜ ˜ skσ 1 i(e jσ 1 + ekσ 1 )Bkσ 1 , jσ 1 B jσ 1 ,kσ 2 Bkσ 2 , jσ 2 . . . B jσ l(σ ) ,kσ 1 × σ =1

(1)  (1) is the cumulative contribution to Z (1) = Z diag 1 + Z off ) of all structures with V encounters, each having a fixed number of stretches l(σ ), σ = 1 . . . V and a fixed distribution of ports { jσ i , kσ i } among the pseudo-orbits; herein each allowed way of drawing contraction lines makes for one structure. The purely diagonal additive term is due to V = 0. Each factor in the product over encounters in (10.9.18) can now be written as a supertrace which automatically takes care of the factor skσ 1 . Still refraining from summing over port labels, I keep the pertinent indices fixed using projection matrices P1 = diag(1, 0), P2 = diag(0, 1), to get

1 (10.9.19) Z (1) (V, {l(σ ), jσ i , kσ i }) = ei(e A +e B −eC −e D )/2 V! 88 V 99 , i(e jσ 1 + ek σ 1 ) Str Pkσ 1 B P jσ 1 B˜ Pkσ 2 . . . P jσ l(σ ) B˜ . × σ =1

All the foregoing choices of signs indeed produce the factor (−1)nC +n D in (10.8.14). To see that I employ a more explicit form of the formal contraction rule (10.9.7). Contraction lines in Eq. (10.9.19) may be drawn either between ports within the same encounters (inside the same supertrace) or from different encounters (supertraces). For these two cases the formal rule can be written in terms of the matrices B and with the Gaussian of (10.9.15) accounted for, as AA @@  δ j j δkk Str(Pk B P j Y )Str (X P j B˜ Pk ) . . . = −

Str(Pk X P j Y ) . . . i(e j + ek )

(10.9.20)

for inter-encounter contractions and @@ AA  δ j j δkk Str(Pk B P j U P j B˜ Pk V ) . . . = −

Str(P j U )Str(Pk V ) . . . i(e j + ek )

(10.9.21)

for intra-encounter contractions. The symbols X, Y, U, V represent matrix products ˜ X = B . . . B, U = B˜ . . . B, V = B . . . B. ˜ The as in (10.9.19) with Y = B˜ . . . B, 1 Kronnecker deltas in the arising link factors −δ j j δkk i(e j +e ) reflect the fact that links k connect ports associated to the same pre-reconnection pseudo-orbit ( j = j ) and the same post-reconnection pseudo-orbit (k = k ). The “averages” on both sides of each contraction rule comprise all structures accessible by drawing further contraction lines; the set of periodic orbits so included by the right-hand-side average can be

456

10 Semiclassical Roles for Classical Orbits

smaller than the set included in the left-hand-side average, since two ports and one link are taken out. I shall prove these contraction rules in the following subsection. The contraction rules can now be used to simplify our expressions for the structure contributions in steps, removing two matrices at a step. This corresponds to removing a link connecting two ports from a periodic-orbit structure, without changing other links. At first sight, the rules (10.9.20) and (10.9.21) do not seem to yield any sign factors. But special attention must be payed to the steps where the contraction line to be removed pertains to a single-link orbit; such a remnant arises when previous steps have removed all other contraction lines from some multiple-link orbit. Then the B and B˜ connected by the last line represent the last two remaining ports of the orbit. Since the orbit is periodic, these ports must be connected not only by the link but ˜ also by an encounter stretch. According to our convention for ordering B’s and B’s the matrices representing these ports must follow each other with just one projection matrix in between. If the orbit in question is an original one, the matrices follow each ˜ The contraction rule (10.9.21) then applies, with U = 1. Then the other like B P j B. first supertrace on the r.h.s. turns into Str P j ; but that supertrace equals 1 for j = 1, i.e., if the orbit belongs to pseudo-orbit A, while for j = 2, i.e., if the orbit belongs to C we have Str P2 = −1. The desired sign factor −1 for each orbit included in C thus arises. An analogous result holds for partner orbits. When the last contraction line associated to a partner orbit is removed the ports connected by that line must also be connected by a partner encounter stretch. The matrix B representing the entrance port and the matrix B˜ representing the exit port must follow each other like B˜ Pk B. Rule (10.9.21) then applies with V = 1 and the second supertrace on the r.h.s. thus equals Str Pk , i. e. −1 if k = 2 and the orbit belongs to D. To conclude, by turning B and B˜ into supermatrices with Grassmannian offdiagonal entries, we have successfully incorporated the sign factor (−1)nC +n D .

10.9.4 Proof of Contraction Rules, Unitary Case An interlude is in order to derive the contraction rules (10.9.20) and (10.9.21) from the raw versions (10.9.7) and (10.9.8). The exponent (10.9.15) permits nonzero contractions only for mutually complex conjugate matrix elements. For Bk j and B˜ j k

(the latter agreeing up to the sign with Bk∗ j ) we have >>

?? ˜ Bk j B˜ j k g(B, B) =−

 sj ˜



δ j j δkk g(B, B) . i( j + k )

(10.9.22)

Here s j is a sign factor equal to 1 if j = 1 and to −1 if j = 2. To understand its origin we must appreciate the rule (10.9.7) for the Bosonic variables ( j = k) yielding − i1 and (10.9.8) for the Fermionic variables ( j = k) yielding i1 ; a addi-

10.9

Semiclassical Construction of a Sigma Model, Unitary Symmetry Class

457

∗ tional factors −1 emerges when j = k = 2 due to the negative sign in B˜ 22 = −B22 ˜ and when j = 1, k = 2 because of minus at the term proportional to B21 B12 in the exponent of (10.9.15). Using Eq. (10.9.22) we obtain the rule (10.9.20) for contractions between B and B˜ in different supertraces (inter-encounter contractions) as

?? ?? >> >> ˜ . . . = Str (P j Y Pk B)Str (P j B˜ Pk X ) . . . Str (Pk B P j Y )Str (Pk X P j B) ??

>> = s j Y jk Bk j s B˜ j k X k j . . . j

 s j δ j j δkk s j Y jk s j X k j . . .

i( j + k )  δ j j δkk =− s j Y jk X k j . . .

i( j + k )  δ j j δkk =− Str (P j Y Pk X ) . . . .

i( j + k ) =−

Here I used the cyclic invariance of the supertrace, wrote out the supertrace in components, with the help of Str (Pk Z ) = sk Z kk for any supermatrix Z , and finally used Eq. (10.9.22). Equation (10.9.21) for intra-encounter contractions follows similarly from ?? >> ?? >> ˜ ˜ Str (Pk B P j U P j B Pk V ) . . . = sk Bk j U j j B j k Vk k . . . > ?? ˜ = sk Bk j B j k U j j Vk k . . .  s j δ j j δkk sk U j j Vkk . . . i( j + k )  δ j j δkk =− Str (P j U )Str (Pk V ) . . . .

i( j + k ) =−

In the second line I could interchange U j j and B˜ j k since a nonzero result arises only for j = j in which case U j j is Bosonic and commutes with all other variables.

10.9.5 Emergence of a Sigma Model To determine Z (1) I sum over all structures of pseudo-orbit quadruplets. In Eq. (10.9.19) all ways of linking ports with fixed labels are already collected. Now all possibilities of assigning ports to pseudo-orbits must be picked up, i. e., the indices jσ i , kσ i in (10.9.19) be summed over. That sum just amounts to dropping the

458

10 Semiclassical Roles for Classical Orbits

projection matrices. To keep the factors i(eσ 1 + eσ 1 ) connected to the indices in the supertrace I once more use the diagonal offset matrices eˆ , eˆ and write Z (1) (V, {l(σ )}) =

ei(e A +e B −eC −e D )/2 V!

>> , V

??   ˜ l(σ ) + eˆ ( B˜ B)l(σ ) . i Str eˆ (B B)

σ =1

It remains to sum over the number V of encounters and over their sizes l(σ ), σ = 1, 2, . . . , V . Including the trivial term V = 0 which takes care of the purely diagonal contribution, I have

Z

(1)

i (e A +e B −eC −e D )/2

=e

∞  1 V! V =0 88

88-

= ei (e A +e B −eC −e D )/2 exp i Str

.V 99

∞  

i Str

˜ l eˆ ( B˜ B)l + eˆ (B B)



l=2

99  ˜ l . eˆ ( B˜ B)l + eˆ (B B)

∞   l=2

Yet one more time I take advantage of the precaution of a large imaginary part of all energy offsets. Due to that restriction the B-integral draws dominant contributions from near the saddle of the integrand at B = 0. In fact, without changing the previous moments of the Bosonic Gaussians by more than corrections of the order e−η it is possible to restrict the integration ranges for the Bosonic variables as |B11 |, |B22 | < 1 and then sum the geometric series in the foregoing exponent, to get    B˜ B i B B˜

˜ d[B, B] exp + eˆ = −e Str eˆ 2 1 − B˜ B 1 − B B˜     ˜ ˜ ˜ exp i Str eˆ 1 + B B + eˆ 1 + B B . (10.9.23) = − d[B, B] 2 1 − B˜ B 1 − B B˜ 

Z

(1)

i (e A +e B −eC −e D )/2

The matrix integral arrived at compactly sums up the asymptotic 1e series. Due to the restriction Im e = η  1 of the present semiclassical derivation it yields only the large-η asypmtotics Z (1) . The integral can be done and the large-η asymtotics exhibited. To that end, I propose to establish a transformation of the integration variables bringing the integral to a Gaussian form. The matrix products B B˜ and B˜ B are respectively diagonalized by U and V , ˜ = V −1 B˜ BV = diag (l B , l F ), U −1 B BU with eigenvalues 0 ≤ l B < 1, l F ≤ 0. The diagonalizing matrices read21

21

More about the transformation achieved by U, V will be presented in Sect. 11.6.7.

(10.9.24)

10.9

Semiclassical Construction of a Sigma Model, Unitary Symmetry Class

459



  −iφ B  η 1 + ηη∗ /2 e 0 U = , η∗ 1 − ηη∗ /2 0 e−iφ F   τ 1 − τ τ ∗ /2 = V † = σz V −1 σz . V = −τ ∗ 1 + τ τ ∗ /2

(10.9.25)

The eight independent parameters in the matrices B, B˜ are thus expressed in terms of eight new variables four of which (l B , l F , φ B , φ F ), are Bosonic and four (η, η∗ , τ, τ ∗ ) Fermionic. It will be useful to know that the two eigenvalues l B , l F have different signs, l B = |B11 |2 + . . . ,

l F = −|B22 |2 + . . . ,

(10.9.26)

where the dots refer to “non-numerical” additions bilinear in the original Grass∗ ∗ , B21 , B21 . The rational functions of B B˜ and B˜ B appearing in the mannians B12 , B12 exponent of the matrix integral (10.9.23) are similarly diagonalized,   lB B B˜ lF B˜ B U = V −1 V = diag , ≡ diag(m B , m F ), 1 − lB 1 − lF 1 − B B˜ 1 − B˜ B (10.9.27) and the eigenvalues again have different signs (of their “numerical parts”), m B > 0, m F < 0, provided the restriction |B11 | < 1 met above is respected. Next, we can rejoice in realizing that the matrices U and V allow for a certain ˜ analogue of singular-value decompositions of B and B, U −1

& & & & B = U diag ( l B , −l F )V −1 , B˜ = U diag ( l B , − −l F )V −1 . (10.9.28) The matrices √ √ √ √ C = U diag ( m B , −m F )V −1 , C˜ = U diag ( m B , − −m F )V −1 (10.9.29) thus have the products

C C˜ =

B B˜ , 1 − B B˜

˜ = CC

B˜ B . 1 − B˜ B

(10.9.30)

Indeed then, the integrand of the matrix integral (10.9.23) becomes a Gaussian ˜ The transformation B, B˜ → C, C˜ has the Jacobian unity such in terms of C, C. ˜ = d[C, C] ˜ with d[C, C] ˜ structured like that the integration measures obey d[B, B] ˜ as given in (10.9.16). The integral in question thus takes the form d[B, B]

460

10 Semiclassical Roles for Classical Orbits

     ˜ exp i Str eˆ CC ˜ + eˆ C C˜ (10.9.31) Z (1) = −ei (e A +e B −eC −e D )/2 d[B, B] 2 = − e 2 ( A + B −C − D )/2  2 d B11 d 2 B22 i{B11 B ∗ ( A + B )+B22 B ∗ (C + D )} 11 22 × e π π  ∗ ∗ ∗ ∗ × d B12 d B12 d B21 d B21 ei{B12 B12 ( B +C )+B21 B21 ( A + D )} . i

(10.9.32)

Admitting an error of the order e−η I can extend the Bosonic integrations over the whole complex planes. The elementary integrals then yield Z (1) ( A ,  B , C ,  D ) = ei ( A + B −C − D )/2

( A +  D )( B + C ) , ( A +  B )(C +  D )

η  1. (10.9.33)

1 can arise as corrections. e Nothing prevents us from continuing the result to real offsets. But then the foregoing Z (1) cannot exhaust the high-energy asymptotics of the generating function Z since the exponentially suppressed Riemann–Siegel type complement remains suppressed. However, the Riemann–Siegel lookalike does yield the full high-energy asymptotics through (10.6.19) as The error being exponentially small, no powers in

Z ( A ,  B , C ,  D ) ∼ e 2 (e A +e B −eC −e D ) i

−e 2 (e A +e B +eC +e D ) i

(eC + e B )(e D + e A ) (e A + e B )(e D + eC ) (e B − e D )(e A − eC ) , (e A + e B )(eC + e D )

(10.9.34) |e|  1,

like in the diagonal approximation (10.7.7). The high-energy asypmtotics of the correlator C(e) thus comes out as in the diagonal approximation (10.7.8), C(e) ∼ (1 − e2ie )/2(ie)2 . Given the absence of any 1 -corrections and assuming the exact correlator analytic in the upper half e-plane, e I can be sure [56] that the high-energy asymtotics fixes the exact correlator and conclude C(e) = (1 − e2ie )/2(ie)2 . The RMT result for the unitary symmetry class is thus recovered semiclassically. ˜ The B, B-integral in (10.9.23) is well known from random-matrix theory where it arises as the so-called rational parametrization of the zero dimensional sigma model, there for real eC , e D and with fully specified integration ranges for the Bosonic variables B11 , B22 , and thus giving the exact generating function in full. Anticipating what will be expounded in Sect. 11.6.7, I just mention here that the contribution Z (1) is associated with the stationary point B = B˜ = 0 (“standard saddle”) while the Riemann–Siegel complement comes from a second stationary point, the “Andreev– Altshuler saddle” B = B˜ = diag(0, ∞).

10.10

Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class

461

10.10 Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class When time reversal invariance reigns close self-encounters become possible where some encounter stretches are almost mutually time reversed rather than being all close in phase space. In configuration space, all stretches of such an encounter still run close to one another but with opposite senses of traversal. If connections inside such an encounter are changed, the links outside still look almost the same in configuration space, but some of them acquire opposite directions. A prime example is the Sieber–Richter pair, see the uppermost left in Fig. 10.11. The new pseudo-orbit quadruplets with oppositely oriented encounter stretches and links now make for non-cancelling contributions to the generating function Z in all orders in 1e . The summation of the relevant quadruplets can again be done with the help of a sigma-model type matrix integral, much in parallel to the above treatment of the unitary symmetry class. The faithfulness of individual dynamical systems to the RMT prediction for the two-point correlator of the level density will thus again find its semiclassical explanantion.

10.10.1 Structures Since mutually time-reversed stretches enter and leave the encounter from different sides, it becomes cumbersome to work with the notions of entrance and exit ports. Instead we arbitrarily refer to the ports on one side of each encounter as “left” ports and those on the opposite side as “right” ports. Encounter stretches may lead either from left to right or from right to left. The links may connect left and left, right and right, or left and right ports. The definition of structures must therefore be modified: (S1’) When numbering encounters one must make sure that in the partner pseudo-orbit the left port of the i-th stretch is always connected to the right port of the (i − 1)-st stretch. (S2’) When connecting ports by links arbitrary connections are permissible such that (2L)! possibilities arise. (S3’) Beyond partitioning n original orbits among the pseudo-orbits ( A, C) and n partner orbits among B, D, now every orbit can be assigned either of

two senses of traversal; there are thus 4n+n structures for each of the choices made in (S1’,S2’). Again, different structures can describe one and the same quadruplet. In particular, the choice of the right and left side of each encounter is arbitrary, and for an overall number V of encounters there are 2V equivalent possibilities of labelling the sides. To avoid overcounting, that factor must be divided out. The starting point for the construction of the sigma model therefore becomes modified from (10.8.14) to

(1) = Z off

+  (−1)nC +n D enc i(e A or C + e B or D ) + , V 2 V! links (−i(e A or C + e B or D )) structures

(10.10.1)

462

10 Semiclassical Roles for Classical Orbits

(1) (1) with Z (1) = Z diag (1 + Z off ) still intact but the diagonal part given by (10.7.9) as

(1) Z diag

=e

i 2 (e A +e B −eC −e D )



(eC + e B )(e D + e A ) (e A + e B )(e D + eC )

2 .

(10.10.2)

10.10.2 Leading-Order Contributions To illustrate the use of the foregoing master formula (10.10.1) I here consider the contributions responsible for the leading non-oscillatory and oscillatory terms in the correlator C(e). To get the first non-oscillatory correction δC(e) ∝ e13 I check the two diagrams with L − V = 1 in the gallery of Fig. 10.11. The one with a single parallel 2-encounter does not contribute, as a reasoning completely parallel to the one in Sect. 10.8.3 reveals. So only the Sieber–Richter diagram with a single antiparallel 2-encounter remains, the uppermost left in Fig. 10.11. That diagram has one encounter (V = 1) and one orbit both before and after reconnection. I first allot these orbits to A and B such that C and D remain empty. There are four equivalent associated structures, in accordance with two possible senses of traversal of the (1) i(e A +e B ) 2 reads 4 2111! [−i(e initial and final orbit, and their joint yield for Z off 2 = i(e +e ) . A B A +e B )] For the three other allotments of orbits to pseudo-orbits different subscripts on the offset variables and the sign factor (−1)nC +n D arise. Altogether the Sieber–Richter (1) results as contribution to Z off   2 1 1 1 1 + − − i e A + eB eC + e D e A + eD eC + e B 2 (e A − eC )(e B − e D )(e A + e B + eC + e D ) = . (10.10.3) i (e A + e B )(e B + eC )(e A + e D )(eC + e D )

(1) Z off,SR ==

(1) (1) The addition to Z (1) is Z diag Z off,SR and the one to C(e) is given by (10.6.12),

 CSR (e) = −2

∂2 Z (1) Z (1) ∂e A ∂e B diag off,SR

 e A,B,C,D =e

.

(1) To save labor it is well to realize that the factor (e A − eC ) (e B − e D ) in Z off,SR vanishes as the four offset variables are all equated. Both derivatives must therefore act on that factor while everywhere else the arguments can be set equal to e. In this way the result

CSR (e) = −2

  2 4e i = 3 4 4 i 2e e

(10.10.4)

10.10

Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class

463

is readily reached; it indeed agrees with the third-order term of the RMT correlator given in (10.6.5); its Fourier transform −2τ 2 was first found by Sieber and Richter [55] as the leading correction to the diagonal approximation, 2τ , for the spectral form factor K (τ ) = 2τ − τ ln(1 + 2τ ), and that discovery opened the door to the complete semiclassical construction of the correlator described here. For the leading oscillatory term, on the other hand, I consider (10.10.5) Z (1) (e A , e B , −e D , −eC ) 2  (e A − eC ) (e B − e D ) i (1) Z off = e 2 (e A +e B +eC +e D ) (e A , e B , −e D , −eC ) , (e A + e B ) (eC + e D ) immediately resorting to real energy offsets. The factor [(e A − eC )(e B − e D )]2 vanishes for e A,B,C,D = e even after application of ∂ 2 /∂e A ∂e B . The oscillatory contri(1) (e A , e B , −e D , −eC ) bution in search must therefore come from the summands in Z off (1) −1 proportional to [(e A − eC )(e B − e D )] . The respective terms in Z off (e A , e B , eC , e D ) −1 behave like [(e A + e D )(eC + e B )] . The contributing diagrams must (i) be specific for the orthogonal case and (ii) contain at least two pre- and two post-reconnection orbits (the denominator indicates that one link was in A before and D after reconnection; the second link was in C before and in B after reconnection). A glance at the gallery of Fig. 10.11 reveals that the only relevant diagram is the “double Sieber–Richter pair” (to be shorthanded as dSR below) in the third row of the column 2:2. The set of structures is further restricted by the demand that of the two pre-reconnection orbits one must be in A and the other in C; after reconnection one orbit must belong to B and the other to D. I arbitrarily name the pre-reconnection orbits γ1 , γ2 and their post-reconnection partners γ¯1 , γ¯2 . Since each of these orbits contains one encounter and two links and each can have two senses of traversal, there are 22 × 22 = 16 structures for each allotment to pseudo-orbits. Four allotments are possible, 1. 2. 3. 4.

{γ1 {γ1 {γ1 {γ1

∈ A, γ2 ∈ C, γ¯1 ∈ C, γ2 ∈ A, γ¯1 ∈ A, γ2 ∈ C, γ¯1 ∈ C, γ2 ∈ A, γ¯1

∈ ∈ ∈ ∈

D, γ¯2 ∈ B}, B, γ¯2 ∈ D}, B, γ¯2 ∈ D}, D, γ¯2 ∈ B}.

Only the first two of these yield the desired net denominator, and they contribute identically to (1) Z off,dSR (e A , e B , eC , e D ) = 2 × 16 ×

=−

[i (e A + e D )] [i (eC + e B )] 22 2! [−i (e A + e D )]2 [−i (eC + e B )]2

4 . (e A + e D ) (eC + e B )

The exchange eC ↔ −e D then gives

(10.10.6)

464

10 Semiclassical Roles for Classical Orbits (2) Z off,dSR (e A , e B , eC , e D ) = −

4 + ... (e A − eC ) (e B − e D )

where the dots stand for terms present in the generating function but irrelevant for the oscillatory part of the correlator. Upon differentiating I finally arrive at the leading oscillatory contribution 

 ∂2 (2) (2) Z diag Z off,SR ∂e A ∂e B e A,B,C,D =e    ∂2 4 = −2 − ∂e A ∂e B (e A − eC ) (e B − e D )

CdSR (e) = −2

e 1 = 4 ei2e , 2e

i 2 (e A +e B +eC +e D )

(e A − eC )2 (e B − e D )2 (e A + e B )2 (eC + e D )2

. e A,B,C,D =e

(10.10.7)

in agreement with (10.6.5); it was first found in a daring foray by Bogomolny and Keating in 1996 [57] and then confirmed within the systematic approach described here in Ref. [51].

10.10.3 Symbols for Ports and Contraction Lines for Links Turning from the leading terms towards doing the sum over all structures I again propose to consider the sequence of ports starting with the first left port of the first encounter, continuing with the first right port, etc. The left ports will be denoted by Bνk,μj and the right ports by B˜ μj,νk . The first and the second pair of indices of B respectively refer to the properties of the port before and after reconnection, and ˜ In particular, similar to the unitary case the Latin index j = 1, 2 vice versa for B. indicates whether the port belongs to the original pseudo-orbit A ( j = 1), or C ( j = 2). The index k reveals whether after reconnection the port belongs to the partner pseudo-orbit B (k = 1) or D (k = 2). As a new element relative to the unitary class, Greek indices μ, ν account for the directions of motion through the port and the attached encounter stretches, both in the pertinent original and partner orbits: If μ = 1 the direction is from left to right in the original orbit; if μ = 2 it is from right to left. The index ν = 1, 2 indicates the same for the partner orbits. The symbols Bνk,μj , B˜ μj,νk can be considered as ˜ whenever convenient a capital Latin letter will be elements of 4 × 4 matrices B, B; employed for the composite index like J = (μ, j). The order of alternating symbols is such that a B and the immediately following B˜ represent ports connected by an encounter stretch of an original orbit. The two ports are therefore traversed in the same direction and hence their indices J = (μ, j) coincide. Similarly each B˜ and the immediately following B represent ports

10.10

Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class

465

connected by a stretch of a partner orbit and the corresponding subscripts K = (ν, k) coincide. (When a B˜ represents the last right port in its encounter, its subscripts ν, k have to coincide with those of the first B.) At any rate, the indices J and K are arranged like in a product of matrices, in fact the product already met in Eq. (10.9.2) but with j, k replaced by J, K . Next, the links can be built in and depicted by “contraction lines” above the symbol sequences; such lines can now connect not only B to B˜ but also B to B and ˜ Any two ports connected by a link must belong to the same pseudo-orbit, B˜ to B. before and after the reconnection and hence their Latin indices must coincide, as in the unitary case. The Greek direction indices require new reasoning. A contraction between a B˜ and a B stands for a link connecting a right port to a left one. In this case, if an orbit leaves one encounter at the right port it must enter the other encounter stretch at the left port. The direction of motion (left to right) is thus the same for both encounter stretches. The same applies for a direction of motion from right to left, and for original and partner orbits alike. Hence for contractions between B˜ and B the Greek subscripts indicating the directions of motion must coincide, μ1 = μ2 , ν1 = ν2 . On the other hand, in the contractions B . . . B and B˜ . . . B˜ the connected ports lie on the same side, and hence their directions of motion must be opposite. We shall use a bar over ν, μ to indicate that the port direction of the motion is flipped like 1  2 (such that μ ¯ = 3 − μ). Again denoting the energy offsets as e1 = e A , e2 = eC , e1 = e B , e2 = e D we have the link contribution  δkk δ j j

δμμ ¯ δν¯ ν

× −i(e j − ek ) δμμ δνν

for

⎧ ⎨B ⎩

νk,μj Bν k ,μ j

and B˜ μj,νk B˜ μ j ,ν k

(10.10.8)

Bνk,μj B˜ μ j ,ν k .

The contribution of an encounter is defined by its first left port and can be written as i(e jσ 1 + ek σ 1 ). And again, there will be a factor (−1)nC +n D depending on the numbers of orbits included in C and D.

10.10.4 Gauss and Wick Trotting along the path laid out in the previous section I seek to represent additive terms in (10.10.1) by contractions in a Gaussian integral. I again stipulate any two symbols connectable by a link to be represented by a pair of mutually complex conjugate variables, up to a sign. Since rule (10.10.8) allows for contractions between two B’s differing in both their subscripts μ and ν I must set Bνk,μj = ±Bν∗¯ k,μ¯ j .

(10.10.9)

The identities so arising for the various choices of the indices can be compactly written as a matrix identity if the Bνk,μj are assembled into a 4 × 4 matrix B. Organized with the help of four 2 × 2 blocks that matrix reads

466

10 Semiclassical Roles for Classical Orbits ?

B=



bp ba

ba∗ b∗p

 ,

(10.10.10)

with the question mark on the equality a reminder of eight as yet unspecified signs. The index p on b p , b∗p signals μ = ν, i.e., the ports associated with these subblocks are traversed in parallel directions by the original and the partner orbits. In contrast, the ports associated with ba , ba∗ have μ = ν and their traversal directions by original and partner orbits are antiparallel. On the other hand, the rule (10.10.8) allows contractions between matrix elements of B and B˜ with identical subscripts. Hence these matrix elements, too, must be chosen as mutually complex conjugate (up to a sign). Since the subscripts in B and B˜ are ordered differently, B˜ must coincide with the adjoint of B (again up to signs), i.e., ? B˜ = B † .

(10.10.11)

A Gaussian-weight integral of a product of mutually conjugate matrix elements will yield a sum over all permissible ways of drawing contraction lines. To obtain ∗ the correct contribution from each link the prefactor of Bμk,ν j Bμk,ν j in the exponent

of the Gaussian again has to be chosen as i(e j + ek ). Furthermore, due to the internal ∗ structure of B the sum over all Bμk,ν j Bμk,ν j in the exponent includes each pair of complex conjugate elements twice, and hence that sum must be divided by 2. With the help of e J ≡ e j and e K ≡ ek (independent  of the Greek part of J, K ) the exponent in the Gaussian can be written as ± 2i J K (e J + e K )B K J B K∗ J . Incorporating also the factor i(e Jσ 1 − e K σ 1 ) for each encounter I have   ˜ ± 2i J,K (e J +e K )BK J BK∗ J (10.10.12) ± d[B, B]e ×

V ,  σ =1

 ±i(e Jσ 1 + e K σ 1 )B K σ 1 ,Jσ 1 B˜ Jσ 1 ,K σ 2 B K σ 2 ,Jσ 2 . . . B˜ Jσ l(σ ) ,K σ 1 .

Each integral of this type includes the correct encounter and link factors for all structures with V encounters involving l(1), l(2) . . . l(V ) encounter stretches.

10.10.5 Signs Towards re-enacting the sign factor (−1)nC +n D and the correct prefactor of the diagonal approximation, I finally specify the matrices B and B˜ in full: The offdiagonal entries in all four blocks of B are chosen Fermionic and the diagonal entries Bosonic, and the signs in (10.10.11) are picked as   b p −ba∗ (10.10.13) , B˜ = σz ⊗ B † ; B= ba −σz b∗p

10.10

Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class

467

the tensor product σz ⊗ B † means that each 2 × 2 block of B † is multiplied with σz ; taking the adjoint of our 4 × 4 supermatrices involves, apart from complex conjugation and transposition in the sense of ordinary matrices, a sign flip in the lower left (Fermi–Bose) entries of all 2 × 2 blocks. Various signs in the Gaussian weight and in the matrix product must still be fixed, and I do that such that both quantities can be written in terms of supertraces. The supertrace is now defined as the sum of the two upper left (Bose-Bose) entries minus the sum of the lower right (Fermi–Fermi) entries of the two diagonal blocks; in other words, the supertrace includes a negative sign for all diagonal elements associated to a Latin index 2 (regardless of the Greek index). The integral (10.10.12) then acquires the form    ˜ exp i Str eˆ B B˜ + Str eˆ B˜ B (. . .) ≡ . . ., (10.10.14) d[B, B] 2 with eˆ , eˆ 4×4 diagonal matrices whose diagonal elements are eˆ J = e j , eˆ K = ek . After eliminating all contraction lines from any integral of this type one is left with the Gaussian integral  1 = =

2

2   ˜ exp i Str eˆ B B˜ + Str eˆ B˜ B = (e1 + e2 ) (e2 + e1 ) d[B, B]

2 2 (e1 + e1 ) (e2 + e1 )2

(e A + e D )2 (eC + e B )2 , (e A + e B )2 (eC + e D )2

(10.10.15)

as in the diagonal factor for time-reversal invariant systems. Upon adding to (10.10.14) the Weyl factor and the factor 1/(2V V !) from (10.10.1) I obtain the contribution to the generating function 1 (10.10.16) Z (1) (V, {l(σ ), Jσ i , K σ i }) = ei(e A +e B −eC −e D )/2 V !2V 99 88 V , i(e Jσ 1 + e K )Str PK σ 1 B PJσ 1 B˜ PK σ 2 . . . PJσ l(σ ) B˜ . × σ =1

σ1

Here PJ is a 4 × 4 projection matrix within which the J -th diagonal element equals 1 while all other elements vanish, and PK is defined analogously. To show that the sign factor (−1)nC +n D is also obtained correctly, the contraction rules (10.9.20) and (10.9.21) must be invoked once more. In fact, two more ˜ which such rules are needed here, for the contractions between two B’s or two B’s represent links connecting two left ports or two right ports, ?? >>



Str (PK B PJ Y )Str (PK B PJ Z ) . . . =

 δ K¯ K δ J¯ J Str (PJ Y PK Z˜ ) . . . −i(e J + e K )

468

10 Semiclassical Roles for Classical Orbits

?? >> Str (PK B PJ Y PK B PJ Z ) . . . =

 δ K¯ K δ J¯ J Str (PK Y˜ PJ¯ Z ) . . . (10.10.17)

−i(e J + e K )

˜ Here Y and Z are products of the type required in (10.10.12) and analogously for B. ˜ The matrix Z˜ differs from Z but has the order of matristarting and ending with B. ces interchanged, and B replaced by B˜ and vice versa. The familiar link factor shows up, notwithstanding the more compact notation J = (μ, j),

J¯ ≡ (μ, ¯ j),

K = (ν, k),

K¯ ≡ (¯ν , k).

(10.10.18)

The Kronecker deltas imply that the connected ports must belong to the same pseudo-orbits and have opposite direction of motion. The four contraction rules can now be employed to stepwise remove the contraction lines corresponding to all links of a structure. Sign factors arise only in the “final” steps where single-link orbits are removed. A final link always connects a left to a right port, or else it could not form a periodic orbit together with one encounter stretch. Hence the rules (10.9.20) and (10.9.21) apply, the ones for removing links ˜ The same argubetween left ports (denoted by B) and right ports (denoted by B). ment as in Sect. 10.9.3 shows that the final step yields a factor Str (PJ ) for each original orbit and a factor Str (PK ) for each partner orbit. The latter supertraces equal −1 if the corresponding lower-case indices are j = 2 (corresponding to the pseudo-orbit C) or k = 2 (corresponding to the pseudo-orbit D), and these factors combine to the desired term (−1)nC +n D .

10.10.6 Proof of Contraction Rules, Orthogonal Case The derivation of the rules (10.9.20) and (10.9.21) already established for the unitary symmetry class carries over directly. The only difference is that J, K are now double indices consisting of the Greek and Latin parts, J = (μ, j), K = (ν, k); the sign factor s J has the values 1 for j = 1 and −1 for j = 2 regardless of the Greek index, and s K is defined analogously. In addition, both B and B˜ now contain pairs of complex conjugate matrix ele˜ become possible. ments, and therefore contractions between two B’s or two B’s The further contraction rules (10.10.17) for contractions between two B’s thus arose ˜ To derive these above, as well as analogous ones for contractions between two B’s. rules I employ the symmetry of B implied by the definition (10.10.13),   0 σz ˜ Σ B˜ t Σ t = −B, Σ = , (10.10.19) Σ B t Σ t = − B, 1 0 where B t is the transpose22 of B. Similarly conjugating the projection matrices PK , PJ we get

22 Transposition of a supermatrix involves interchanging the indices of the matrix elements and afterwards flipping the sign of the lower left elements in each 2 × 2 block.

10.10

Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class

Σ PK Σ t = PK¯ ,

Σ PJ Σ t = PJ¯ ;

469

(10.10.20)

Now I consider the matrix products Y and Z in (10.10.17) which involve alternat˜ and interspersed projection matrices. Under transpoing sequences of B’s and B’s sition and conjugation with Σ these products have (i) the order of matrices inverted (due to the transposition), (ii) B replaced by B˜ and vice versa (due to (10.10.19)), and (iii) the subscripts J and K of all projection matrices replaced by J¯ and K¯ (due ˜ are equal the sign factors from to (10.10.20)). Since the numbers of B’s and B’s (10.10.19) all mutually compensate. Thus equipped I attack the rule for inter-encounter contractions between two B’s in (10.10.17). By using the invariance of the supertrace both under transposition of its argument and under conjugation with Σ we have ?? >> Str (PK B PJ Y )Str (PK B PJ Z ) . . . >> ?? = Str (PK B PJ Y )Str (Σ Z t PJ B t PK Σ t ) . . . >> ?? = Str (PK B PJ Y )Str ( Z˜ PJ¯ B˜ PK¯ ) . . . =−

 δ J¯ ,J δ K¯ ,K Str (PJ Y PK Z˜ ) . . . .

i( J +  K )

(10.10.21)

Here in the third line Z˜ denotes the product obtained from Z by steps (i)-(iii) above. The first of the rules in (10.10.17) is thus established. I proceed to the intra-encounter contractions Str (PK B PJ Y PK B PJ Z ) . . .. The contracted matrix elements are B K J and B K J , the latter coinciding up to a sign with B K∗¯ , J¯ . Hence the only relevant case is J = J¯ , K = K¯ and I get >>

??

>> ?? Str (PK B PJ Y PK¯ B PJ¯ Z ) . . . = s K B K J (Y PK¯ B) J J¯ Z J¯ K . . . (10.10.22)

with  t (Y PK¯ B) J J¯ = Σ t (Σ B t Σ t )(Σ PK¯ Σ t )(ΣY t Σ t )Σ J J¯   = Σ t B˜ PK Y˜ Σ J¯ J = Σ Jt¯ ,J B˜ J,K Y˜ K , J¯ Σ J¯ ,J = s J B˜ J,K Y˜ K , J¯ .

(10.10.23)

Here I first replaced the matrix sequence by itself doubly transposed; note that double transposition of a supermatrix (not the identity operation!) leaves its Bosonic

470

10 Semiclassical Roles for Classical Orbits

elements like the one with the subscripts J J¯ unchanged. Then I invoked Eqs. (10.10.19) and (10.10.20) and noted that for the matrix elements at hand transposition just amounts to interchanging the two subscripts. In the third line I exploited the fact that only those elements of Σ, Σ t are nonzero for which one subscript equals another one barred, i.e., the port directions of motion are opposite. Finally, I employed the identity Σ Jt¯ ,J Σ J¯ ,J = s J . Altogether I get >> ?? Str (PK B PJ Y PK B PJ Z ) . . . ?? >> = δ J¯ ,J δ K¯ ,K s K B K J s J B˜ J,K Y˜ K , J¯ Z J¯ ,K . . .

 δ J¯ ,J δ K¯ ,K

˜ s K Y K , J¯ Z J¯ ,K . . .

i( J +  K )  δ J¯ ,J δ K¯ ,K

Str (PK Y˜ PJ¯ Z ) . . . , =−

i( J +  K )

=−

(10.10.24)

the intra-encounter contraction rule in (10.10.17); the intermediate steps rely on (10.9.22) and the quantity Y˜ defined in analogy to Z˜ . ˜ as the Rules analogous to (10.10.17) also hold for contractions between two B’s, interested reader will easily check.

10.10.7 Sigma Model Summing over structures as in Sect. 10.9.5 and again using the assumed largeness of the imaginary part of all energy offsets we get the generating function cZ

(1)

i (e A +e B −eC −e D )/2

=e



   i B B˜ B˜ B

˜ d[B, B] exp + eˆ , Str eˆ 2 1 − B˜ B 1 − B B˜ (10.10.25)

equal in appearance as (10.9.23) for the unitary symmetry class, up to a factor 12 in the exponent. The matrix integral again sums up the contributions from all structures which in its raw form is an asymptotic expansion in inverse powers of energy offsets. The integration domain must include the stationary point B = B˜ = 0 and let the eigenvalues of 1 = B B˜ and 1 = B˜ B be positive; it needs no further specification due to the underlying assumption η  1. To see that the prediction of the Gaussian orthogonal ensemble (GOE) of randommatrix theory is recovered I can argue like in the unitary case. The sigma model of the GOE yields the same matrix integral, two important differences apart. First, ˜ second, the integration domain is fully specified for all Bosonic elements of B, B;

10.11

Outlook

471

the offsets are unrestricted and can even be real. The large-e asymptotics of the integral is again dominated by two saddles of the exponent both of which need to be accounted for unless the offsets have large imaginary parts. The standard saddle at B = B˜ = 0 yields a contribution dominated by small B and identical to the above semiclassical Z (1) . The contribution of the Andreev–Altshuler saddle yields the Riemann–Siegel complement. Our semiclassical results thus recover the highenergy asymptotics of the generating function in agreement with RMT. The same must then be true for the correlator C(e). Moreover, an analytic function can, under rather general conditions which I dare to assume fulfilled, be restored from its asymtotic series by Borel summation [56]. That method involves the term-by-term Fourier transform of the asymptotic expansion leading to a converging series, followed by the inverse Fourier transform of the resulting analytic function. The first stage of the Borel summation applied to the two components of the correlation function gives the spectral form factor K (τ ) both for τ < 1 (from Z (1) ) and τ > 1 (from the Riemann–Siegel complement). The inverse Fourier transform recovers the closed RMT expression for the correlation function in Eq. (10.6.2), for all energies .

10.11 Outlook The interplay of orbit-based semiclassics and the nonlinear sigma model can be expected to become, and in fact has already become helpful in tackling further challenges, beyond the two-point correlator of the level density for the unitary and the orthogonal symmetry classes considered in this chapter. The symplectic symmetry class awaits treatment of the oscillatory part of the two-point function [50, 58, 59]. Transitions between the symmetry classes are largely understood [60–64]. Higherorder correlations of the level density have thus far been treated only for the unitary class [65]. A somewhat more ambitious goal will be the density of level spacings. Arithmetical billiards and their pseudo-arithmetical variants [66] have recently found a semiclassical explanation of their different spectral statistics [67]. Both types of dynamics exhibit full classical chaos and share the peculiarity of exponential degeneracy of the actions of periodic orbits. Quantum mechanically, the arithmetical case has Poissonian level statistics due to intrinsically quantum Hecke symmetries; in the pseudo-arithmetical case no Hecke symmetries reign, and the level statistics is Wigner-Dyson. The surprisingly simple semiclassical explanation of the difference builds on the Maslov phase factors appearing in the Gutzwiller trace formula. The phase factors vary randomly within each action multiplet of pseudo-arithmetical dynamics and thus make the exponential action degeneracy ineffective; the semiclassical reasoning presented in this chapter then goes through. On the other hand, the Maslov phase factors do not vary within (almost all) action multiplets of arithmetical systems such that all orbits in an action multiplet make identical contributions and therefore yield non-universal spectral fluctuations.

472

10 Semiclassical Roles for Classical Orbits

Periodic orbits with periods of the order of the Ehrenfest time [68, 69] or shorter can be investigated for their influence on system specific behavior without RMT counterpart. Transport through ballistic chaotic conductors [7–14] including the “full counting statistics” [70] and localization in long thin wires [15] have been treated successfully. The relative status of the ballistic sigma model and periodic-orbit theory [71] begins to be understood. An important step towards understanding the so called Andreev gap in mesoscopic normal/superconducting hybrid structures in terms of periodic orbits was recently taken in [72]. More semiclassical work on the new nonstandard symmetry classes will surely come forth.

10.12 Mixed Phase Space Up to this point, we have mostly taken for granted that the classical dynamics under consideration are hyperbolic, i.e., have no stable periodic orbits. Generic systems, however, are neither integrable nor fully chaotic in the sense mentioned but rather fill their “mixed” phase spaces with chaotic as well as stable regions. I shall not enter an extensive discussion of the rich behavior but confine myself to a few remarks and references. If a regular region is so small that it is not resolved by a Planck cell, it leaves but feeble signatures in quantum behavior. For differences to fully hyperbolic systems to become sizable, the islands of regular motion must contain a good fraction of the total number of Planck cells at all visited. Chaotic regions are populated by hyperbolic periodic orbits that allow for semiclassical treatment by Gutzwiller’s trace formula. Islands of regular motion, on the other hand, have central elliptic periodic orbits as their backbones; these are surrounded by chains of further elliptic and hyperbolic orbits. Under refined resolution, that phase space structure appears repeated to ever finer scales in a self-similar fashion: there is an infinite hierarchy of islands within islands [73]. Needless to say, quantum mechanics is immune to such excesses, and Planck’s constant sets the limit of resolution. Some spectral characteristics can, at least summarily, be understood by accounting for the separate chaotic and regular regions with their phase space volumes. According to a rather successful hypothesis by Berry and Robnik [74], for instance, the spacing distribution P(S) for the spectrum of a system with ω = Ωc /Ω and ω = 1 − ω = Ωr /Ω as phase-space fractions is approximated by superimposing two independent ladders of levels, one Poissonian and the other Wignerian, in the way explained in Sect. 5.6. The resulting spacing distribution reads, for β = 1, c P(S) = ω2 e−ωS erfc

√      π π π ωS + 2ωω + ω3 S exp −ωS − ω2 S 2 . 2 2 4 (10.12.1)

10.12

Mixed Phase Space

473

As a control parameter is varied to steer a dynamical system from regular toward increasingly chaotic behavior, one passes through bifurcations at which new periodic orbits are born. All periodic orbits, elliptic or hyperbolic, with periods up to half the Heisenberg time TH are relevant for a semiclassical description of spectral properties. The number of times a single Planck cell (located in a chaos dominated region) is visited by periodic orbits with periods up to some value T grows larger than unity once T exceeds the Ehrenfest time TE . At T ≈ TH /2, a typical Planck cell is crowded by exponentially many periodic orbits. We still do not really know how to deal with such a multitude of orbits which tends to give a quantum mechanically illegitimate structure to Planck cells. Worse yet, since all of these orbits must have arisen through bifurcations the cell in question must typically contain many orbits that have just arisen in bifurcations. Fortunately, some progress has been achieved lately in dealing with phase-space structures like chains of islands around central orbits. Such local structures are classically generated by collective actions that can be classified by so-called normal forms typical for the bifurcation at which the chain structure is born from the central orbit. Normal forms themselves are characterized by their codimension, i.e., the number of controllable parameters needed for their location: Codimension-one bifurcations are seen when a single parameter is varied; if one gets close to some bifurcation by changing a single parameter but does not quite hit, one may zero in by fine-tuning a second parameter and then has located a case of codimension two. Codimension-one bifurcations were discussed by Meyer [75] and codimension-two ones more recently by Schomerus [76]. Phase-space structures generated by a normal-form action collectively contribute to the trace of the quantum propagator through terms of the form AeiS/ where S is the normal form in question and A is a suitable prefactor. Such collective contributions uniformly regularize the sum of single-orbit terms that diverge at the underlying bifurcation. Such uniform approximations were pioneered by Ozorio de Almeida and Hannay [77] and brought to recent fruition by Tomsovic, Grinberg, and Ullmo [78, 79] and Schomerus and Sieber [76, 80–83]. It is well to acknowledge that uniform approximations have a long history in optics and in catastrophe theory [84]. A most interesting phenomenon related to bifurcations is the semiclassical relevance of so-called ghost orbits. These are complex solutions of the nonlinear classical equations of motion which of course have no classical reality to them; they arise as saddle-point contributions to integral representations of quantum propagators, with equal formal right as the stationary-phase contributions of real periodic orbits, provided the original path of integration can legitimately be deformed so as to pass over the corresponding saddles; that condition always rules out ghost orbits whose complex actions have a positive real part such that eiS/ would diverge for  → 0. Contributions from ghost orbits are usually suppressed exponentially in the latter limit except in some neighborhood of a bifurcation where the competition of the two “limits”  → 0 and ImS → 0 may render them important, sometimes even so far away from the bifurcation that the ghost can be accounted for with a separate term AeiS/ rather than including it in a collective normal-form term [85].

474

10 Semiclassical Roles for Classical Orbits

If Gutzwiller’s trace formula is augmented by contributions from stable orbits, ghost orbits and clusters of orbits making up phase-space structures characteristic of bifurcations, reasonable semiclassical approximations for whole spectra become accessible for the kicked top (with not too large Hilbert spaces), and the mean error for a level is a small single-digit percentage of the mean spacing [86]. For general systems, and in particular for infite dimensional Hilbert spaces, no competitive semiclassical schemes for calculating spectra is available as yet. The traces tn = Tr F n of powers of the Floquet operator of periodically driven systems and thus the form factor display anomalously large variations in their dependence on control parameters as a bifurcation is passed. The same is true for fluctuations of the level staircase. Every type of bifurcation produces a peaked contribution, and the maximum amplitude follows a power law with a bifurcation specific exponent for its -dependence [87–90]. Interesting consequences of the coexistence of regular and chaotic motion can be seen in the time evolution of the classical phase-space density, which obeys the Liouville equation in the Hamiltonian case. The pertinent time-evolution operator, commonly called the Frobenius–Perron operator is unitary w.r.t. the Hilbert space of functions one may define on phase space. Thus, the spectrum of the Frobenius– Perron operator lies on the unit circle in the complex plane. However, the resolvent of the Frobenius–Perron operator may have poles inside the unit circle on a second Riemann sheet; these so-called Pollicott–Ruelle resonances [91] have a solid physical meaning as the rates of probability loss from the phase space regions supporting the corresponding eigenfunctions. These resonances can rather simply be detected by looking on phase space with limited resolution. Such blurring automatically arises when the infinite unitary Frobenius–Perron operator is cut to finite size, say N × N , in a basis whose functions are ordered by resolution [92, 93]. Of course, the N × N approximant of the Frobenius–Perron operator is non-unitary and has eigenvalues not exceeding unity in modulus. Those eigenvalues of the finite approximating matrix which are insensitive to variations in N once N is sufficiently large, turn out to be Pollicott-Ruelle resonances. The corresponding eigenfunctions are located on and immediately around elliptic periodic orbits for resonances approaching unity in modulus for large N ; resonances smaller than unity in modulus, on the other hand, have eigenfunctions supported by the unstable manifolds of hyperbolic periodic orbits; at any rate, these eigenfunctions are strongly scarred [93]. Once a resonance is identified, it can be recovered through the so-called cycle expansion, i.e., a representation of the spectral determinant in the fashion explained in Sect. 10.5, where the periodic orbits included are those visible in the scars (and their repetitions). These classical findings have quantum analogues. Instead of the phase-space density, one must employ any of the so-called quasi-probability densities that are defined so as to have the phase space coordinates as independent variables and to represent the density operator; examples are the Wigner and Husimi or Glauber’s Q-function [94]. The Liouville-von Neumann equation for the density operator may be written as an evolution equation for the quasi-probability used, say the Q-function. Thus, a Q propagator arises whose classical analogue is the Frobenius–Perron operator. In

10.13

Problems

475

contrast to the classical Frobenius–Perron operator, the Q propagator is an N Q2 × N Q2 matrix whose dimension is fixed through Weyl’s law by the number of Planck cells contained in the classically accessible phase-space volume Ω, i. e. N Q = Ωf . The spectrum of the Q propagator must lie on the unit circle due to the unitarity time evolution of the density operator. It turns out that classical and quantum dynamics become indistinguishable if looked upon with a resolution much coarser than a Planck cell: N × N approximants of the Frobenius–Perron operator and the Q propagator with N  Nq yield the same resonances and scarred eigenfunctions [95], which is a rather intuitive result indeed. Of the host of observations of mixed phase spaces in real systems, at least some examples deserve mention. Conductivity measurements for two-dimensional antidot structures in semiconductors [96] have revealed classical [97] and semiclassical [98] manifestations of mixed phase spaces. Magneto-transport through chaotic quantum wells has recently attracted interest [99], as has the radiation pattern emerging from mesoscopic chaotic resonators [100]. The mass asymmetry in nuclear fission has found a semiclassical interpretation [101]. Atomic hydrogen in a strong magnetic field remains a prime challenge for atomic spectroscopy [102]. Finally, quasi-particle excitations of Bose–Einstein condensates trapped in anisotropic (even axially symmetric) parabolic potentials are nonintegrable if their energies are comparable to the chemical potential [103]; these latter phenomena make for a particularly nice enrichment of known chaotic waves, inasmuch as the underlying wave equation is the nonlinear Gross–Pitaevski equation.

10.13 Problems 10.1 Retrace your first steps into WKB terrain: Entering Schr¨odinger’s equation iψ˙ = ( p2 /2m + V (x))ψ with the ansatz ψ = A(x, t)eiS(x,t)/ show that to leading order in  the phase S/ obeys the Hamilton–Jacobi equation S˙ + H (x, ∇ S) = 0 while the next-to-leading order yields the continuity equation ˙ + div j = 0 for the probability density  = |A|2 and the probability current density j = ∇ S/m. 10.2 Modify the Van Vleck propagator (10.2.19) and the trace formula (10.3.5) for f degrees of freedom. 10.3 The baker map operates on the unit square as phase space in two steps, like a baker on incompressible dough: Compression to half height in the q-direction and double breadth in p is followed by stacking the right half of the resulting rectangle on top of the left. Show graphically that the number of intersections of the nth iterate of a line of constant q has 2n intersections with any line of constant p. 10.4 Transcribing the reasoning of Sect. 10.3.2 from discrete to continuous time shows that free motion has minimal action. You will enjoy being led in this classical problem to the Schr¨odinger equation of a particle in a box.

476

10 Semiclassical Roles for Classical Orbits

mω 10.5 Show that the harmonic oscillator has (1) the action S(q, q , t) = 2 sin ωt 2

2

[(q + q ) cos ωt − 2qq ], (2) a Van Vleck propagator coinciding with the exact one and shrinking to a delta function every half period, and (3) a conjugate point every half period. After having studied Sect. 10.4, draw out the fate of the initial Lagrangian manifold.

10.6 The Kepler ellipse can be parametrically represented by r = a(1 − e cos x ) : ma 3 t= (x − e sin x ) GM where the radial coordinate r is reckoned from a focus, a is the semimajor axis, e the excentricity, m the reduced and M the & total mass, and G the gravitational constant. Using a = −Gm M/2E and e = 1 + 2E L 2 /m 3 M 2 G 2 with E the energy and L the angular momentum, find the configuration-space caustics and check that ∂ 2 S/∂t 2 = −∂ E/∂t diverges thereon. Hint: Look at the turning points of the radial oscillation. Note that for f > 1, in contrast to oscillations of a single degree of freedom, the turning points of one coordinate are not characterized by vanishing kinetic energy. 10.7 Show that the symplectic form (10.4.1) is independent of the coordinates used to compute it. Hint: Use a generating function F(Q, q) to transform canonically from q, p to Q, P.  10.8 Use the definition (E) = i δ(E − E i ) of the level density of autonomous systems to write the form factor as a sum of unimodular terms e−i(Ei −E j )TH τ/ . Before embarking on the few lines of calculation, think about what expression for the Heisenberg time TH you will come up with. 10.9 A particle in a one-dimensional box has the energy spectrum E j = j 2 where for notational convenience 2 /2m is set equal to unity. Use the infinite-product representation of the sine, sin z = z

∞  , j=1

z2 1− 2 2 π j

 (10.13.1)

to show that a convergent zeta function √ sin π E { A j (E − j )} = ζ (E) = √ π E j=1 ∞ ,

2

results from A j = −1/E j . Note that the energy levels are the zeros of ζ (E).

References

477

10.10 Consider the harmonic oscillator with the spectrum E = j − 1/2 for j = 1, 2, . . .. Use Euler’s infinite product for the Gamma function ∞

, 1 = zeγ z Γ (z) j=1

 1+

z j



e−z/j

 (10.13.2)

and the product (10.13.1) once more to show that the regularizer A(E, E j ) = −

1 Ej +

1 2

exp

E+

1 2 E j + 12

yields the zeta function ζ (E) =

1 γ ( E+ 1 ) 2 Γ e π

 E+

   1 1 sin π E + 2 2

where γ is Euler’s constant. Check that ζ (E) has the eigenvalues as its zeros, i.e., that the additional zeros of the sine are cancelled by the poles of the Gamma function. 10.11 Rederive the Riemann–Siegel lookalike (10.5.28) in a simplified manner by neglecting orbit repetitions already in the exponentiated orbit sum (10.5.10) and expand the exponential in powers of the periodic-orbit sum. Does the stability amplitude FP now differ for hyperbolic orbits with and without reflection? 10.12 Recover the diagonal approximation (10.7.6) starting from the pseudo-orbit expansion (10.6.20) for Z (1) . Drop the restriction TTC ,TD 0, evaluating the square root on the lower lip of the cut such that upon dropping ImE − , we have the saddle points a± = E/2 ∓ i{λ2 − (E/2)2 }1/2 . They are mutual complex conjugates and lie in the complex a plane on the circle of radius λ around the origin. The integrand in (11.2.20) vanishes for a → ±∞ on the real axis and has no singularities at finite a except for a pole at a = E − , right below the real axis. The original path of integration along the real axis can be deformed so as to climb along a path of constant phase of the integrand over the saddle at a− , without crossing the said pole; a path of constant phase from −∞ to +∞ over the saddle at a+ also exists but cannot be deformed back to the original path without crossing the pole. Thus, only the saddle at a− contributes.

488

11 Superanalysis for Random-Matrix Theory

 (2) ( E2 )2 > λ2 : Expanding the square root for  → 0+ we have a± = (E/2) 1 ± &    1 − (2λ/E)2 − i(/2) 1 ± 1/ 1 − (2λ/E)2 . Again, the saddle point a+ lies below the real axis while a− lies above such that only the latter can contribute. Thus, for all cases we conclude that

&



6

∞

−N f (a)

da A(a)e −∞

≈

2π A(a− )e−N f (a− ) , N f

(a− )

(11.2.24)

where the phase of the square root in the denominator is defined by the steepest descent direction of integration across the saddle point. A little surprise is waiting for us in the integrals over b in (11.2.20), 

∞

−∞

dbB(b)e−N g(b)

with

g(b) = b2 /2λ2 − ln(E − − ib) .

(11.2.25)

Now b = −iE is a zero of the integrand rather than&the location of a singularity. Saddles are encountered for ib± = a± = (E − /2) 1 ± 1 − (2λ/E − )2 . The saddle points b± are obtained by a clockwise (π/2)-rotation of the a± about the origin in the complex a plane. Two cases again arise: lie symmetrically to the imaginary axis of the com(1) (E/2)2 < λ2 : The saddles& plex a-plane at b± = −iE/2 ∓ λ2 − (E/2)2 . They are degenerate in height since Reg(b+ ) = Reg(b− ). Since the integrand has no singularity at any finite point b, the path of integration can be deformed so as to climb uphill from −∞ to b+ , then descend to the zero at −iE, climb again to b− , and finally descend toward +∞. That deformed path may be chosen as one of constant phase, except at the zero of the integrand where no phase can be defined, such that, the phase jumps upon passing through the exceptional point. 2 2 (2) (E/2) &> λ : In thatcase both saddle points are purely imaginary, b± =  −i(E/2) 1 ± 1 − (2λ/E)2 , and b+ is further away from the origin than b− ; see Fig. 11.1. The saddle at b+ is higher than that at b− since Reg(b+ ) < Reg(b− ), and one might thus naively expect only b+ to be relevant. But surprisingly, b+ does not enter at all, as the following closer inspection reveals. The phase of the integrand, i.e., Img(x + iy) = x y/λ2 + arctan[x/(E + y)], vanishes along the imaginary axis. That path of the vanishing phase passes through both saddles and the zero of the integrand; along it the integrand has a minimum at b− and a maximum at b+ . Another path of constant vanishing phase goes through each saddle; along the one through b− , the modulus of the integrand has a maximum at b− ; that latter path extends to ±∞ + i0 and makes b− the only contributing saddle. No excursion from b− to b+ can be steered toward +∞ + i0 without climbing higher after reaching b+ , except for the inconsequential excursion returning to b− along the imaginary axis. To summarize the discussion of the b-integral, I note that

11.2

Semicircle Law for the Gaussian Unitary Ensemble

489

Fig. 11.1 Contour plot of the exponent −g(b) from (8.2.25). The saddles at b± are marked as ⊕ and %. Height is indicated in shades of grey, √ ranging from black (low) to white (high). Left for case (1), i.e., (E/2)2 < λ2 , with E = 1, λ = 1/ 2. The two saddles are of equal height. The path of integration climbs up the valley from the left through b− and descends to the zero of exp(−N g(b)) on the imaginary axis, then again uphill to √ b+ , and therefrom √ down the valley toward +∞. Right for case (2), i.e., (E/2)2 > λ2 , with E = 2 + 0.1, λ = 1/ 2. We recognize that the saddle at b+ lies higher than that at b− and also see the logarithmic “hole” below b+ . As described in the text the path of integration climbs up the valley from the left toward the saddle at b− and descends on the other side into the valley to the right



∞ −∞

dbB(b)e−N g(b) ≈

⎧ ⎪ ⎪  % 2π ⎪ ˆ ⎪ ˆ −N g(b) ⎪ ⎨ ˆ B(b)e N g

(b) ˆ + ,b− b=b

⎪ ⎪ ⎪ % ⎪ ⎪ ⎩

2π B(b− )e−N g(b− ) N g

(b− )

for ( E2 )2 < λ2 (11.2.26) for ( E2 )2 > λ2 .

Combining the a- and b-integrals, we first treat the case labelled (2) above, i.e., (E/2)2 > λ2 . Since only the saddles at a− and b− contribute, we get G(E − ) ≈

1



N +1 N − − λ2 (E − a− )(E − − ib− )



2π(E − − a− ) 6    (2π)2 exp −N f (a− ) + g(b− ) . × N 2 f

(a− )g

(b− )

(11.2.27)

However, due to a− = ib− we have f (a− ) + g(b− ) = 0 such that the exponential factor equals unity. Similarly, f

(a− ) = g

(b− ) = λ−2 − (E −−a− )−2 whereupon the −1/2  curly bracket above cancels as N −1 f

(a− )g

(b− ) {. . .} = 1+O(1/N ). But since corrections of relative order 1/N are already& made in the saddle-point approximation, −1   G(E − ) = (E − − a− )−1 ≈ (E −/2) 1 + 1 − (2λ/E − )2 . Setting  = 0, we

490

11 Superanalysis for Random-Matrix Theory

&   −1 finally get the purely real result G(E − ) ≈ (E/2) 1 + 1 − (2λ/E)2 . Thus, the 2 2 average density vanishes, (E) ≈ 0 for (E/2) > λ . In the hope of a finite mean density of levels, we finally turn to case (1), (E/2)2 < λ2 . According to the above findings, we confront contributions from the two pairs of saddles {a− , b− } and {a− , b+ }. The first pair is dealt with in full analogy with the discussion of the pair {a− , b− } under case (2) and furnishes & corresponding −1  G(E − ) with E/2 − i λ2 − (E/2)2 . The pair a− , b+ is incapable of providing anything of asymptotic weight: The argument of the exponential in (11.2.27) is  2  2 /(2λ2 ) + ln a+ − a+ /(2λ2 ) − ln a− replaced with −N f (a− ) + g(b+ ) = −N a− ∗ which is purely imaginary due to a− = a+ , such that the exponential remains unimodular however large N . Collecting to the   all other N -dependent factors according  model of (11.2.27), we get N −1 . . . = N −1 N /λ2 −(N +1)/(a+ a− ) = −1/N λ2 , i.e., an overall weight of order 1/N . Indeed then, only the pair {a− , b− } contributes in the limit N → ∞, just as in case (2). It remains to collect the limiting form of the averaged Green function,

lim

N →∞

G(E − )

=

⎧ ⎨ ⎩

√1

(E/2)(1+ E/2−i

1−(2λ/E)2

√ 12

λ −(E/2)2

for

( E2 )2 > λ2

for

( E2 )2 < λ2 ,

(11.2.28)

and, after taking the imaginary part, Wigner’s semicircle law, lim (E) =

N →∞

 0 1 πλ2

&

λ2 − (E/2)2

for for

( E2 )2 > λ2 ( E2 )2 < λ2 .

(11.2.29)

Needless to say, the edges of the semicircular level density at E = ±2λ are rounded off at finite N , as may be studied by scrutinizing next-to-leading-order corrections. The importance of the semicircle law is somewhat indirect: It must be scaled out of the energy axis before the predictions of random-matrix theory for spectral fluctuations take on their universal form to which generic dynamical systems are so amazingly faithful.

11.3 Superalgebra 11.3.1 Motivation and Generators of Grassmann Algebras A compaction of our notation is indicated before we can generalize the above discussion of the mean density of levels (E) to two-point correlation functions like (E)(E ). We need to generalize some basic notions of algebra to accommodate both commuting and anticommuting variables as components of vectors and elements of matrices. We had previously (see Sect. 4.13) introduced Grassmann variables η1 , . . . , η N with the properties ηi η j + η j ηi = 0 and shall stick to these as the “generators of a

11.3

Superalgebra

491

2 N -dimensional Grassmann algebra”. Whenever convenient, we shall, as already done in the previous section, complement these N generators by N more such, η1∗ , . . . , η∗N , so as to simply have a 22N -dimensional Grassmann algebra. An internal structure is provided to the latter by defining complex conjugation as (ηi )∗ = ηi∗ and (ηi∗ )∗ = −ηi .

11.3.2 Supervectors, Supermatrices We shall have to handle (2M)-component vectors with M commuting components S1 , . . . , S M and M anticommuting components χ1 , . . . , χ M ,  S Φ= , χ

⎛ ⎞ S1 ⎜ .. ⎟ S=⎝.⎠, SM

⎛ ⎞ χ1 ⎜ .. ⎟ χ =⎝.⎠.

(11.3.1)

χM

Quite intentionally, I have denoted the components χi with a Greek letter difto fering from the one reserved for the generators ηi since it will be necessary  to be odd functions of the generators like, e.g., χ = c η + allow the χ i i i j j j  jkl ci jkl η j ηk ηl . . . with commuting coefficients c. Similarly, the commuting charprevent the appearance of η’s but restricts the Si to be even in acter of the Si does not the η’s, like Si = z i + jk z i jk η j ηk +. . .. If a verbal distinction between commuting numbers without and with additions even in the η’s is indicated I shall call the former numerical; an addition even in the Grassmannians is a nilpotent quantity;1 the coefficients c and z in the above characterizations of anticommuting χ ’s and commuting S’s are meant to be numerical. In somewhat loose but intuitive jargon, I shall sometimes refer to commuting variables as bosonic and anticommuting ones as fermionic. At any rate, a vector of the structure (11.3.1) is called a supervector. If the need ever arises, one can generalize the definition of a supervector such as to allow for the number of elements of the bosonic and fermionic subvectors to be different, M B for S and M F for χ . Moreover, M F need not coincide with the number N of generators of the Grassmann algebra. Unless explicitly stated otherwise, I shall in the following take M B = M F = M. Supermatrices are employed to relate supervectors linearly. We introduce such 2M × 2M matrices as        aσ aσ S aS + σ χ F= , FΦ = = . (11.3.2) ρ b ρ b χ ρ S + bχ For FΦ to have the same structure as Φ, where the first M components are bosonic and the last M ones are fermionic, of the four M × M blocks in F, the 1 A nilpotent quantity has vanishing powers with integer exponents upward of some smallest positive one.

492

11 Superanalysis for Random-Matrix Theory

matrices a and b must have bosonic elements while σ and ρ must have fermionic elements. It is customary to speak of a as the Bose–Bose block or simply the Bose block, of b as the Fermi–Fermi or simply the Fermi block, of σ as the Bose–Fermi block, and of ρ as the Fermi–Bose block. For products of supermatrices, the usual rules of matrix multiplication must hold. (If M B = M F , a is M B × M B , b is M F × M F , while the two blocks with fermionic elements are rectangular matrices, σ an M B × M F one and ρ an M F × M B one.) We need to define transposition for supermatrices. Since it is desirable to retain the usual rule (F1 F2 )T = F2T F1T , we must modify the familiar rule for transposing purely bosonic matrices to  T   a˜ ρ˜ aσ = ; F = ρ b −σ˜ b˜ T

(11.3.3)

As done previously in this book, I reserve the tilde to denote the familiar matrix transposition and shall use the superscript T for transposition of supermatrices that includes the unfamiliar minus sign in the Fermi–Bose block. A further price for retaining the usual rule for transposition of products is that double transposition is not the identity operation, 

 a −σ  F. = (F ) = −ρ b T T

(11.3.4)

A useful extension of the notion of a trace to supermatrices is the supertrace StrF = Tra − Trb =

M 

(amm − bmm ) .

(11.3.5)

m=1

The minus sign in front of the trace of the Fermi block is beneficial by bringing about the usual cyclic invariance for the supertrace, StrF1 F2 = StrF2 F1 ,

(11.3.6)

since StrF1 F2 = Tr(a1 a2 + σ1 ρ2 − ρ1 σ2 − b1 b2 ) = Tr(a1 a2 + σ1 ρ2 + σ2 ρ1 − b1 b2 ) is obviously symmetric in the labels 1 and 2. It follows that StrM F M −1 = StrF where M is another supermatrix and M −1 is its inverse. A most welcome consequence of the cyclic invariance of the supertrace is the following identity for the logarithm of a product, Str ln F G = Str ln F + Str ln G

(11.3.7)

whose proof parallels that of the corresponding identity for ordinary matrices. Sketching that proof provides an opportunity to recall the definition of the logarithm as a series,

11.3

Superalgebra

493

∞   (−1)n ln F = ln 1 + (F − 1) = − (F − 1)n , n n=1

(11.3.8)

up to analytic continuation for F outside the range of convergence. To display unity as the reference matrix about which the expansion goes I momentarily introduce F = 1 + f, G = 1 + F −1 g such that F G = 1 + f + g. Thus, the identity to be proven may be written as Str ln(1 + f + xg) = Str ln(1 + f ) + Str ln[1 + x(1 + f )−1 g] ?

(11.3.9)

where x is a real parameter which can obviously be sneaked in without gain or loss of generality, but with profit in convenience. Expanding the x-dependent logarithms about the unit matrix, we transform our identity to −

∞  (−1)n n=1

n

n ?

Str( f + xg) = Str ln(1 + f ) −

∞  (−1)n n=1

n

Str



n x g . (11.3.10) 1+ f

When differentiating with respect to x, we benefit from the cyclic invariance of the trace and rid ourselves of the factors (1/n) in the nth terms of the expansions, such that the expansions take the forms of geometric series, 1 1 1 ? g= g. 1 + f + xg 1 + (1 + f )−1 xg 1 + f

(11.3.11)

But now the question mark can be dropped since the two sides do equal one another due to (ab)−1 = b−1 a −1 . We can also wave good-bye to the question mark in (11.3.9); the functions of x on the two sides have coinciding derivatives for all x and take the same value for x = 0. Needless to say, our proof of (11.3.7) applies to traces of logarithms of products of ordinary matrices as well.

11.3.3 Superdeterminants The definition of a superdeterminant SdetF to be given presently is meant to preserve three familiar properties of usual determinants: (1) a supermatrix F and its transpose F T should have the same superdeterminant, SdetF = SdetF T , (2) the superdeterminant of a product should be the product of the superdeterminants of the factor supermatrices, SdetF1 F2 = SdetF1 SdetF2 ; and (3) for a supermatrix, the logarithm of its superdeterminant should equal the supertrace of its logarithm, ln SdetF = Str ln F. We shall find these requirements met when the superdeterminant is defined as the ratio of usual determinants

494

11 Superanalysis for Random-Matrix Theory

 SdetF = Sdet

a σ ρ b



= det(a − σ b−1 ρ)/ det b

if

det b0 = 0 (11.3.12)

where b0 denotes the numerical part of the Fermi block b. If det b0 = 0, the superdeterminant of F does not exist. Its inverse may, however, be given a meaning, (SdetF)−1 = det(b − ρa −1 σ )/ det a

if

det a0 = 0 ,

(11.3.13)

provided the numerical part a0 of the Bose block is nonsingular. If both a0 and b0 are nonsingular, the compatibility of the two definitions must be proven. No work is required to that end if F is block diagonal with σ = ρ = 0, so that the superdeterminant is the ratio of the determinants of the Bose and Fermi blocks, SdetF =

det a det b

for

σ = ρ = 0,

(11.3.14)

a case of some importance for applications. The compatibility proof for general F is a good opportunity to shape up for less trivial adventures to follow later. Under the assumption mentioned, we may rewrite (11.3.12) as SdetF = det(1 − a −1 σ b−1 ρ) det a/ det b and (11.3.13) as (SdetF)−1 = det(1 − b−1 ρa −1 σ ) det b/ det a such that we need to show the equality of the two  −1 usual N × N determinants det(1 − a −1 σ b−1 ρ) and det(1 − b−1 ρa −1 σ ) or, equivalently, the equality of Tr ln(1 − a −1 σ b−1 ρ) with −Tr ln(1 − b−1 ρa −1 σ ). To that end, we expand both logarithms and need to verify that Tr(a −1 σ b−1 ρ)n = −Tr(b−1 ρa −1 σ )n .

(11.3.15)

But this can be checked by rewriting the l.h.s. in two steps. The first relies on the cyclic invariance of the usual trace, l.h.s. = Trσ [b−1 ρ(a −1 σ b−1 ρ)n−1 a −1 ]. No less easy even though a little unfamiliar is the second step. Again under the protection of the trace operation, I want to change the order of the factors σ and [. . .]; but since both factor matrices are now fermionic, a minus sign results, Trσ [. . .] = −Tr[. . .]σ , simply because the matrix elements of σ and [. . .] anticommute; the fermionic character of [. . .] follows from the fact that the latter matrix involves an odd number of fermionic factors ρ and σ ; with the minus sign generated by shifting σ to the right, we have indeed arrived at the equality (11.3.15) and thus at the compatibility of (11.3.12) and (11.3.13). The proof of the equality of the superdeterminants of F and F T , SdetF = SdetF T

(11.3.16)

is a nice aside left to the reader. I shall, however, pause to show that indeed ln SdetF = Str ln F .

(11.3.17)

11.3

Superalgebra

495

To that end, I write the general supermatrix F as a product,  F=

a ρ

σ b

 =

 a 0

 0 1 b b−1 ρ

 a −1 σ , 1

(11.3.18)

and use (11.3.7) for the supertrace of its logarithm,  a Str ln F = Str ln 0

  0 1 + Str ln −1 b b ρ

 a −1 σ . 1

(11.3.19)

The first of the supertraces appearing on the r.h.s. is simply evaluated as  a Str ln 0

  0 ln a = Str b 0

 det a 0 = Tr ln a − Tr lnb = ln , ln b det b

(11.3.20)

whereas for the second, we must invoke once more the definition of the logarithm as a series; momentarily abbreviating as ν = a −1 σ, μ = b−1 ρ,  Str ln

1 μ

   ∞ (−1)n−1 0 ν Str = μ 1 n n=1

n ν . 0

But only even-order terms of the series have nonvanishing supertraces such that =−

 ∞  1 (νμ)n Str 0 2n n=1

 ∞  1  0 Tr(νμ)n − Tr(μν)n = − n (μν) 2n n=1

∞  1 =− Tr(νμ)n = Tr ln(1 − νμ) = ln det(1 − νμ) . n n=1

(11.3.21)

 Putting (11.3.20) and (11.3.21) into (11.3.19), we get Str ln F = ln det(a −  σ b−1 ρ)/ det b = ln SdetF, the desired result. Finally, the product rule Sdet(F1 F2 ) = (SdetF1 ) (SdetF2 )

(11.3.22)

follows from the above as ln SdetF1 F2 = Str ln F1 F2 = Str ln F1 + Str ln F2 = ln SdetF1 + ln SdetF2 = ln SdetF1 SdetF2 .

11.3.4 Complex Scalar Product, Hermitian and Unitary Supermatrices As we have already seen when deriving the semicircle law in the previous section, it is sometimes desirable to associate complex conjugates with Grassmann variables.

496

11 Superanalysis for Random-Matrix Theory

Our present goal of setting up a suitable superalgebraic framework means that we need to define complex conjugation of supervectors. Well then, we shall work with Φ=

 S , χ

Φ∗ =

 ∗ S , χ∗

(Φ ∗ )∗ =



 S , −χ

(11.3.23)

whereby we simply double the number of generators of the Grassmann algebra formed by the Fermionic components to 2M F . As already done in Sect. 11.2, the definition of complex conjugation is completed by requiring that (χi∗ )∗ = −χi , (χi χ j )∗ = χi∗ χ ∗j , and (χi + χ j )∗ = χi∗ + χ ∗j . It follows that χi∗ χi is real. However, χi + χi∗ is not real since its complex conjugate is −χi + χi∗ . We shall employ the scalar product

(Φ1 , Φ2 ) =

M  (S1i∗ S2i + χ1i∗ χ2i )

(11.3.24)

i=1

which enjoys the familiar property (Φ1 , Φ2 )∗ = (Φ2 , Φ1 ). The Hermitian conjugate or “adjoint” of a supermatrix F is obtained by taking both the complex conjugate and the transpose, F † = (F T )∗ = (F ∗ )T ;

(11.3.25)

this is useful because of (FΦ1 , Φ2 ) = (Φ1 , F † Φ2 ). We should note that in contrast to twofold transposition, twofold Hermitian conjugation is the identity operation, F †† =

 a ρ

††  † σ a = b −σ˜ ∗

†  a ρ˜ ∗ = −ρ ∗∗ b†

 −σ ∗∗ = F. b

(11.3.26)

Incidentally, the foregoing implies that for the Bose–Fermi and Fermi–Bose blocks, we can maintain the usual definition of Hermitian conjugation, σ † = σ˜ ∗ , which implies σ †† = −σ . A Hermitian supermatrix can be defined as usual as a supermatrix equalling its own adjoint; it has the structure H = H† =

 a ρ

 ρ† b

where

a † = a, b† = b .

(11.3.27)

Unitary supermatrices preserve the scalar product of supervectors, (U Φ1 , U Φ2 ) = (Φ1 , Φ2 ) ,

UU † = U † U = 1 .

(11.3.28)

11.3

Superalgebra

497

11.3.5 Diagonalizing Supermatrices Special attention is due to Hermitian supermatrices. The eigenvalue problem H Φ = hΦ yields coupled homogeneous equations for the bosonic and fermionic components S and χ or, after eliminating of one of them,  1 ρ S=0 b−h

(11.3.29)

 1 † ρ χ = 0. a−h

(11.3.30)

 a − h − ρ†

 b−h−ρ

The special structure of these equations forbids the appearance of too many eigenvalues: the first makes sense only if the numerical part (b − h)num of b − h has a nonvanishing determinant and thus provides M B eigenvalues for the Bose block; similarly, the second requires det(a − h)num = 0 and gives M F eigenvalues for the Fermi block. To reveal this structure explicitly it is well to consider the simplest case M B = M F = 1 for which (11.3.29) may be rewritten as the quadratic equation h 2 − (a + b)h + ab − ρ ∗ ρ = 0; one of the two solutions, h Bghost = b − ρ ∗ ρ/(a − b), must be discarded since h Bghost − b has a vanishing numerical part, i.e., is nilpotent. Similarly, (11.3.30) has the spurious solution h Fghost = a + ρρ ∗ /(a − b) such that we are left with one eigenvalue each for the Bose and Fermi blocks, h B = a + ρ ∗ ρ/(a − b), h F = b + ρ ∗ ρ/(a − b) , for M B = M F = 1 .

(11.3.31)

Needless to say, no eigenvalue exists if a = b. An eigenvalue is called positive if its numerical part is positive. Momentarily staying with M B = M F = 1, we may wonder which unitary supermatrix diagonalizes H ; it is H = U diag(h B , h F )U −1 with U=

 1 + ηη∗/2 η∗

 η , 1 − ηη∗/2

U −1 =

 1 + ηη∗/2 −η∗

 −η (11.3.32) 1 − ηη∗/2

where η = −ρ ∗ /(a − b), η∗ = +ρ/(a − b), and U −1 = U † . Obviously again, for ρ ∗ = 0, nilpotency of a − b would preclude diagonalizability. Our expressions for the eigenvalues h B and h F , the matrices U and U −1 , and for η and η∗ remain valid when b is continued to arbitrary complex values; they are in fact most often used for imaginary b. Even though H then ceases to be Hermitian and U to be unitary, U still diagonalizes H . However, unitarity of U can be restored by redefining complex conjugation for Grassmannians such that η and η∗ remain complex conjugates with b an arbitrary complex number (See Problems 11.3 and 11.4).

498

11 Superanalysis for Random-Matrix Theory

11.4 Superintegrals 11.4.1 Some Bookkeeping for Ordinary Gaussian Integrals To properly build up the notion of superintegrals as integrals over both commuting and anticommuting variables, it is well to recall the familiar ordinary Gaussian integral   , M d 2 Si i=1

π

M    Si∗ ai j S j = exp − i, j=1

1 , det a

(11.4.1)

where the range of integration is the full complex plane for each of the M complex integration variables Si and the M × M matrix a must have a positive real part for convergence. The differential d 2 Si is meant as dReSi dImSi . For maximal convenience in changing integration variables, it is sometimes advantageous to think of the differential volume element as an ordered antisymmetric product, the so-called wedge product,2 d 2 Si = dReSi ∧ dImSi = −dImSi ∧ dReSi = (2i)−1 d Si∗ ∧ d Si = −(2i)−1 d Si ∧ d Si∗ ; the variables Si and Si∗ , as well as their differentials d Si and d Si∗ , may then, with some caution, be regarded as independent such that the complex Si -plane is tiled with area elements d Si∗ ∧ d Si = −d Si ∧ d Si∗ . One bonus of this bookkeeping device lies in the possibility of changing integration variables in the sense of independent analytic continuation in both Si and Si∗ , so as to give up the kinship between Si and Si∗ as mutual complex conjugates. The simplest such change shifts one, but not the other, of these variables by a constant complex number. Thus, e.g., 

d 2 S e−S

∗

(S+c)

=π.

(11.4.2)

For any change of the integration variables, the artifice of the wedge product readily yields the familiar transformation of the differential volume element, M , i=1

d 2 Si = det

M M  S∗, S  , ∂(S ∗ , S) , 2 d z = J d 2 zi . i ∂(z ∗ , z) i=1 z ∗ , z i=1

(11.4.3)

In particular, for a linear transformation which keeps the Si∗ unchanged but reshuffles the Si ,

2 Readers not previously familiar with the wedge product of differentials of commuting variables have every right to be momentarily confused. They have the choice of either spending a quiet hour with, e.g., Ref. [5], or simply ignoring the following remark and convincing themselves of the correctness of (11.4.2) in some other way, most simply by expanding the integrand in powers of c.

11.4

Superintegrals

499

zi =



ai j S j

=⇒

J

j

S z

=

1 . det a

(11.4.4)

In fact, (11.4.1) can be seen as an application of (11.4.4): With the latter transformation introduced in the Gaussian integral, we read Si∗ and z i as complex conjugates $ $  M  ∗ M and get (2π i)−1 d S ∗ ∧ dz exp(−S ∗ z) = π −1 d 2 Se−S S = 1. With these  remarks in mind, we may change the integration variables in (11.4.1)  as Si → Si − j (a −1 )i j z j , Si∗ → Si∗ − j z ∗j (a −1 ) ji to get (det a)

  , M d 2 Si i=1

≡

@ exp

π

M M     Si∗ ai j S j exp (Si∗ z i + z i∗ Si ) exp − i, j=1

i=1

M M 3 4A  (Si∗ z i + z i∗ Si ) = exp z i∗ (a −1 )i j z j . i=1

(11.4.5)

i, j=1

The angular brackets (. . .) in the foregoing identity suggest a change of perspective, hopefully not too upsetting to the reader;3 they are a shorthand for the average of (. . .) w.r.t. the normalized Gaussian distribution appearing in the first member. In fact, we may read the quantity in (11.4.5) as the moment generating function for that Gaussian. A nice application is Wick’s theorem, which we find by differentiating w.r.t. the auxiliary variables {z i , z i∗ } and subsequently setting these to zero; the simplest cases read Si S ∗j  = (a −1 )i j Si S j Sk∗ Sl∗  = Si Sk∗ S j Sl∗  + Si Sl∗ S j Sk∗  = (a −1 )ik (a −1 ) jl + (a −1 )il (a −1 ) jk ,

(11.4.6)

and the general case is offered to the interested reader as Problem 11.5.

11.4.2 Recalling Grassmann Integrals By now, we have often used the Grassmann analogue (4.13.1) of (11.4.1),  , M n=1

 dχn∗ dχn



exp −

1...M 

 χi∗ bi j χ j = det b ,

(11.4.7)

ij

3 Truly daring readers might even enjoy contemplating the kinship of (11.4.5) as well as its superanalytic generalizations below (see (11.4.9) and (11.4.16) with the Hubbard–Stratonovich transformation.

500

11 Superanalysis for Random-Matrix Theory

which holds for any M × M matrix b and, like all Grassmann integrals, does not require boundaries to become definite. To tune in to what is to follow, I propose once more to prove that integral representation of det b, in a way differing from the  b χ original one of Sect. 4.13. Changing integration variables as ρi = i j j while j holding on to the χ ∗ ’s and employing the Jacobian (4.13.17),   χ J = det b ρ

=⇒

,

dχi∗ dχi = det b

,

i

dχi∗ dρi ,

(11.4.8)

i

we see the Gaussian integral (11.4.7) yielding the determinant det b, up to the incon M  $ M $ dχ ∗ dρ(1 − χ ∗ ρ) = 1. sequential factor dχ ∗ dρ exp(−χ ∗ ρ) = change variables in (11.4.7), χi → χi −  of∗ integration  The−1fruit of another ∗ ∗ −1 (b ) η , χ → χ − η (b ) with constant Grassmannians ηi , ηi∗ , is i j j ji i i j j j worthy of respectful consideration,

(det b)−1

 , M

1...M  1...M   dχn∗ dχn exp − χi∗ bi j χ j + (χi∗ ηi + ηi∗ χi )

n=1

≡

@



ij

i

M  A  1...M  exp (χi∗ ηi + ηi∗ χi ) = exp ηi∗ (b−1 )i j η j . i=1

(11.4.9)

ij

We confront the Grassmannian analogue of the moment generating function (11.4.5) that invites pushing the analogy to ordinary Gaussian integrals toward “means” like χi χ ∗j . By differentiating (11.4.9) w.r.t. to some of the η’s and η∗ ’s and setting all of these “auxiliary” parameters to zero thereafter, we get χi  = χi∗  = 0 ,

χi χ ∗j  = (b−1 )i j =

∂ ln det b ; ∂b ji

(11.4.10)

the latter identity is, of course, an old friend from determinantology (see, e.g., (7.4.5)) and may show that the “probabilistic” perspective in which I am indulging here does have useful implications, its admittedly frivolous appearance notwithstanding. Upon taking more derivatives, we encounter the Grassmannian version of Wick’s theorem, e.g., χi χ j χk∗ χl∗  = −χi χk∗ χ j χl∗  + χi χl∗ χ j χk∗  .

(11.4.11)

It is well to note that the different powers of det a and det b in the Jacobians (11.4.4) and (11.4.8) and thus in the Gaussian integrals (11.4.1) and (11.4.7) determine the starting point of superanalysis in random-matrix theory.

11.4

Superintegrals

501

11.4.3 Gaussian Superintegrals The simplest Gaussian superintegral is just the product of the Gaussian integral representations for det b and 1/ det a,   −1 det b a0 (11.4.12) = Sdet 0b det a   , M    d 2 Si ∗ Si∗ ai j S j + ηi∗ bi j η j , dηi dηi exp − = π ij i=1 which I hurry to compact with the help of Φ = 

S η

and F =

a 0 0b

dΦ ∗ dΦ exp(−Φ † FΦ) = (SdetF)−1 .

as (11.4.13)

We have already used this in Sect. 11.2 for the generating function Z in (11.2.3) and (11.2.6). There, as again below, we profit from having lifted the matrices a, b to an exponent which makes subsequent averages over these matrices easy to implement, provided that their elements are independent Gaussian random numbers. We should not forget the existence condition a + a † > 0. Once and for all, I have introduced here the differential volume element for the integration over a supervector Φ, dΦ ∗ dΦ =

M , d 2 Si i=1

π

dηi∗ dηi =

M , d S ∗ ∧ d Si i

i=1

2π i

dηi∗ dηi .

(11.4.14)

If ever needed, one may allow for the numbers M B and M F of bosonic and fermionic components of Φ to be different. We need to talk about transformations of integration variables. As long as we stick to separate transformations for bosonic and fermionic variables (i.e., forbid mixing of the two types) and further restrict ourselves to linear reshufflings of the fermionic variables, we just assemble the respective Jacobians (11.4.4) and (11.4.8) as     ∗ ∂(S ∗ , S) ∂(χ ∗ , χ ) ∂(S ∗ , S, η∗ , η) S , S, η∗ , η = det = Sdet . (11.4.15) J z ∗ , z, χ ∗ , χ ∂(z ∗ , z) ∂(η∗ , η) ∂(z ∗ , z, χ ∗ , χ ) The final form of the Jacobian, often called the Berezinian, also holds for transformations mixing bosonic and fermionic variables; of course, only such transformations are to be admitted which lead to M B new bosonic and M F fermionic variables; in particular, the new bosonic variables may have additive pieces even in the old fermionic ones. For a proof of the general validity of the last expression for the Berezinian, I refer the reader to Berezin’s book [6].

502

11 Superanalysis for Random-Matrix Theory

The compact version (11.4.13) of our superintegral is actually more general than the original, (11.4.12), inasmuch as it does not require the supermatrix F to be block diagonal with vanishing Bose–Fermi and Fermi–Bose blocks. To see this, we simply replace the integration variables Φ by Ψ = the Φ ∗ unchanged.  while leaving  Φ FΦ −1 The Berezinian of that transformation is J Ψ = (SdetF) ; in the remaining inte$ gral dΦ ∗ dΨ exp(−Φ † Ψ ) we may, in the sense of analytic continuation already alluded to in the previous subsections, reidentify Ψ with the complex conjugate of Φ, whereupon the integral takes the form (11.4.12) with F = 1 and is therefore † equal to unity. Of course, the Bose–Bose block FB B of F must obey FB B + FB B > 0 for the integral (11.4.13) to exist. It is both interesting and useful to extend Wick’s theorem to superanalysis, i.e., to “moments” of a “normalized Gaussian distribution” (Sdet F) exp(−Φ † FΦ). In analogy with our above reasoning for the bosonic and fermionic moment-generating functions (11.4.5) and (11.4.9), we avail ourselves of a superanalytic one by chang ∗ −1 (F ) ing integration variables in (11.4.13) as Φi → Φi − 2M i j Ψ j , Φi → j=1  ∗ −1 Φi∗ − 2M j=1 Ψ j (F ) ji to get  (SdetF)

dΦ ∗ dΦ exp(−Φ † FΦ) exp (Φ † Ψ + Ψ † Φ)

≡ exp (Φ † Ψ + Ψ † Φ) = exp(Ψ † F −1 Ψ ) .

(11.4.16)

By differentiating w.r.t. the auxiliary variables {Ψi , Ψi∗ }, we obtain the superanalytic generalization of (11.4.6) and (11.4.11), Φi Φ ∗j Φk Φl∗  = Φi Φ ∗j Φk Φl∗  + (−1)? Φi Φl∗ Φk Φ ∗j  = (F −1 )i j (F −1 )kl + (−1)? (F −1 )il (F −1 )k j ;

(11.4.17)

the unspecified sign factor is −1 if an odd number of commutations of fermionic variables is involved in establishing the order of indices ilk j from i jkl and +1 otherwise. Needless to say, that identity encompasses the purely bosonic (11.4.6) and the purely fermionic (11.4.11) as special cases.

11.4.4 Some Properties of General Superintegrals One more issue comes up with changes of integration variables in integrals like   d M B S d M F χ f (S, χ ) , (11.4.18) I = R

where d M B S = d S M B . . . d S2 d S1 , d M F χ = dχ M F . . . dχ2 dχ1 and where the M B bosonic integration variables Si , as well as their range of integration R, are purely numerical, i.e., contain no even nilpotent admixtures. According to (4.13.7) and (4.13.8), I is well defined provided that the ordinary integral remaining converges,

11.4

Superintegrals

503

once the Grassmannian one is done. We may, however, think of a change of the integration variables which brings in new bosonic variables with nilpotent admixtures even in the Grassmannians χi , formally S = S(z, χ ). The question arises as to what the original integration range R turns into and what is to be understood by the new bosonic part of the differential volume element. Berezin realized that consistency with the previously established rule (11.4.15) of changing integration variables requires us simply to ignore the nilpotent admixtures of the z i everywhere in the integrand, in d M B S, as well as in the new integration range, provided that the function f (S, χ ) and all of its derivatives vanish on the boundary of the original range R. Referring the interested reader to [7] for the general case, I confine myself here to sketching the consistency proof for M B = 1, M F = 2. Then, the integral (11.4.18) takes the form 



b

I =

dS

dχ2 dχ1 f (S, χ )

(11.4.19)

a

where S, a, b are purely numerical. The transformation S = z + Z (z)χ1 χ2

(11.4.20)

has the Jacobian d S/dz = 1 + Z (z)χ1 χ2 . Accepting Berezin’s rule that the new boundaries should still be a, b, we need to find the conditions for the integral 



b

I˜ =

dz

dχ2 dχ1

a

dS f (z + Z (z)χ1 χ2 , χ ) dz

(11.4.21)

to equal I in (11.4.19). Expanding the integrand, we get f (z + Z (z)χ1 χ2 , χ ) = (z,0) Z (z)χ1 χ2 , and thus f (z, χ ) + ∂ f ∂z 



dS  ∂ f (z, 0) f (z, χ ) + Z (z)χ1 χ2 dz ∂z a  b ∂ f (z, 0)Z (z) =I+ dz ∂z a = I + f (b, 0)Z (b) − f (a, 0)Z (a) .

I˜ =

b

dz

dχ2 dχ1

Obviously, now, I˜ = I provided f = 0 at the boundaries S = a, b. More caution is indicated when the integrand does not vanish at the boundaries of the range R, as the following example shows. Let the integrand f (S, χ ) in (11.4.19) be g(S + χ1 χ2 ) = g(S) + g (S)χ1 χ2 . The integral comes out as I = g(b) − g(a). If one insists in shifting the bosonic integration variable to z = S + χ1 χ2 , one should rewrite the integral by introducing unit step functions,

504

11 Superanalysis for Random-Matrix Theory

 ∞  I = d S dχ2 dχ1 g(S + χ1 χ2 )Θ(b − S)Θ(S − a) −∞  ∞  = dz dχ2 dχ1 g(z)Θ(b − z + χ1 χ2 )Θ(z − a − χ1 χ2 ) −∞  ∞    = dz dχ2 dχ1 g(z) Θ(b−z) + δ(b−z)χ1 χ2 Θ(z −a) − δ(z −a)χ1 χ2 −∞

= g(b) − g(a) .

(11.4.22)

Fortunately, we shall be concerned mostly with integrands that vanish, together with all of their derivatives, at the boundary of the integration range R and in that benign case, one enjoys the further rules 



∂ ( f g) = 0 , ∂ Si R   ∂ d MB S d MF χ ( f g) = 0 , ∂χi R d MB S

d MF χ

(11.4.23)

which are useful for integration by parts.

11.4.5 Integrals over Supermatrices, Parisi–Sourlas–Efetov–Wegner Theorem We shall have ample opportunity to deal with integrals over manifolds of nonHermitian supermatrices Q=

 ∗ aρ ρ ib

(11.4.24)

with real ordinary numbers a, b and a pair of Grassmannians ρ, ρ ∗ . In particular, for integrands of the form f (StrQ, StrQ 2 ) that vanish for a → ∞, b → ∞, we shall need the identity 

dadb ∗ dρ dρ . 2π (11.4.25) This is mostly referred to as the Parisi–Sourlas–Efetov–Wegner (PSEW) theorem, in acknowledgment of its historical emergence 4 [1, 8–11]. An important special case is I =

d Q f (Str Q, Str Q 2 ) = f (0, 0)

where

dQ =

4 Useful generalizations, in particular to higher dimensions, and a watertight proof can be found in Ref. [11].

11.4

Superintegrals

505



  1 d Q exp − Str Q 2 = 1 , 2

(11.4.26)

and the correctness of that latter$ identity is obvious from StrQ 2 = a 2 + b2 + 2ρ ∗ ρ. An immediate generalization is d Q exp (−c Str Q 2 ) = 1 with an arbitrary positive parameter c. To prove the general case (11.4.25), we need to recall from Sect. 11.3.5 that Q can be diagonalized as Q = U diag(q B , iq F )U −1 with U given in (11.3.32), but η = −ρ ∗ /(a − ib), η∗ = ρ/(a − ib). The eigenvalues read qB = a +

ρ∗ρ , a − ib

iq F = ib +

ρ∗ρ . a − ib

(11.4.27)

To do the superintegral in (11.4.25), it is convenient to change integration variables from a, b, ρ, ρ ∗ to q B , q F and the Grassmannian angles η, η∗ . Evaluating the Berezinian for that transformation, we get the measure dQ =

dq B dq F dη∗ dη . 2π(q B − iq F )2

(11.4.28)

The integral in search thus takes the form  I =

dq B dq F dη∗ dη f (q B − iq F , q B2 + q F2 ) 2π(q B − iq F )2

(11.4.29)

which is discomfortingly undefined: the bosonic integral diverges due to the pole in the measure while the Grassmannian integral vanishes. The following trick [11, 12] helps to give a meaning to the unspeakable 0 · ∞. One sneaks the factor exp (−cStr Q 2 ) into the integrand and considers the c-dependence of the integral  I (c) = =

d Q f (Str Q, Str Q 2 ) exp (−c Str Q 2 ) 

(11.4.30)

dq B dq F dη∗ dη f (q B − iq F , q B2 + q F2 ) exp {−c (q B2 + q F2 )} 2π (q B − iq F )2

which taken at face value, is as undefined for any value of c as for c = 0. However, the situation improves for the derivative d I (c)/dc if we differentiate under the integral. Due to q B2 + q F2 = (q B + iq F )(q B − iq F ), the pole of the integrand at the origin of the (q B − q F )-plane now cancels such that the still vanishing Grassmann integral enforces d I (c)/dc = 0. We conclude that I (c) itself is independent of c and may thus be evaluated√at, say, c → ∞. The latter limit is accessible after the transformation Q → Q/ 2c. Since the Berezinian of that latter transformation equals unity,

506

11 Superanalysis for Random-Matrix Theory



√ 1 d Q f [Str Q/ 2c, Str Q 2 /(2c)] exp (− Str Q 2 ) c→∞ 2  1 = f (0, 0) d Q exp (− Str Q 2 ) = f (0, 0) . (11.4.31) 2

I = lim

11.5 The Semicircle Law Revisited To illustrate the conciseness of the superanalytic formulation, I propose to resketch the derivation of the semicircle law using the newly established language. The average generating function (11.2.7) can be written as the superintegral  Z (E − , j) =

  dΦ ∗ dΦ exp − iΦ † FΦ = Sdet−1 F ,

F = Eˆ ⊗ 1 N − 1ˆ ⊗ H − Jˆ ⊗ 1 N

(11.5.1)

  + N d 2 Si ∗ dηi dηi . over the 2N -component supervector Φ = ηS with dΦ ∗ dΦ = i=1 π The three summands in the 2N × 2N matrix F have been written as direct products of 2 × 2 and N × N matrices, a hat distinguishes the former,  ˆE = E − 1 0

 0 = E − 1ˆ , 1

 Jˆ =

j 0

 0 . 0

(11.5.2)

The GUE average (11.2.11) reads 3 1 4 3 N λ2 4 ˜2 , exp{i Φ † 1ˆ ⊗ H Φ} = exp − (Φ † 1ˆ ⊗ H Φ)2 = exp − Str Q 2  2 ∗   ∗ 1 z z z η m m m m m ˜ = m (11.5.3) Q ∗ ∗ N m ηm z m m ηm η m and yields  Z (E − , j) =

3  4 2 ˜2 . ˜ +λ Q dΦ ∗ dΦ exp − N Str i( Eˆ − Jˆ ) Q 2

(11.5.4)

This demands a Hubbard-Stratonovich transformation for us to come back to a Gaussian superintegral over the supervector Φ. The resulting representation (11.2.14) of the average generating function can be seen as a fourfold superintegral over the elements of a 2 × 2 supermatrix   ∗ ˆ = a σ , Q σ ib

ˆ 2 = a 2 + b2 + 2σ ∗ σ , Str Q

ˆ = dadb dσ ∗ dσ , (11.5.5) dQ 2π

11.5

The Semicircle Law Revisited

507

and takes the form  Z (E − ,

j) =  = =



ˆ e− 2λ2 Str Qˆ dQ N

2



ˆ Q− ˆ Jˆ )φ −iφ † ( E−

∗

N

dφ dφ e

ˆ e− 2λ2 Str Qˆ Sdet−N ( Eˆ − Q ˆ − Jˆ ) dQ N

2

(11.5.6)

3  4 ˆ exp − N Str 1 Q ˆ − Jˆ ) . ˆ 2 + ln( Eˆ − Q dQ 2λ2

Here the original 4N -fold superintegral over the components of the supervector Φ and their conjugates could be replaced by the N th power of a fourfold integral over the two-component supervector φ = ηz . Needless to say, all matrices appearing in (11.5.6) are 2 × 2. While in Sect. 11.2 we proceeded by doing the two Grassmann integrals over ˆ rigorously and reserved a saddle-point approximation for the skew elements of Q the integral over the diagonal elements, here we shall treat all variables equally and propose a saddle-point approximation for the whole superintegral  Z (E − , j) =

ˆ Jˆ ) ˆ e−N A( Q, dQ ,

ˆ Jˆ ) = Str A( Q,

 1 ˆ − Jˆ ) . ˆ 2 + ln ( Eˆ − Q Q 2λ2

(11.5.7)

Before embarking on that adventure, it is well to pause by remarking that the foregoing integral must equal unity if we set Jˆ = 0,  Z (E − , 0) =

ˆ e−N A( Q,0) = 1, dQ ˆ

(11.5.8)

since unity is indeed the value of a block diagonal superdeterminant whose Bose and Fermi blocks are identical, cf (11.2.3) or (11.4.12). This is relevant since due to (11.2.4) we eventually want to set j and thus the matrix Jˆ equal to zero, 

 G(E − ) =

ˆ Str dQ

Jˆ 1 ˆ j Eˆ − Q

 e−N A( Q,0) . ˆ

(11.5.9)

Now, the spirit of the saddle-point approximation, we may appeal to the assumed largeness of N and pull the preexponential factor out of the integral and evaluate it at ˆ Then, the remaining superintegral ˆ 0 of the supermatrix Q. the saddle-point value Q yields unity, and we are left with  G(E − ) = Str

Jˆ 1 ˆ0 j Eˆ − Q

 .

(11.5.10)

508

11 Superanalysis for Random-Matrix Theory

ˆ 0 , we must solve the saddle-point equation To find Q 

 ˆ ˆ0 Q 1 1 Q − = . (11.5.11) = 0 =⇒ 2 2 ˆ ˆ ˆ ˆ0 λ λ E−Q E−Q ˆ 0 diagonalizable, we may start looking for diagonal solutions; there Assuming Q is indeed no undue loss of generality in that since all other solutions are accessible from diagonal ones by unitary transformations. For diagonal saddles, however, the saddle-point equation becomes an ordinary quadratic equation and yields two solutions (cf (11.2.23)) &   (11.5.12) Q ± = (E − /2) 1 ± 1 − (2λ/E − )2 . ˆ 0) = Str δ Q ˆ δ A( Q,

Precisely as in Sect. 11.2 we must evaluate the square root in Q ± = a± = ib± on the lower lip of the cut along the real axis from −2λ to +2λ in the complex energy plane. Concentrating immediately on (E/2)2 < λ2 , i.e., case (1) of Sect. 11.2.4 & 2 Q ± = a± = ib± = E/2 ∓ i λ − (E/2)2 . Four possibilities arise for the diagonal 2 × 2 matrix in search, 

Q+ 0

0 Q+



 ,

Q+ 0

0 Q−



 ,

Q− 0

0 Q+



 ,

Q− 0

0 Q−

 .

(11.5.13) According to (11.5.10) and (11.5.11), each of the first two would contribute Q + /λ2 , and the last two Q − /λ2 to the average Green function. Therefore, the first two must be discarded right away since the imaginary part of ImG(E − ) = π (E) must be positive. This corresponds with our discarding the saddle a+ in Sect. 11.2.4, arguing that the original contour of integration could not be continuously deformed so as to pass through that saddle without crossing a singularity. The question remains whether both of the last two contribute or only one of them and then which. To get the answer, we scrutinize the Gaussian superintegral over the fluctuations around ˆ 0) of the action, i.e., the second-order term the saddle. The second variation δ 2 A( Q, ˆ 0) − A( Q ˆ 0 , 0), reads ˆ of A( Q ˆ 0 + δ Q, in δ Q . ˆQ ˆ 0 )2 ˆ 2 1 (δ Q (δ Q) 2 ˆ δ A( Q 0 , 0) = Str − (11.5.14) 2 λ2 λ4 ˆ 0 . Denoting by δ 2 A−− the result and must be evaluated for the two candidates for Q for diag(Q − , Q − ) and by δ 2 A−+ that for diag(Q − , Q + ) (for simplicity at E = 0), 1 ˆ 2 = 1 (δa 2 + δb2 + δσ ∗ δσ ) , Str (δ Q) λ2 λ2 1 = 2 (δa 2 + δb2 ) . λ

δ 2 A−− = δ 2 A−+

(11.5.15)

Obviously saddle does not contribute at all (to leading order in $ now, the second ˆ exp(−N δ 2 A+− ) = 0, such that we arrive at N ) since dδ Q

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

G(E − ) =

 & 1 − 2 − (E − /2)2 /2 + i λ E λ2

for

509

(E/2)2 < λ2 ,

(11.5.16)

the result already established by a less outlandish calculation in Sect. 11.2; see (11.2.28). The reader is invited to go through the analogous reasoning for (E/2)2 > λ2 . At any rate, the semicircle law follows by taking the imaginary part. Incidentally, it is quite remarkable that the nonscalar saddle diag(Q − , Q + ) does not contribute since it would come with a whole manifold of companions: Indeed, a ˆ solving the saddle-point equation gives rise to further solutions U QU ˆ † with any Q † ˆ is not ˆ ˆ unitary transformation U , provided that Q = U QU , i.e., provided that Q proportional to the unit matrix. As a final remark on the saddle diag (Q − , Q + ), I mention that it corresponds to the saddle {a− , b+ } of Sect. 11.2.4 which there, too, did not contribute in leading order, due to the vanishing of the preexponential factor at the saddle. Should the previously uninitiated suffer from a little dizziness due to our use of the saddle-point approximation for a superintegral, cure might come from Problem 11.7.

11.6 The Two-Point Function of the Gaussian Unitary Ensemble We proceed to calculate Dyson’s two-point cluster function for the Gaussian unitary ensemble and show that it equals its counterpart for the circular unitary ensemble (see Sect. 4.14),  YGUE (e) = YCUE (e) =

sin π e πe

2 .

(11.6.1)

As in Sect. 11.2, I shall start with the Green function G(E) = N −1 Tr(E − H )−1 but now must consider the GUE average of the product of two such, G(E1 )G(E2 ). We shall see (in the beginning of the next Subsection) that average behaves differently, depending on whether the two complex energies lie on the same side or on different sides of the real axis. In the former case, the average product tends to the product of averages, N →∞

G(E 1 + iδ)G(E 2 + iδ) −→ G(E 1 + iδ) G(E 2 + iδ) ,

(11.6.2)

while in the latter case G(E 1 + iδ)G(E 2 − iδ) tends, apart from normalization, to a function of the single variable e = (E 1 − E 2 )N 

E + E  1 2 , 2

(11.6.3)

510

11 Superanalysis for Random-Matrix Theory

i.e., the energy difference measured in units of the mean level spacing at the center energy. Anticipating the validity of the factorization (11.6.2) and employing (11.2.2), we can immediately check the connected density-density correlator,    Δ(E 1 )Δ(E 2 ) = (E 1 ) − (E 1 ) (E 2 ) − (E 2 ) , to be given by the real part of the connected version ΔG(E 1 − i0+ )ΔG(E 2 + i0+ ) of the averaged product of two Green functions, Δ(E 1 )Δ(E 2 ) =

1 Re ΔG(E 1 − i0+ )ΔG(E 2 + i0+ ) . 2π 2

(11.6.4)

11.6.1 The Generating Function The product of two Green functions can be obtained from the generating function det(E 3 − H ) det(E 4 − H ) det(E 1− − H ) det(E 2+ − H ) det i(E 3 − H ) det i(E 4 − H ) = (−1) N det i(E 1− − H ) det(−i)(E 2+ − H )

Z (E 1− , E 2+ , E 3 , E 4 ) =

(11.6.5)

by differentiating as G(E 1− )G(E 2+ ) =

1 ∂ 2 Z (E 1− , E 2+ , E 3 , E 4 )  .  E 1 =E 3 ,E 2 =E 4 N2 ∂ E1∂ E2

(11.6.6)

This is analogous to (11.2.3) and (11.2.4), and the proof parallels that in (11.2.5): Each differentiation brings about one factor G and the determinants cancel pairwise once we set E 1 = E 3 , E 2 = E 4 . A final comment on the definition is in order. In the end, we shall be interested in (the limit of) real energy arguments. For the moment it is imperative, however, to place the energy arguments E 1− , E 2+ of the spectral determinants in the denominator away from and on different sides of the energy axis. Such precaution is not necessary for E 3 , E 4 which may be arbitray complex. Silly as it may appear to have sneaked various factors ±i in the last member of (11.6.5) (since they obviously cancel against the factor (−1) N ), such precaution pays, similarly as in Sect. 11.2, once we employ the Gaussian superintegral (11.4.12) to represent quotients of determinants,    † d 2 z 1i ∗ † dη1i dη1i exp i z 1 (H − E 1− )z 1 + η1 (H − E 3 )η1 Z = (−1) π i=1    , N  d 2 z 2i ∗ † † × dη2i dη2i exp i − z 2 (H − E 2+ )z 2 + η2 (H − E 4 )η2 , (11.6.7) π i=1  , N

N

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

511

with bosonic vectors z 1 , z 2 and fermionic vectors η1 , η2 , all having N components. Indeed, the i’s now conspire to ensure convergence of the bosonic Gaussian integrals, just as in (11.2.6). To save space, we introduce the (4N )-component supervector ⎛ ⎞ ⎛ ⎞ Φ1 z1 ⎜z 2 ⎟ ⎜Φ2 ⎟ ⎟ ⎜ ⎟ Φ=⎜ ⎝η1 ⎠ = ⎝Φ3 ⎠ η2 Φ4

(11.6.8)

such that the differential volume element dΦ ∗ dΦ comprises those in the preceding integral, (11.6.7). Writing the energy variables with their infinitesimal imaginary offsets as E α + i(−1)α 0+ , α = 1, 2, 3, 4 and employing the symbols L α where L 1 = L 3 = L 4 = −L 2 = 1 to accommodate the funny looking convergence ensurer L 2 = −1 in the exponent, we can write the generating function in the compact form

  4 Z = (−1) N dΦ ∗ dΦ exp i

α=1

  Φα† L α H − E α − i(−1)α 0+ Φα .

(11.6.9)

All is set now for the average over the GUE; the superhocus-pocus has served to move the random Hamiltonian matrix H upstairs into an exponent. Recalling the variances (11.2.9) and the Gaussian average of exponentials (11.2.10), in the spirit of (11.2.11), 

 2 1  † Φα L α H Φ α = exp − α=1 2 . 2 1 † † † † = exp − z 1 H z 1 − z 2 H z 2 + η1 H η1 + η2 H η2 (11.6.10) 2    λ2  ∗ ∗ Lα Lβ Φαm Φαn Φβn Φβm . = exp − 2N α,β m,n

  4 exp i

† Φ α L α H Φα



Once at work down-sizing bulky expressions, why not continue with 4 × 4 matrices L = diag(1, −1, 1, 1) ,

Eˆ = diag(E 1− , E 2+ , E 3 , E 4 ) ,

N  ∗ ˜ αβ = 1 Φαm Φβm Q N m=1

(11.6.11)

and write the averaged generating function analogously to (11.5.4) as  3 2 4  ˆ = (−1) N dΦ ∗ dΦ exp N Str − i Eˆ Q ˜ L − λ (Q ˜ L)2 . Z ( E) 2

(11.6.12)

512

11 Superanalysis for Random-Matrix Theory

˜ =Q ˜ † is Hermitian and has a non-negative Bose–Bose block. The supermatrix Q It remains to do the 8N -fold superintegral in (11.6.12) which, unfortunately, contains a quartic expression in the integration variables of the exponent. Taking a short breath, it will be good to realize that at the corresponding stage of the calculation of the mean Green function in Sect. 11.2, we had to employ the Hubbard–Stratonovich transformation to return to Gaussian integrals.

11.6.2 Unitary Versus Hyperbolic Symmetry Had we placed all energy arguments Eα of the generating function on one and the same side of the real axis, like Eα = E − , we would have arrived at (11.6.12) with all ˆ then the integrand would be invariant under Q ˜ → U QU ˜ † four L α = 1, i.e., L = 1; † with an arbitrary unitary supermatrix since StrU MU = StrM. I strongly recommend that the reader consider that case and run through all steps of Sect. 11.5 to check that everything goes through, the doubling of the dimension of the matri˜ Q ˆ notwithstanding. It is precisely because of that dimension doubling that ces Q, the saddle-point integral gives the average product G(E 1− )G(E 2− ) as the product of averages G(E 1− ) G(E 2− ), as claimed in (11.6.2) . The case is less trivial when the energy arguments lie on both sides of the real axis, Eα = E α + i(−1)α 0+ . Then the exponential in the superintegral (11.6.12) would be invariant under the transformation ˜ → T QT ˜ † Q

(11.6.13)

where a 4 × 4 supermatrix T satisfies T †LT = L

(11.6.14)

if Re Eˆ were proportional to the 4 × 4 unit matrix. Of course, we need to allow for E 1 = E 3 = E 2 = E 4 such that the “hyperbolic” alias “pseudounitary” symmetry (11.6.13) and (11.6.14) does not hold rigorously. But since the cumulant function in search decays to zero on an energy scale of the order of the mean level spacing which for the normalization of H chosen is of the order 1/N , the relevant difference of the two energies in question is very small, E 1 − E 2 = O(1/N ), and the hyperbolic symmetry does indeed hold to leading order in N . The name “hyperbolic” symmetry should be commented on. If L were the unit matrix, the restriction (11.6.14) would render T unitary and the ensuing unitary symmetry of the exponent (11.6.12) would be the one pertaining to the disconnected part G(E 1 ) G(E 2 ) of the two-point function, as discussed above. It is precisely the single negative element in the Bose block of L = diag(1, −1, 1, 1) that replaces the unitary symmetry with the pseudounitary alias hyperbolic one we need to deal with when after the cumulant function. I shall refer to supermatrices T with the property

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

513

T † L T = L as pseudounitary. The reader is invited to check that these matrices form a group (see Problem 11.9). ˜ L in (11.6.12) can For use later, I propose to show that the 4 × 4 supermatrix Q be diagonalized by a pseudounitary matrix. That diagonalizability arises due to the ˜ ˜ =Q ˜ † and the positivity of the Bose block of Q. Hermiticity Q Since L is diagonal and has the 2×2 unit matrix in the Fermi block, we may save space and labor and still retain the essence of the proof by first restricting ourselves to the toy problem of ordinary 2 × 2 matrices. For these, it is easy to see that the manifold M2 of Hermitian matrices Q = Q † for which Q L with L = diag (1, −1) is pseudounitarily diagonalizable is M2 :

Q = Q† =

 ∗ ac , c b

 |c|2
0. Direct calculation yields S † L S L = 10 01 √ det S > 0, we can renormalize as T = S/ det S such that T is indeed pseudounitary, T † L T = L. Positive matrices Q fall in our manifold M2 : Positivity of Q requires a > 0, b > 0, ab > |c|2 and is compatible with |c|2 < (a + b)2 /4 since ab ≤ (a + b)2 /4 = (a − b)2 /4 + ab. On the other hand, the case of negative radicand would still leave Q L diagonalizable by the above S, with complex eigenvalues q± ; but since the off-diagonal elements S12 = S21 also become complex, pseudounitarity is lost. Finally, a vanishing radicand yields det S = 0 and altogether precludes diagonalizability. To proceed from the toy problem to the manifold M4 of 4 × 4 supermatrices diagonalizable by pseudounitary transformations, we take two steps. The first is to let Q = Q † be 4×4 and block diagonal with the Bose block from M2 . Then, the toy argument goes through practically unchanged where T is also block diagonal; the

514

11 Superanalysis for Random-Matrix Theory †

toy problem teaches us that the Bose block of T is pseudounitary, TB B L B B TB B = L B B ; the Fermi block TF F is unitary since the Hermitian 2 × 2 matrix (Q L) F F = Q F F L F F = Q F F can be diagonalized by a unitary 2 × 2 matrix. Then, the 4 × 4 matrix T itself is pseudounitary in the sense T † L T = L. The second step of elevation amounts to showing that a 4 × 4 matrix Q L can be brought to block diagonal form by a pseudounitary matrix T . It is indeed easy to see that the job required is done by &



T =

T

†

& =

 1 − γ †γ k & γ † , −γ k 1 − γ kγ †

(11.6.17)

 1 − kγ † γ & −kγ † . γ 1 − γ kγ †

(11.6.18)

Here, I have given the typographically less voluminous name k to the 2 × 2 block L B B = diag(1, −1) ≡ k and called γ and γ † a pair of 2 × 2 matrices with four independent anticommuting entries each, viz., γi j and (γ † )i j = γ ji∗ , such that γ ji∗ is complex conjugate to γ ji . There are eight independent parameters in T

and T † ; these are all Grassmannians and just right in number to make the equations (T † Q L T ) B F = (T † Q L T ) F B = 0 generically solvable. Pseudounitarity, on the other hand, is built into (11.6.17) and (11.6.18). As block diagonality is thus achieved, the Bose and Fermi blocks and their eigenvalues will have changed only by acquiring nilpotent additions; such additions cannot foul up the positivity of the Bose block since positivity of an eigenvalue is a property of its numerical part. Further preparation for our intended use of the above results is to be undertaken. To ensure convergence of certain Gaussian integrals, we shall need to analytically continue the 4 × 4 matrices Q such that their Fermi-Fermi blocks QFF become anti-Hermitian rather than Hermitian, in analogy with what we had to do in (11.5.5) (where we put b → ib). Then, we can still employ the matrices T and T † to bring Q L to block diagonal form. If we stick to the conventional definition of complex conjugation for Grassmannians, the pseudounitarity of T is lost. However, according to the model of the discussion in the concluding paragraph of Sect. 11.3.5, we may restore pseudounitarity by appropriately redefining complex conjugation of Grassmannians; in the following, we shall tacitly imagine that such restoration is implemented together with the continuation of QFF to anti-Hermiticity. Finally, we turn to the Hubbard–Stratonovich transformation needed to turn the integral over the supervectors Φ, Φ ∗ in (11.6.12) into a Gaussian. We cannot uncritically employ the usual formula which would involve an unrestricted integral over an auxiliary 4 × 4 supermatrix. The correct procedure was pioneered in a rather different context by Sch¨afer and Wegner [13], later adapted to superanalytic needs by Efetov [1], and comprehensively discussed in Ref. [2]; it amounts to restricting the integration range for the auxiliary supermatrix to the manifold M4 . To explain the essence of the idea, it suffices once more to consider ordinary 2 × 2 matrices and prove the integral identity

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

 ˜ L)2 = − exp − 12 tr( Q

 M2

515

 ˜L d Q exp − 12 tr(Q L)2 + i trQ L Q

(11.6.19)

˜ ∈ M2 ; as usual, the integration measure is d Q = dadbd 2 c/(2π 2 ). for matrices Q The minus sign signals that the integral is far from being a usual Gaussian. For the ˜ L to diagonal form by a suitable pseudounitary transformaproof, we first bring Q −1 ˜ tion, T Q L T = diag(q+ , q− ). Subsequently, when T is absorbed in the integration variable Q, neither the integration range nor the measure is changed. The remainder of the proof is a straightforward calculation, 

∞



−∞

 d 2 c exp − 12 (a 2 + b2 − 2|c|2 ) + i(aq+ − bq− )

db

2 |c|2 0 , 2 2 where the dots refer to nilpotent stuff. The inequality does indeed ensure convergence for the integral in (11.6.22). To achieve minor embellishments in(11.6.22), we invoke Str ln(iL Eˆ − iL Q L) = ln Sdet(L Eˆ − L Q L) = ln SdetL + ln Sdet( Eˆ − Q L) and SdetL = −1 and even use Q L rather than Q as integration variables. We confront  ˆ =− Z ( E)

3  1   4 2 ˆ −Q . d Q exp − N Str Q + ln E 2λ2 M4 L

(11.6.24)

The very appearance of this integral invites a saddle-point approximation. In contrast to previous appearances of that approximation, a whole manifold of saddle points is lurking around the corner, so we ought to proceed with care. First, we are interested in E 1 − E 2 = O(1/N ) and make such smallness manifest ˜ E˜ − Q)−1 where E˜ = by writing Str ln( Eˆ − Q) = Str ln( E˜ − Q) + Str( Eˆ − E)( diag(E − , E + , E − , E + ) and the energy E is suitably chosen such that Eˆ − E˜ = O(1/N ). The diagonal matrix E˜ differs from E 1ˆ only by the various imaginary infinitesimals; wherever these latter are dispensable, I shall indulge in the shorthand E˜ → E 1ˆ → E. With these conventions,  ˆ =− Z ( E) A(Q) = Str

   d Q exp[−N A(Q)] exp − N Str Eˆ − E˜

 1   Q 2 + ln E˜ − Q 2 2λ

1 E˜ − Q (11.6.25)

where the first exponential enjoys the pseudounitary symmetry discussed in the previous subsection; the second exponential, on the other hand, is effectively independent of N and thus immaterial for the saddle-point equation 1 Q = λ2 E−Q

(11.6.26)

which is identical in appearance with that found for the Green function G(E − ), (11.5.11). In analogy to that previous case, we may start looking for diagonal solutions and eventually extend to nondiagonal ones by pseudounitary transformations. But while we were previously led to a “scalar” saddle, i.e., a matrix proportional to the unit matrix, a nonscalar saddle is waiting for us now. To find a suitable solution

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

517

Q s of (11.6.26), it is well to take maximum benefit from what we already know; therefore, we look at the two-point function that results from taking derivatives w.r.t. E 1 and E 2 and then setting E 1 = E 3 , E 2 = E 4 , E = (E 1 + E 2 )/2, G(E 1− )G(E 2+ ) =

 d Q exp[−N A(Q)] ×

(11.6.27)

3 4  Q 11 Q 22 ˆ − E)( ˜ E˜ − Q)−1 . exp − N Str ( E λ2 λ2

The factors Q 11 /λ2 and Q 22 /λ2 correspond to G(E 1− ) and G(E 2+ ), respectively, such that the principal suspect of being the relevant saddle is5 & Q s = E/2 + iΛ λ2 − (E/2)2 , Λ = diag(1, −1, 1, −1) ;

(11.6.28) (11.6.29)

indeed, the two positive entries correspond to the saddle previously found for the advanced Green function G(E − ), and the two negative ones would have resulted had we chosen to evaluate the retarded Green function G(E + ); moreover, it is reasonable that the Fermi block in Λ should equal the Bose block since we might as well generate the two-point function by taking derivatives w.r.t. E 3 and E 4 . To elevate the principal suspect (11.6.28) and (11.6.29) to the doubtless exclusive relevant diagonal saddle, we must secure solid evidence, rather than rely on the intuitive reasoning just put forth. After all, there are sixteen possibilities for assigning a sign to the square root term in the four diagonal elements. I shall come back to that task at the end of this subsection. Inasmuch as the saddle (11.6.28) and (11.6.29) is not proportional to the unit matrix, it gives rise to a whole manifold of saddle points since it is accompanied by the following other solutions of the saddle-point equation: & Q s (T ) = T Q s T −1 = E/2 + iT ΛT −1 λ2 − (E/2)2 ,

(11.6.30)

where T is any pseudounitary matrix. Accounting for all of these saddle points amounts to an integration to which we must turn now. The manifold of distinct saddle points is not quite as voluminous as the group of pseudounitary 4 × 4 supermatrices T (which is commonly called U (1, 1/2); the dash may be said to separate, in view of the definitions (11.6.14) of pseudounitarity and (11.6.11) of the diagonal matrix L, the 2 × 2 Bose–Bose block of L where the pseudounitarity is rooted from the 2 × 2 Fermi–Fermi block where unitarity reigns). For instance, Q 0 = T ΛT −1 is unchanged if a given T is multiplied from the right by any diagonal unitary 4×4 matrix. The latter matrices form the subgroup

5 In specifying the saddle (11.6.28), we have immediately restricted ourselves to the energy interval |E/2| < λ to which the spectrum is confined by the semicircle law.

518

11 Superanalysis for Random-Matrix Theory

U (1)×U (1)×U (1)×U (1), and the integration range in question can thus be reduced to the coset space U (1, 1/2)/U (1) × U (1) × U (1) × U (1). Actually, the group of pseudounitary matrices commuting with Λ is larger than just the one of the diagonal matrices such that the integration range must be further reduced. To that end, it is convenient to reorder our 4 × 4 matrices. We agreed on  Q=

QBB QBF QFB QFF



Eˆ = diag(E 1− , E 2+ , E 3 , E 4 )

and

(11.6.31)

with 2 × 2 blocks Q B B etc.; that convention is sometimes referred to as the “Bose– Fermi” notation. Now, we permute the four rows and columns according to 1234 → 1324 such that E˜ → diag(E − , E − , E + , E + ) Eˆ → diag (E 1− , E 3 , E 2+ , E 4 ) Λ → diag (1, 1, −1, −1) L → diag (1, 1, −1, 1)

(11.6.32)

˜ E, ˆ Λ, L , Q. Again, the matrices Q can be but shall retain the previous names E, written as 2 × 2 matrices where all four elements are 2 × 2 supermatrices; the upper left block then refers to “advanced” Green functions where ImE < 0, and the lower right one to “retarded” functions where ImE > 0; therefore, the reordering in question is referred to as the “advanced-retarded” notation. Whenever suitable, we shall write Λ and L as 2 × 2 matrices with 2 × 2 elements as well,   1 0 Λ= , 0 −1

  1 0 L= , 0 −k

  1 0 k= . 0 −1

(11.6.33)

With that reordering done, we may imagine Q block diagonalized by a suitable pseudounitary matrix T0 ,  Q = T0

 Qa 0 T −1 , 0 Qr 0

(11.6.34)

where Q a , Q r are not yet diagonal. Of the sixteen independent parameters in Q (yes, sixteen indeed, since we started from a Hermitian 4 × 4 matrix; neither the analytic continuation to an anti-Hermitian Fermi block nor the subsequent multiplication with L could change that number), eight are accommodated in the blocks Q a , Q r and eight in the matrix T0 which brings about the block diagonalization. We shall see in Sect. 11.6.4 that the integral over blocks Q a , Q r is taken care of by the saddle-point approximation and that the matrices T0 in effect span the saddle-point manifold. Then, it is also clear which group, larger than U (1)×U (1)×U (1)×U (1), leaves Λ = diag(1, 1, −1, −1) invariant: It is U (2)×U (1, 1), formed by block diagonal matrices with one unitary and one pseudounitary 2 × 2 supermatrix along the

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

519

diagonal; all such matrices indeed commute with Λ. Consequently, the integration manifold for T0 is the eight-parameter coset space remaining when the group of pseudounitary 4 × 4 matrices is deprived of the subgroup U (1/1) × U (1/1)6 . We may parametrize the block diagonalizing matrix as7 T0 =

√ 1 + Γ † kΓ Γ

† √ Γ k 1 + Γ Γ †k

 (11.6.35)

such that the eight parameters in T0 are located within a 2 × 2 supermatrix Γ and its adjoint Γ † . We easily verify that † T0

√ 1 + Γ † kΓ = kΓ

√

Γ† 1 + kΓ Γ †

 ,

T0−1 = T0 |Γ →−Γ

(11.6.36)

†

as well as pseudounitarity, T0 L T0 = L. It is worth noting that with our new ordering of rows and columns, the supermatrix rule of adjunction reigns only within the supermatrix Γ . An important benefit of the parametrization (11.6.35) and (11.6.36) is that all four 2 × 2 blocks within T0 can be diagonalized simultaneously with the help of a unitary 2 × 2 supermatrix U and a pseudounitary one V , i.e., U † U = 1 and V † kV = k. Indeed, if such a pair yields Γ = V tU −1

with

t=

  tB 0 0 tF

(11.6.37)

we immediately have Γ † k = U t ∗ kV −1 ,

Γ † kΓ = U t ∗ ktU −1 ,

Γ Γ † k = V t ∗ tkV −1 . (11.6.38)

Moreover, T0 takes the simple form T0 =

  √    −1 U 0 1 + kt ∗ t √ kt ∗ 0 U . 0 V 0 V −1 t 1 + kt ∗ t

(11.6.39)

We may conclude that the eigenvalue t F is bounded in modulus, |t F |2 ≤ 1, whereas t B is unbounded. Thus, our saddle-point manifold U (1, 1/2)/U (1/1) × U (1/1) is revealed as noncompact. To count parameters, we note that the complex eigenvalues accommodate four, whereas the remaining four sit in the diagonalizing 2 × 2 supermatrices U and V . These matrices can be chosen as   It is well to note that for 2 supermatrices ρa ξb no difference between unitary and pseudounitary needs to be made; the apparent difference is undone by redefining complex conjugation of the Grassmannians as ρ ∗ → −ρ ∗ , ξ ∗ → −ξ ∗ .

6

This is similar in structure to T in (11.6.17); note that we have reordered rows and columns in between.

7

520

11 Superanalysis for Random-Matrix Theory

√ U= √ V=

1+ρρ ∗ ρ∗ 1−σ σ ∗ iσ ∗

  ∗ √ ρ ∗ = 1+ρρ∗ /2 ρ 1+ρ ρ √ iσ 1−σ ∗σ



 1−σ σ ∗ /2 = iσ ∗

  0 ρ = exp 1+ρ ∗ρ/2 ρ∗ 



0 iσ = exp 1−σ ∗σ/2 iσ ∗

ρ 0

 (11.6.40)  iσ 0

with Grassmannians ρ, ρ ∗ , σ, σ ∗ ; the unitarity U † U = 1 and the pseudounitarity V † kV = k are readily checked. Anticipating Sect. 11.6.5, we note for instant use the integration measure (qaB − qrB )2 (qaF − qrF )2 (qaB − qrF )2 (qaF − qrB )2 ≡ d Q a d Q r dμ(T0 ) F(Q a , Q r ) ;

d Q = d Q a d Q r dμ(T0 )

(11.6.41)

here, d Q a , d Q r denote the product of differentials of the respective four matrix elements which may be thought of as replaced by the product of the (bosonic) eigenvalue differentials dqaB dqaF dqrB dqrF and of the differentials of the Grassmannian “angles” in the diagonalizing matrices and the Berezinian for the diagonalization; finally, dμ(T0 ) is the integration measure of the coset space spanned by the T0 characterized above, dμ(T0 ) =

d 2 t B d 2 t F dρ ∗ dρdiσ ∗ diσ . π 2 (|t B |2 + |t F |2 )2

(11.6.42)

Finally, we return to the question why we could disregard all but one of the diagonal saddles, viz., (11.6.28) and (11.6.29). To that end, we consider fluctuations δ Q around the diagonal saddles. These can be split into “soft modes” that remain within a symmetry-induced manifold around a given diagonal saddle and “hard modes” transverse to that manifold. Only the hard modes can be integrated over by the saddle-point approximation and should be checked for their contribution to the two-point function. We identify these hard modes from (11.6.34) as δ⊥ Q = T

  δ Qa 0 T −1 0 δ Qr

(11.6.43)

where T is any of the pseudounitary transformations that lead from a diagonal saddle-point matrix to any block diagonal matrix through (11.6.34). Checking the contributions of those hard modes for all sixteen diagonal saddles is more work than is advisable to display here [2]. I resort to a more effective, albeit somewhat frivolous, procedure and indiscriminately admit arbitrary fluctuations  δQ =

δa δσ † δσ δib

 (Bose−Fermi notation)

(11.6.44)

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

521

with 2 × 2 blocks δa etc.. Asin Sect. 11.5, we may inspect the second variation δ 2 A of the action A(Q) = Str Q 2 /(2λ2 ) + ln( E˜ − Q) ; note that the action A(Q) in (11.6.25) is the same function of Q as the action A(Q, J = 0) in (11.5.7); thus, the second variation is still given by (11.5.14). Here, we insert the suspects, diag (Ba , Br , Fa , Fr ) with Ba,r = ±1, Fa,r = ±1, of which that where Ba = Fa = 1, Br = Fr = −1 is our principal one. In evaluating the supertrace of the present 4 × 4 supermatrices, we may, as in Sect. 11.5, choose E = 0, just to avoid unnecessarily lengthy formulae, and thus get ∗ δaar (11.6.45) λ2 δ 2 A(Ba , Br , Fa , Fr ) = (δaaa )2 + (δarr )2 + (1 + Ba Br )δaar 2 2 ∗ δbar +(δbaa ) + (δbrr ) + (1 + F a F r )δbar ∗ +(1 + Ba Fa )δσaa δσaa + (1 + Br Fr )δσrr∗ δσrr ∗ ∗ +(1 + Ba Fr )δσra δσra + (1 + Br Fa )δσar δσar .

To keep the further effort within reasonable limits, I declare the two Bose entries, Ba = −Br = 1, nonnegotiable and appeal to our previous experience: In Sects. 11.2.4 and 11.5 the saddles diag(a+ , ib± ) = diag(Q + , Q ± ) could not contribute to G(E − ) (just as diag(a− , ib± ) = diag(Q − , Q ± ) not to G(E + )) since the original path of integration could not be deformed so as to pass through them without crossing a singularity of the integrand. It remains to check on the four possibilities Fa = ±1, Fr = ±1, of which Fa = −Fr = 1 corresponds to our principal suspect. The four cases give rise to the following four second variations: (i) F = B = diag (1, −1), ∗ δσaa + 2δσrr∗ δσrr , λ2 δ 2 A+− = (δaaa )2 + (δarr )2 + (δbaa )2 + (δbrr )2 + 2δσaa

(ii) F = −B = diag (−1, 1), ∗ ∗ λ2 δ 2 A−+ = (δaaa )2 + (δarr )2 + (δbaa )2 + (δbrr )2 + 2δσra δσra + 2δσar δσar ,

(iii) F = 1, ∗ ∗ λ2 δ 2 A++ = δaaa2 + δarr2 + δbaa2 + δbrr2 + 2δbar2 + 2δσaa δσaa + 2δσra δσra ,

(iv) F = −1, ∗ λ2 δ 2 A−− = δaaa2 + δarr2 + δbaa2 + δbrr2 + 2δbar2 + 2δσrr∗ δσrr + 2δσar δσar .

When attempting to do the Gaussian integrals over the fluctuations δ Q with the (negative of the) foregoing quadratic forms in the exponents, we must pay a price for not having restricted δ Q to hard modes: Each missing commuting variable in a quadratic form gives rise to a factor ∞, and each missing Grassmann variable to a factor 0. Thus, cases (i,ii) formally have the weight (∞ × 0)2 , whereas cases (iii,iv) have the “smaller” weight ∞ × 02 . We shall presently convince ourselves

522

11 Superanalysis for Random-Matrix Theory

that each undefined product (∞ × 0) in these weights actually has the value 1 such that cases (iii,iv) do not contribute at all while case (i), i.e. the principal suspect, and case (ii) give identical contributions. To see two diagonal saddles with vanishing contributions justifies having discarded them above. But to find case (ii) as of equal weight as our principal suspect is a worry we must ease. The diagonal saddle from case (ii) lies on the manifold generated pseudounitarily from the principal suspect and therefore is automatically accounted for when integrating over the said manifold. To check on that claim, we must specify the pseudounitary matrix relating the two saddles. It is easy to verify that ⎛

1 ⎜0 T =⎜ ⎝0 0

0 0 0 1

0 0 1 0

⎞ 0 −1 ⎟ ⎟ 0 ⎠ 0

(advanced-retarded notation)

(11.6.46)

  is indeed of the structure (11.6.35) with Γ = 00 01 and yields (note that the advancedretarded notation is used) T diag (1, 1, −1, −1)T −1 = diag (1, −1, −1, 1). Now, the finite value unity of the seemingly undefined products ∞×0 is checked at leisure in a way reminiscent of our “proof” of the Parisi–Sourlas–Efetov–Wegner theorem in Sect. 11.4.5: We sneak a parameter c < 1 into those quadratic terms of the second variation (11.6.45) which would otherwise vanish for a given diag∗ ∗ δaar → (1 + cBa Br )δaar δaar . Then, onal saddle in the fashion (1 + Ba Br )δaar the Gaussian superintegrals for cases (i,ii) equal unity, independently of c; cases (iii,iv), however, similarly yield vanishing superintegrals; the auxiliary parameter c can eventually approach unity.

11.6.4 Implementing the Zero-Dimensional Sigma Model Having put the integration measure d Q and the integration range U (1, 1/2)/U (1/1)× U (1/1) at our disposal, now we can do the integrals in the two-point function (11.6.27). We employ the reordered matrices and write G(E 1− )G(E 2+ )

 =  ×  ×





 Q 2a − d Q a exp −N Str + ln(E − Q a ) (11.6.47) 2λ2

 2  Qr + + ln(E − Q ) F(Q a , Q r ) d Q r exp −N Str r 2λ2  ˆ E˜ − Q)−1 Q aB B Q rB B . dμ(T0 ) exp N Str ( E˜ − E)( λ2 λ2

Here, Q aB B , Q rB B are the Bose–Bose entries of the 2 × 2 supermatrices Q a , Q r . The exponentials in the first and second lines involve only Q a and Q r . Both of these exponentials are U (1/1) invariant such that we confront, with respect to both

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

523

Q a and Q r , a saddle-point integral analogous to that encountered for the average Green function in Sects. 11.2 and 11.5. In particular, (11.5.7) and (11.5.8) apply (with J = 0), since the prefactor F(Q a , Q r ) equals unity at the saddle point; indeed, at the saddle (11.6.28), yields s s = qaF = qaB

E1 2

s s qrB = qrF =

E2 2

⎫ % + i λ2 − ( E21 )2 = λ2 G(E 1− )⎬ % =⇒ F s (Q a , Q r ) = 1 . (11.6.48) + ⎭ E2 2 2 2 − i λ − ( 2 ) = λ G(E 2 )

The foregoing integrals over Q a and Q r are thus both equal to unity. Denoting by & Q s (T0 ) = T0 Q s T0−1 = E/2 + iT0 ΛT0−1 λ2 − (E/2)2 = E/2 + iπ λ2 (E) T0 ΛT0−1

(11.6.49)

the general member of the saddle-point manifold, we write the remaining integral over the coset space spanned by the matrices T0 as G(E 1− )G(E 2+ ) ∼



×

 s ˆ Q (T0 ) dμ(T0 ) exp N Str( E˜ − E) λ2  s  s Q (T0 ) 1B B Q (T0 ) 2B B λ2

λ2

(11.6.50)

which constitutes, as already announced above, Efetov’s zero-dimensional supermatrix sigma model. Now, the exponential has an argument of order unity, as N → ∞, due to E − Eˆ = O(1/N ), and this is why we cannot invoke any further largeN approximation in doing the integral over T0 . The asymptotic-equality sign “∼” indicates the loss of a correction of relative weight 1/N in the saddle-point approximation which is finally implemented here; the reader is asked to keep that loss in mind when the equality sign is used again in the following. It may be helpful to underscore the fact that on the saddle manifold spanned by the Q s (T0 ) the eigenvalues are fixed as the entries of the diagonal matrix Q s = ¯ The manifold is exhausted as the “angles” in the diagonalizing E/2 + iπ λ2 (E)Λ. matrix T0 vary. I would also like to repeat that the matrices T0 do not span the whole group of pseudounitary matrices for two reasons. First, the subgroup of diagonal unitary 4 × 4 matrices must be taken out. Second, due to the double degeneracy of the eigenvalues, all block diagonal matrices with invertible 2 × 2 matrices commute with Λ and must therefore be taken out as well. I now recall from (11.6.4) that we are really after the connected two-point function ΔG(E 1− )ΔG(E 2+ ) = G(E 1− )G(E 2+ ) − G(E 1− ) G(E 2+ ). When subtracting the product of averages G(E 1− ) G(E 2+ ) from both sides of (11.6.47) and (11.6.50), we Qs Qs can pull that product into the integrands such that it is the term λaB2 B λrB2 B from which the subtraction is made. Such manipulation is justified by the identity

524

11 Superanalysis for Random-Matrix Theory

1 = Z (E 1− , E 2+ , E 1− , E 2+ )  3  Q2 4 a − + ln(E − Q ) = d Q a exp − N Str a 2λ2  3  Q2 4 r + × d Q r exp − N Str + ln(E − Q ) F(Q a , Q r ) r 2λ2   ˆ − Q)−1 (11.6.51) × dμ(T0 ) exp N Str( E˜ − E)(E

which is obvious from the definition (11.6.5) of the generating function. That identity remains intact, to leading order in N , when the integrations over Q a and Q r are done in the saddle-point approximation, 

 Q s (T0 )  ˜ ˆ ∼ 1, dμ(T0 ) exp N Str( E − E)  λ2 E 3 =E 1 ,E 2 =E 4 

(11.6.52)

whereupon the connected two-point function can be written as ΔG(E 1− )ΔG(E 2+ )

 =

 s ˆ Q (T0 ) dμ(T0 ) exp N Str( E˜ − E) (11.6.53) λ2 5   Q s (T0 ) 1B B Q s (T0 ) 2B B − G(E 1− ) G(E 2+ ) . × λ2 λ2

Now, I propose to reveal explicitly the dependence of the integrand on the integration variables t B , t F , ρ, ρ ∗ , σ, σ ∗ . First turning to the supertrace in the exponent, we realize, with the help of (11.6.49), that it must be proportional to E 1− − E 2+ since both Λ and the unit matrix have vanishing supertraces. To fully evaluate that supertrace we insert the parametrization (11.6.39) and (11.6.40) of T0 and ˆ ΛT −1 = −iπ (E) (E − − ˆ Q s (T2 0 ) ] = iπ (E)Str( E˜ − E)T thus get Str[( E˜ − E) 1 λ  20 0 2  + −1 − + E 2 )/2 StrΛT0 ΛT0 = −i2π(E)(E 1 − E 2 ) |t B | + |t F | . At this point, the familiar spacing variable e from (11.6.3) comes into play with a negative imaginary infinitesimal attached to it, and the exponential in (11.6.53) takes the form  s    ˆ Q (T0 ) = exp −i2πe− |t B |2 + |t F |2 . exp N Str( E˜ − E) λ2

(11.6.54)

The absence of the Grassmannians ρ, ρ ∗ , σ, σ ∗ from the exponential (11.6.54) is noteworthy since it allows us to discard everything in the curly bracket in (11.6.54) except the term proportional to the product ρρ ∗ σ σ ∗ . To find that term, once more we invoke the parametrization (11.6.39) and (11.6.40) of T0 and get

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

  Q 1B B /λ2 = G(E 1− ) + i2π(E 1 ) U ktt ∗ U −1 B B ,   Q 2B B /λ2 = G(E 2+ ) − i2π(E 2 ) V ktt ∗ V −1 B B ,     U ktt ∗ U −1 B B = |t B |2 + |t B |2 + |t F |2 ρρ ∗ ,     V ktt ∗ V −1 B B = |t B |2 − |t B |2 + |t F |2 σ σ ∗ .

525

(11.6.55)

Now, we see that the desired term in the prefactor in (11.6.54) is proportional  2 2 to |t B | + |t F |2 which factor cancels against the denominator in the integration measure (11.6.42). By doing the fourfold Grassmann integral, we get 

d 2tF d 2tB π |t F |≤1 π    × exp − i2π |t B |2 + |t F |2 e− .

ΔG(E 1− )ΔG(E 2+ ) = (2π )2 (E 1 )(E 2 )

(11.6.56)

Herein, t B ranges over the whole complex plane and this is why the negative imaginary infinitesimal addition to the spacing variable had to be taken along; the restriction of t F to the unit disc around the origin of the complex t F plane comes with the parametrization (11.6.39) and was already noted above. With these remarks in mind, we evaluate the integrals and, after taking the real part according to (11.6.4), find the two-point cluster function of the GUE, Δ(E 1 )Δ(E 2 ) = δ(e) − (E 1 )(E 2 )



sin π e πe

2 = δ(e) − YGU E (e) ,

(11.6.57)

which equals that of the circular unitary ensemble derived in (4.14.3).

11.6.5 Integration Measure of the Nonlinear Sigma Model To fulfill the promise of deriving the integration measure (11.6.41) and (11.6.42) for the saddle-point manifold, we start with some $ warm-up exercises: (i) The first is to return to an integral dΦ ∗ dΦ f (Φ, Φ ∗ ) over a supervector   Φ = ηS and change the integration variables to a new complex conjugate pair of supervectors Φ , Φ ∗ according to Φ = Φ(Φ ). We had convinced ourselves of ∗ dΦ ∗ dΦ = dΦ ∗ dΦ J ( ΦΦ ∗ )J ( ΦΦ ) with the Berezinian J(

Φ ) = Sdet Φ



∂ Φj ∂Φi

 ≡ Sdet DiTj = Sdet Di j ;

(11.6.58)

∂ see (11.4.15). The supermatrix of derivatives D T = ∂Φ

Φ also determines the transformation of the elementary squared length in the space of supervectors, (δΦ, δΦ) =    ∗

T

δΦ δΦ . Let δΦ = δΦ D = D δΦ i i ij i j j i j ji j , where transposition is, of

526

11 Superanalysis for Random-Matrix Theory

course, meant in the supermatrix sense to account for the anticommutativity of a fermionic Di j with a fermionic δΦ j . Then, the squared length element reads (δΦ, δΦ) = (DδΦ , DδΦ ) = (δΦ , D † DδΦ ) ≡ (δΦ , GδΦ ) .

(11.6.59)

The “metric” supermatrix G = D † D occurring herein has as its superdeterminant the square of the Berezinian under discussion, SdetG = Sdet D † D = |Sdet D|2 = J



Φ ∗, Φ Φ ∗, Φ



  2  Φ  . =  J Φ 

(11.6.60)

Thus, we may say that the Berezinian is related to the metric in superspace. (ii) In a second warm-up, we exploit the connection between metric and Berezinian for an integral over a Hermitian 2 × 2 supermatrix,  I =

d H f (H ) , 

H = dH =

a ρ∗ ρ b

 = U

  hB 0 U −1 0 hF

(11.6.61)

1 dadbdρ ∗ dρ . π

As new integration variables, we wish to employ the two eigenvalues h B , h F and the Fermionic “angles” η∗ , η in the diagonalizing matrix U , given in (11.3.31) and (11.3.32). A straightforward calculation yields the Berezinian (see Problem 11.11) Sdet D = Sdet

1 ∂(a, b, ρ ∗ , ρ) . = ∂(h B , h F , η∗ , η) (h B − h F )2

(11.6.62)

A shortcut to that result involves the metric in the space of supermatrices engendered by the squared length element 2  (δ H, δ H ) ≡ Str (δ H )2 = Str δU hU −1 + U δhU −1 + U hδU −1  2 = Str [U −1 δU, h] + δh   = Str [U −1 δU, h]2 + (δh)2 .

(11.6.63)

Now filling in U, U −1 from (11.3.32), 2  2 2   Str (δ H )2 = δh B − δh F + 2 h B − h F δη∗ δη

(11.6.64)

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

⎛ 1 0 0   ⎜0 −1 0 ∗ = δh B , δh F , δη , δη ⎜ ⎝0 0 0 (h B 0 0 −(h B − h F )2 ⎛ ⎞ δh B   ⎜δh F⎟ ∗ ⎟ = δh B , δh F , δη , δη G ⎜ ⎝δη∗⎠ . δη

527

⎞⎛ ⎞ 0 δh B ⎟⎜δh F⎟ 0 ⎟⎜ ⎟ − h F )2⎠⎝δη∗⎠ 0 δη

But since the integration measure in the present case is d H , rather than dΦdΦ ∗ as before, the superdeterminant of the metric matrix G in the above length element is the square of the Jacobian (11.6.62). The reader might appreciate the greater ease of the “metric way” of getting that Jacobian, compared to straightforwardly evaluating the superdeterminant of the derivative matrix D. With so much lure laid out, now we go the metric way toward the principal goal of this subsection, derivation of the integration measure (11.6.41) and (11.6.42). As in the second warm-up above, the problem is to change integration variables from the matrix elements of the 4 × 4 matrix Q to the eigenvalues qaB , qaF , qrB , qrF and the twelve parameters in the diagonalizing matrix T . The latter number is twelve even though the group U (1, 1/2) of pseudounitary matrices has sixteen parameters since, as remarked before, if some matrix T diagonalizes Q, Q = T qT −1 , so does any T = T e−iφ with φ a real diagonal matrix; thus the set of matrices relevant for the envisaged change of integration variables is the coset space U (1, 1/2)/U (1) × U (1) × U (1) × U (1). We had argued further that that set can be split into the eight-parameter coset space U (1, 1/2)/U (1, 1) × U (1, 1) of the block diagonalizing matrices T0 relevant to the saddle-point manifold, see (11.6.34), and the four-parameter space of matrices T1 diagonalizing the 2 × 2 blocks Q a , Q r (i.e., the space U (1, 1) × U (1, 1)/U (1) × U (1) × U (1) × U (1)),   Qa 0 (11.6.65) Q = T0 T −1 = T0 T1 qT1−1 T0−1 = T qT −1 0 Qr 0   u0 T1 = , 0v

and

(11.6.66)

with unitary u and pseudounitary v, the latter both 2×2 . To start our metric walk, we write the increment of Q = T qT −1 as δ Q = [δT T −1 , Q] + T δqT −1 and develop a 16 × 16 metric matrix G in (δ Q, δ Q) = (δ Q , Gδ Q ) where δ Q is made up of the increments δq and T −1 δT . In analogy with (11.6.63),   Str(δ Q)2 = Str [T −1 δT, q]2 + (δq)2 =

(11.6.67)

1...4 

1...4 

i= j

i

(−ki )(T −1 δT )i j (qi − q j )2 (T −1 δT ) ji +

ki (δqi )2

528

11 Superanalysis for Random-Matrix Theory

where ki is equal to +1 or −1 when the label i is, respectively, bosonic or fermionic. The latter sign factor stems, of course, from the definition of the supertrace. Obviously, then, the metric matrix G is block diagonal, a fully diagonal 4 × 4 block pertains to the eigenvalue increments δqi , and a 12 × 12 block is for the (T −1 δT )i j . The superdeterminant SdetG factorizes accordingly, and the 4 × 4 block equals unity. Thus we need to worry only about the remaining 12 × 12 supermatrix which I choose to call G T . We meet M B = 4 bosonic labels and M B = 8 fermionic labels in G T since the eigenvalues are all bosonic. Inasmuch as we may choose the twelve independent increments (T −1 δT )i j with i = j as increments of the parameters in T , we immediately conclude from the second line in (11.6.63) that G T is itself block diagonal, with six 2 × 2 blocks, since a given (T −1 δT )i j couples only to (T −1 δT ) ji . Thus, the metric matrix has the + 4ki k j , whose square superdeterminant (see Problem 11.2) SdetG T = i1...4 = j (qi − q j ) root is the Berezinian in8 ⎡ ⎤⎡ ⎤   , , , 2ki k j ⎦ ⎣ −1 ⎣ ⎦ dQ = (qi − q j ) dqi . T dT (11.6.68) i= j

ij

i= j

i

As an intermediate result, we have the integration measure of the full coset space U (1, 1/2)/U (1) × U (1) × U (1) × U (1) dμ(T ) =

,

T −1 dT

 ij

.

(11.6.69)

i= j

Completely analogous reasoning yields an analogous result for the change of variables from matrix elements to eigenvalues and “angles” in the diagonalizing matrices u, v of the blocks Q a , Q r . By combining these results with (6.10.68) we get (11.6.41). Note that all intrablock eigenvalue differences are then absorbed in d Q a , d Q r such that only interblock eigenvalue differences remain on display, together with the yet to be determined integration measure dμ(T0 ) of the reduced coset space U (1, 1/2)/U (1/1) × U (1/1). When accounting for T = T0 T1 , yet another transformation of variables must be effected. It is well to keep in mind the restrictions on the pseudounitary matrices T0 and T1 which bring about the appropriate number of independent parameters. Pseudounitarity itself allows us to consider all diagonal elements as dependent on  † the off-diagonal ones: Indeed, T † L T = L yields |Tii |2 L ii = L ii − j(=i) Ti j L j j T ji , and this fixes the moduli of the diagonal elements; the phases may be chosen arbitrarily anyway, as already mentioned. Moreover, the matrix T1 needed to diagonalize the 2 × 2 blocks Q a and Q r can be chosen block diagonal itself and involves four fermionic parameters, two each for u and v, see (11.6.66); the matrix T0 which 8 The product of increments (T −1 dT ) must be read as a wedge product, as far as the bosonic ij increments are concerned.

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

529

brings Q to block diagonal form has diagonal 2 × 2 blocks T011 and T022 which depend on the two off-diagonal blocks T012 and T021 ; that structure is obvious from the explicit form (11.6.35); the number of independent parameters in T0 is thus eight, as stated previously; four each are bosonic and fermionic. With these preliminary remarks in mind, we go the metric way toward the Berezinian for our final transformation and consider 2  Str (T −1 δT )ind   −1   −1    2 u u δu 0 u0 0 −1 + = Str δT (11.6.70) T 0 0 v ind 0 v −1 δv 0 v −1 0 

  2   2  −1 12 −1  −1 −1 21 = str δuu ind + δvv ind + 2 T0 δT0 ind (T0 δT0 )ind . To attract the reader’s attention to the change of the dimension of the supermatrices from four to two in the foregoing chain, I employ two slightly differing symbols for the supertrace; the subscript “ind” reminds us of the need to drop the diagonal elements of δuu −1 and δvv −1 and even the two diagonal blocks (T0−1 δT0 )11 and (T0−1 δT0 )22 , to keep the correct number of independent increments. Thus, it is obvious once more that the Berezinian for the envisaged change of variables (from the twelve independent increments in T −1 δT to the independent increments in u −1 δu, v −1 δv, and T0−1 δT0 ) factorizes in a 4 × 4 part pertaining to the parameters in u and v and an 8 × 8 superdeterminant pertaining to dμ(T0 ); the latter is determined by the last term in (11.6.70). To finally nail down dμ(T0 ), we inspect the last term in (11.6.70) and, using the structure (11.6.39) of T0 , note that 3&  1 + k|t|2 (U −1 δU )kt ∗ + kδt ∗ 4 & & −kt ∗ (V −1 δV ) 1 + k|t|2 − kt ∗ δ 1 + k|t|2 V −1 , 3&   −1 21 1 + k|t|2 (V −1 δV )t + δt T0 δT0 = V 4 & & −t(U −1 δU ) 1 + k|t|2 − tδ 1 + k|t|2 U −1 .



T0−1 δT0

12

=U

The reader will recall that t = diag(t B , t F ) is the diagonalized version of T0 and thus harbors four real bosonic parameters. Two independent fermionic parameters live in U and V each, as is indeed obvious from the explicit form (11.6.40). As independent increments in the matrices (U −1 δU ) and (V −1 δV ), thus we may take their off-diagonal parts, just as was the case for the blocks u −1 δu and v −1 δv above. Before inserting their explicit forms in the above off-diagonal blocks of T0−1 δT0 , it is well to simplify the latter a bit by introducing intermediate variables momentarily replacing t B and t F ,

530

11 Superanalysis for Random-Matrix Theory

s=&

t 1+

k|t|2

, i.e.,

sB = &

tB 1 + |t B

|2

, sF = &

tF 1 − |t F |2

.

(11.6.71)

Thus, these blocks take the form  

T0−1 δT0 T0−1 δT0

12 21

&  & = U 1+k|t|2 (U −1 δU )ks ∗ −ks ∗ (V −1 δV )+kδs ∗ 1+k|t|2 V −1 , & &  = V 1+k|t|2 − s(U −1 δU ) + (V −1 δV )s + δs 1+k|t|2 U −1 .

The square-root factors which here are diagonal 2 × 2 & surround the curly brackets & matrices with entries S B ≡ 1 + |t B |2 and S F ≡ 1 − |t F |2 ; they will eventually drop out of the Berezinian we are after since that Berezinian will turn out to be a block diagonal superdeterminant with a 4 × 4 Bose block and an equally large Fermi block both of which involve S B and S F only in a common and thus cancelling prefactor (S B S F )4 . To see this, we need to evaluate the last term in (11.6.70). An easy little calculation starting from (11.6.40) establishes (U

−1



δU )ind

0 δρ = δρ ∗ 0

 ,

(V

−1



δV )ind

0 iδσ = iδσ ∗ 0

 (11.6.72)

and some more patient scribbling accounting for T0 , as in (11.6.39), yields 12  str T0−1 δT0 ind (T0−1 δT0 )21 ind = S B4 δs B∗ δs B + S B2 S F2 (s F∗ δρ + s B∗ δiσ )(s F δρ ∗ − s B δiσ ∗ )

(11.6.73)

+S F4 δs F∗ δs F + S B2 S F2 (s B∗ δρ ∗ + s F∗ δiσ ∗ )(s B δρ − s F δiσ ) . The promised block structure of the Berezinian and the cancelling of the factors S B and S F therein is now manifest. The Fermi block involves the matrix ⎛

⎞ 0 |s F |2 − |s B |2 0 −2s B s F∗ ⎜|s B |2 − |s F |2 ⎟ 0 −2s B∗ s F 0 ⎜ ⎟ ∗ 2 2⎠ ⎝ 0 |s F | − |s B | 0 2s B s F 2s B s F∗ 0 |s B |2 − |s F |2 0

(11.6.74)

which contributes the factor (|s B |2 + |s F |2 )−2 to the Berezinian. Straightening out various factors 2 and π cavalierly left unnoticed along the way, we get dμ(T0 ) =

d 2sB d 2sF ∗ 1 dρ dρdiσ ∗ diσ . (|s B |2 + |s F |2 )2 π π

(11.6.75)

An elementary final transformation dispatches the auxiliary variables s B and s F in favor of the eigenvalues t B and t F and yields the reward for our efforts and (hopefully, not too much) suffering,

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

dμ(T0 ) =

531

d 2tB d 2tF ∗ 1 dρ dρdiσ ∗ diσ , (|t B |2 + |t F |2 )2 π π

(11.6.76)

which was previously announced in (11.6.42) and used in Sect. 11.6.4. Thus, the superanalytic derivation of the two-point cluster function is completed.

11.6.6 Back to the Generating Function ˆ In Sect. 11.6.4, we shifted our attention away from the generating function Z ( E) toward the two-point function, an observable indicator of spectral fluctuations. The generating function is worthy of interest itself, and I propose to derive its explicit form now. For convenience I define dimensionless energy offsets e1,2,3,4 from the center energy E by9 diag(E 1− , E 2+ , E 3 , E 4 ) = E +

  1 diag e1− , e2+ , e3 , e4 ; 2π N ¯

(11.6.77)

these offsets will be assumed independent of N and the center energy placed within the support of Wigner’s semicircle. The generating function can then be written as Z (e1 , e2 , e3 , e4 , E) = e

−

E 2 4π (E)λ ¯

+

(e1− +e2+ −e3 −e4 )

−

Z 0 (e1 , e2 , e3 , e4 ) = e 2 (e2 −e1 −e3 +e4 ) i

+

(e3 − e1− )(e2+ − e4 ) (e2+ − e1− )(e3 − e4 )

−

+ e 2 (e2 −e1 −e4 +e3 ) i

Z 0 (e1 , e2 , e3 , e4 ) , (11.6.78) (11.6.79)

(e4 − e1− )(e2+ − e3 ) . (e2+ − e1− )(e4 − e3 )

In conformity with the definition (11.6.5) we find the generating function symmetric under the exchange of e3 and e4 , Z (e1 , e2 , e3 , e4 , E) = Z (e1 , e2 , e4 , e3 , E); one sometimes refers to that property as to the Weyl symmetry. The E-dependent exponential is obviously non-universal since it involves the semicircular mean density . ¯ It is, however, irrelevant for the connected two-point correlator since it collapses to unity after differentiating and pairwise equating the offsets, e3 = e1− , e4 = e2+ ; it could be suppressed by placing the center energy in the middle of the spectrum, E = 0, and that suppression does not alter the universal result for the twopoint function. The only benefit of keeping the nonuniversal factor is the fact that Z retains the property of reducing to the generating function (11.2.3) of the mean Green function G when we put either e3 = e1− or e4 = e2+ , NE

Z (e1− , e2+ , e3 , e2+ , E) = e 4π λ¯ 2

(e3 −e1− )

Z (e1− , e2+ , e1− , e4 , E) = e − 2π G(E N

9

i

) (e2+ −e4 )

− ∗

−

e 2 (e3 −e1 ) = e − 2π G(E ,

N

−

)(e1− −e3 )

(11.6.80)

¯ see (11.2.1) and (11.2.2). Throughout this chapter the mean level density is written as N ρ(E),

532

11 Superanalysis for Random-Matrix Theory

with the mean (advanced) Green function G(E − ) given in (11.5.16). For the derivation of the foregoing Z I pick up the thread after (11.6.25). Upon using the integration measure (11.6.41) and (11.6.42) and doing the saddle-point integrals over Q r and Q a as in Sects. 11.2 and 11.5, I obtain in the fashion of (11.6.50) 



 Q s (T0 ) ˜ ˆ dμ(T0 ) exp N Str ( E − E) λ2     ˆ 0 G( E)T ˜ −1 . = dμ(T0 ) exp N Str ( E˜ − E)T 0 

ˆ = Z ( E)

(11.6.81)

Here the average Green function with the matrix argument E˜ is to be read as   ˜ = diag G(E − , G(E − , G(E + , G(E + ) G( E)  d E ρ(E ) + iπ ρ(E)Λ . =P E − E

(11.6.82)

The further procedure parallels the calculation of the correlator (11.6.57), up to a certain subtlety worthy of being highlighted. When the saddle manifold (11.6.49) is −1 ¯ invoked, Q s (T0 ) = E/2+iπ λ2 (E)T 0 ΛT0 , the first term E/2 immediately entails the non-universal exponential in the result (11.6.78), and we confront −

E

(e− +e+ −e −e )

3 4 2 2 1 ¯ Z = e 4π (E)λ Z0 ,   i Z 0 = dμ(T0 ) exp − StrT0 ΛT0−1 diag(e1− , e3 , e2+ , e4 ) ; 2

(11.6.83)

here the advanced-retarded notation (11.6.32) is employed. To do the remaining integral over the saddle-point manifold I use the parametrization (11.6.39) and (11.6.40) of the matrices T0 and the integration measure dμ(T0 ) given in (11.6.76), to get Z0 = e

− 2i (e1− −e3 −e2+ +e4 )



d 2 t B d 2 t F dρ ∗ dρdiσ ∗ diσ π 2 (|t B |2 + |t F |2 )2 −

+

× e−i(e1 −e2 )|t B | e−i(e3 −e4 )|t F | − ∗ 2 2 × e−i(e1 −e3 )ρρ (|t B | )+|t F | ) + ∗ 2 2 × e−i(e2 −e4 )σ σ (|t B | )+|t F | ) . 2

2

(11.6.84)

The subtle point announced above is the singularity of the integrand at |t B |2 = |t F |2 = 0 due to the integration measure. That singularity did not bother us in the calculation of the two-point function of the previous section since the term (|t B |2 ) + |t F |2 )−2 was cancelled after the differentiations w.r.t. E 1 and E 2 . Now the singularity makes for an additive contribution to the integral which I shall determine

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

533

presently10 . For the moment naively doing the Grassmann integrals and then the remaining ordinary Gaussian integrals over t B and t F I get Z0 =

+ (e1− − e3 )(e2+ − e4 )  − i (e− +e3 −e+ −e4 ) i − 2 + e− 2 (e1 −e3 −e2 +e4 ) + . . . . (11.6.85) e 2 1 − + (e1 − e2 )(e3 − e4 )

Not visible yet is the symmetry under e3 ↔ e4 ; it becomes manifest once the missing piece “. . .”, due to the singularity mentioned, is added. That piece + i − reads . . . = e− 2 (e1 −e3 −e2 +e4 ) , as the following argument reveals. The superintegral (11.6.84) involves the four offset variables only in the form of differences. We may consider the three combinations e1− − e2+ , e2+ − e3 , e3 − e4 as independent variables. Then differentiating w.r.t. e2+ − e3 in (11.6.84) we get an integral representation for the derivative whithout singularity, since (|t B |2 + |t F |2 ) cancels. Therefore, the missing piece “. . .” in (11.6.85) does not contribute to the derivative. The missing piece can now be seen as an integration “constant”, a function of the two variables e1− − e2+ , e3 − e4 ; it is uniquely determined by imposing the symmetry under e3 ↔ e4 The result announced in (11.6.78) and (11.6.79) is thus established.

11.6.7 Rational Parametrization of the Sigma Model The manifold of saddles encountered in the sigma model need not be parametrized as in the previous subsections, and I would now like to discuss an alternative, the rational parametrization which was employed in the semiclassical construction of the sigma model in Chap. 10. In fact, that parametrization is well known [3] to be particularly useful for perturbative purposes, like the periodic-orbit based semiclassical treatment. I start with the generating function Z as given in (11.6.81). It may be well to recall that the manifold of matrices Q s (T0 ) arises by conjugating the diagonal “standard” saddle Λ = diag(1, 1, −1, −1) ,

(11.6.86)

with pseudounitary matrices T0 as Q s (T0 ) = T0 Q s T0−1 ,

& Q s = E/2 + iΛ λ2 − (E/2)2 ; (11.6.87)

here the diagonal matrix Λ is written in the advanced-retarded notation to which I shall adhere throughout the present subsection. As explained in Sect. 11.6.3, the

10 The said singularity becomes effective when the second and third exponential in (11.6.84) are both replaced with unity; the Bosonic integrals then diverge while the Fermionic ones vanish; the situation is similar to the one met with in our discussion of the Parisi–Sourlas–Efetov–Wegner theorem in Sect. 11.4.5.

534

11 Superanalysis for Random-Matrix Theory

matrices T0 are not allowed to range within the full group U (1, 1/2) of pseudounitary supermatrices but are restricted to the coset space U (1, 1/2)/U (1/1) × U (1/1) deprived of all 4 × 4 matrices commuting with Λ. Before attempting to parametrize the said coset space it is well to shed unnecessay ballast and set the center energy at the middle of the spectrum, E = 0 and go for Z 0 = Z | E=0 as defined in (11.6.83). Moreover, since the principal goal of the present section is to make contact with the semiclassical treatment, I switch to the notation employed for the energy offsets in Eq. (10.6.10) of Chap. 10 and write e1− = −e B , e2+ = e A , e3 = e D , e4 = −eC whereupon the starting point (11.6.83) takes the appearance  Z0 =

−1

dμ(T0 ) e 2 Str T0 ΛT0 i

diag(e B ,e D ,−e A ,−eC )

.

(11.6.88)

To construct theannounced parametrization I write the matrix T0 with the help of  Taa Tar 2×2 blocks, T0 = Tra Trr . Due to the double degeneracy of the two eigenvalues ±1 of Λ, the matrix Q = T0 ΛT0−1 remains unchanged if T0 is replaced  by the prod −1 uct T0 K with K any invertible block diagonal matrix. I choose K = Taa0 Trr0−1 to   endow T0 K with diagonal blocks equal to unity, T0 K = B1˜ B1 with B = Tar Trr−1     † 0 0 −1 and B˜ = Tra Taa . I now invoke the pseudounitarity T0 10 −k T0 = 10 −k with k = diag(1, −1); that property entails the matrix (T0 K )† L(T0 K ) = K † L K =  −1 † −1 (Taa ) (Taa ) 0 to be block diagonal. By imposing that block diagonality −(Trr−1 )† k(Trr−1 0 †    on the product B1˜ B1 L B1˜ B1 one finds the offdiagonal blocks of T0 K related as B˜ = k B † . Now, since T0 and T0 K are equivalent in defining the saddle manifold I rename T0 K as T0 and henceforth work with11  T0 =

1 B B˜ 1



T0−1

,

 =

˜ −1 −B(1 − B˜ B)−1  (1 − B B) . ˜ − B B) ˜ −1 − B(1 (1 − B˜ B)−1

(11.6.89)

The intervening 2 × 2 supermatrices will be written as B=

a μ νb

 ,

B˜ =

a∗ ν∗ μ∗ −b∗



= k B†,

 k=

 1 0 . 0 −1

(11.6.90)

The saddle-point manifold thus results as  T0 ΛT0−1 =

11

1+B B˜ 1−B B˜

˜ − B B) ˜ −1 2 B(1

−2B(1 − B˜ B)−1 B˜ B − 1+ 1− B˜ B

But note that T0 K and thus the new T0 is not pseudounitary.

 (11.6.91)

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

535

˜ and the integral over the manifold becomes one over the 2 × 2 matrices B, B,  Z0 = −

  ˜  1 + B˜ B i

1 + BB ˜ ˆ ˆ +e , d[B, B] exp Str e 2 1 − B˜ B 1 − B B˜

(11.6.92)

with the diagonal 2 × 2 matrices of energy offsets eˆ = diag(e A , eC ) ,

eˆ = diag(e B , e D ) .

(11.6.93)

Precisely that integral was established semiclassically, see (10.9.23), there with the flat integration measure ˜ = dμ(T0 ) = d[B, B]

d 2a d 2b ∗ dμ dμdν ∗ dν ; π π

(11.6.94)

the reader will recall that the precise integration range could not be semiclassically determined. For the present RMT context, the flat measure will be confirmed and the integration range fixed as |a|2 < 1 and |b|2 < ∞ .

(11.6.95)

I shall rightaway work with the measure (11.6.94) and the range (11.6.95), postponing the derivation to further below. The rational-parametrization matrix integral (11.6.92) is not any more difficult to do than the equivalent counterpart in Sect. 11.6.6, and the calculation proceeds quite analogously. The key is the observation that the 2 × 2 matrices B B˜ and B˜ B both have the eigenvalues (aν ∗ − b∗ μ)(a ∗ ν − bμ∗ ) ≥ 0, |a|2 + |b|2 + μμ∗ − νν ∗ (aν ∗ − b∗ μ)(a ∗ ν − bμ∗ ) l F = −|b|2 + νν ∗ + 2 ≤ 0; |a| + |b|2 + μμ∗ − νν ∗ l B = |a|2 + μμ∗ +

(11.6.96) (11.6.97)

the Bose–Bose eigenvalue l B is positive while the Fermi–Fermi one, l F , is negative. The diagonalizing matrices are unitary for B B˜ and pseudounitary for B˜ B, ˜ = V −1 B BV ˜ = U −1 B BU



lB 0 0 lF

 ,

(11.6.98)

and read   −iφ B   ∗ 0 e η 1 + ηη2 ∗ , U= η∗ 1 − ηη2 0 e−iφ F

 V =

∗

τ 1 − τ 2τ ∗ −τ ∗ 1 + ηη2



536

11 Superanalysis for Random-Matrix Theory

aν ∗ − b∗ μ , + |b|2 + μμ∗ − νν ∗ a ∗ μ + bν ∗ τ =− 2 . |a| + |b|2 + μμ∗ − νν ∗ η=−

|a|2

(11.6.99)

The foregoing U is a product of two unitary matrices the right one of which is diagonal and thus not needed for the diagonalization; the purpose of that matrix BV , which “sinrather is to provide the matrix U −1√ √ is also diagonal, with positive −1 −1 ˜ BV = diag(+ l , + −l ); on the other hand, V gular values”, U BU = B F √ √ diag(+ l B , − −l F ); the phases φ B , φ F needed for the purpose in question differ from the phases of the original Bosonic elements a, b of B by suitable nilpotent additions (which will not be of further relevance). Now, the eight new variables l B , l F , φ B , φ F , η, η∗ , τ, τ ∗ can be employed as integration variables instead of the ˜ The pertinent Berezinian whose calculation I forgo yields original elements of B, B. the integration measure ∗ ∗ ˜ = dl B dl F dφ B dφ F dη dηdτ dτ , d[B, B] 4π 2 (l B − l F )2

(11.6.100)

and the integration ranges become 0 < lB ≤ 1 ,

−∞ < l F < 0 ,

0 ≤ φ B , φ F ≤ 2π .

(11.6.101)

In terms of the new variables the integral (11.6.92) takes the form, after doing the trivial phase integrals, 

Z0 =



1



−∞

1 (l − l F )2 B 0  0 i 1 + lB 1 + lF × exp (e A + e B ) − (eC + e D ) (11.6.102) 2 1 − lB 1 − lF    1 + lF  1 + lB i − (e A − eC )τ τ ∗ + (e B − e D )ηη∗ . × exp 2 1 − lB 1 − lF dl B

dl F

dη∗ dηdτ ∗ dτ

The rational functions of l B , l F appearing in the two foregoing exponents repre˜ B˜ B. sent, of course, the rational functions of B B, The same subtlety now arises as for the integral (11.6.84) in the previous subsection: the singularity of the integrand at l B = l F = 0 makes for an additive contribution, and I deal with that point exactly as for (11.6.84). I first do the Grassmann integrals and find the remaining integrand for the integrals over l B , l F freed of the singularity at l B = l F = 0,

11.6

The Two-Point Function of the Gaussian Unitary Ensemble



1

Z 0 = (e A − eC )(e B − e D )  ×

0 0 −∞

i 1+l B dl B 2 1−l B (e A +e B ) e (1 − l B )2

537

(11.6.103)

1+l F dl F −i (e +e ) e 2 1−l F C D + . . . . 2 (1 − l F )

The Bosonic integrals become elementary in terms of the integration variables B F , n F = 1+l and yield n B = 1+l 1−l B 1−l F Z0 =

(e A − eC )(e B − e D )  i (e A +e B −eC −e D ) i − e 2 (e A +e B +eC +e D ) + . . . . (11.6.104) e2 (e A + e B )(eC + e D )

The dots . . . stand for the additive contribution of the singularity mentioned which is determined quite elegantly by realizing that Z 0 can be regarded as a function of the three independent variables (e A + e B ), (e A − eC ), (e B − e D ); the fourth combination, (eC + e D ), then becomes a linear combination of the three variables chosen as independent. Taking the derivative of Z 0 in (11.6.102) w.r.t., say, (e A +e B ), an integrand arises which is no longer singular at n B = n F = 0 since one power of (n B − n F ) cancels, and therefore that derivative is correctly obtained from the foregoing expression (11.6.104) without the correction . . .. The latter correction can thus be seen as an integration constant K (e A − eC , e B − e D ) independent of i e A + e B . That integration constant is fixed as K = e 2 (e A +e B −eC −e D ) by imposing the Weyl symmetry Z 0 (e A , e B , eC , e D ) = Z 0 (e A , e B , −e D , −eC ). The familiar result Z0 =

(e A + e D )(e B + eC ) i (e A +e B −eC −e D ) e2 (e A + e B )(eC + e D ) (e A − eC )(e B − e D ) i (e A +e B +eC +e D ) − e2 (e A + e B )(eC + e D )

(11.6.105)

is thus recovered. I shall not sneak away without filling the promise of showing the correctness of the flat measure (11.6.94) and the integration range (11.6.95). The clue will be a geometric interpretation for the complex Bosonic variables a and b: They respectively parametrize the hyperboloid H 2 and the two-sphere S 2 .   It is helpful to momentarily imagine the matrix T0 = B1˜ B1 rewritten in   the Bose–Fermi notation. The Bose–Bose block a1∗ a1 , the Fermi–Fermi block       1 b 0 μ 0 ν , the Bose–Fermi block , and the Fermi–Bose block then arise. ∗ ∗ ∗ −b 1 ν 0 μ 0 The integration range for a can thus be determined by checking the Bose–Bose block alone. Since the pseudounitarity of T0 resides in that block the integration range is the coset space U (1, 1)/U (1) × U (1) ≡ H 2 . Similarly, b appears only in the Fermi–Fermi block w.r.t. which T0 behaves unitarily such that the integration range is the coset space U (2)/U (1) × U (1) ≡ S 2 . It remains to reveal S 2 and H 2 as sphere and hyperboloid, an elementary exercise in Lie groups.

538

11 Superanalysis for Random-Matrix Theory

First to the sphere I note that U (2) consists of matrices of the form  turning  α β e with 0 ≤ ψ < 2π and the complex entries α, β restricted as −β ∗ α ∗ iψ

|α|2 + |β|2 = 1; the latter restriction is enforced by α = cos θ2 ei f , β = sin θ2 ei( f +φ) with 0 ≤ θ ≤ π and 0 ≤ φ, f < 2π . Upon identifying all of those matrices differing by the overall real phase ψ I arrive at the coset space U (2)/U (1). The further restriction to U (2)/U (1) × U (1) means identifying all matrices differing in the relative phase rows, i.e. in f , such that only the two parameters  the two  between cos θ2 sin θ eiφ α β θ, φ remain, −β ∗ α∗ = − sin θ e−iφ cos2 θ . The resulting coset space is obviously 2

2

isomorphic to the sphere S 2 parametrized by the polar angle θ and the azimuth φ or the complex stereographic projection variable z = tan θ2 eiφ . In terms of the latter we get the coset space spanned by matrices of the form &

1 1 + |z|2



1 z −z ∗ 1

 ∈ U (2)/U (1) × U (1) .

(11.6.106)

The natural measure for the sphere and thus for the coset space is sin θ4πdθdφ = d2z . In view of (11.6.103) and since the eigenvalue l F as given in (11.6.96) π (1+|z|2 )2 has the numerical part −|b|2 the matrix element b must be identified with the stereographic projection variable z. Indeed, in (11.6.103) the denominator (1 + |b|2 ) appears explicitely, as a contribution of the Grassmann integrals. We have thus 2 ascertained the flat measure dπb as well as the full complex b-plane as the integration range. The geometric interpretation of the coset space U (1, 1)/U (1) × U (1) is reached   analogously. The group U (1, 1) is spanned by matrices of the form eiψ βα∗ αβ∗

1

cosh 2t

x3

x3 1

θ

θ 2

θ 2 –1

sin θ z = tan θ2

–1

sinh t

z = tan 2θ = tanh 2t

Fig. 11.2 Stereographic projections of sphere S 2 (left) and (upper half of ) hyperboloid H 2 (right). Plots in planes of constant azimuth. For S 2 , projection onto equatorial plane from south pole. For H 2 , projection from center onto plane tangential to pericenter

11.6

The Two-Point Function of the Gaussian Unitary Ensemble

539

with two complex entries restricted by pseudounitarity as |α|2 − |β|2 = 1 or α = cosh 2t ei f , β = sinh 2t ei( f +φ) with 0 ≤ t < ∞ (see Fig. 11.2). Dropping the overall phase factor I get the coset space U (1, 1)/U (1), and by dropping the relative phase f I arrive at U (1, 1)/U (1) × U (1). Instead of by t, φ that latter set can be parametrized by the complex variable z = eiφ tanh 2t which ranges in the unit disk |z|2 < 1. The pertinent matrices have the form 1

& 1 − |z|2



1 z z∗ 1

 ∈ U (1, 1)/U (1) × U (1) .

(11.6.107) 2

1+|z| 2Re z 2Im z The Cartesian coordinates x 1 = 1−|z| 2 , x 2 = 1−|z|2 , x 3 = 1−|z|2 are related as x12 +x22 −x32 = −1, z 3 > 0. The coset space in question thus indeed turns out isomorphic to the upper sheet H 2 of the hyperboloid and z a complex stereographic project dtdφ d2z tion variable. Now invoking sinh4π = 4(1−|z| 2 )2 , once more inspecting (11.6.103), and appreciating the numerical part |a|2 of the eigenvalue l B as given by (11.6.96) I can identify the matrix element a with the stereographic projection variable z. The 2 flat measure dπa and the unit-disk integration range |a|2 < 1 are thus confirmed.

11.6.8 High-Energy Asymptotics The rigorous result (11.6.105) or (11.6.79) reveals an additive structure for the generating function, Z 0 = Z (1) + Z (2) , Z (2) (e A , e B , eC , e D ) = Z (1) (e A , e B , −e D , −eC ) .

(11.6.108)

It is a remarkable privilege of the unitary ensemble that the two terms Z (1) and Z are each due to a saddle-point contribution to the high-energy asymptotics. To identify the respective saddles I observe that the exponent in the rationalparametrization integral (11.6.92) is an even function of the Fermionic variables μ, μ∗ , ν, ν ∗ and therefore only vanishing saddle-point “values” make sense for them. It follows that saddle-point values for the Bosonic variables a, b can be looked ˜ The superfor by dropping the off-diagonal Fermionic entries in the matrices B, B. 1+|a|2 1−|b|2 trace in the exponent then simplifies to (e A +e B ) 1−|a|2 −(eC +e D ) 1+|b|2 . Its stationary points within the integration ranges (11.6.95) read (2)

a=b=0 a = 0,b = ∞

standard saddle , Andreev–Altshuler saddle

(11.6.109) (11.6.110)

and can easily be seen to be antipodal poles of the sphere S 2 parametrized by the Fermi–Fermi variable b.

540

11 Superanalysis for Random-Matrix Theory

The so-called standard saddle (11.6.109) yields the first term in (11.6.105), Z (1) (e A , e B , eC , e D ), as is readily found by dropping all powers of B B˜ and B˜ B higher than the first in the exponent in (11.6.92). The Andreev–Altshuler saddle (11.6.110) must therefore provide Z (2) (e A , e B , eC , e D ) = Z (1) (e A , e B , −e D , −eC ). Even though the foregoing argument does establish the contribution of the Andreev–Altshuler saddle B AA = diag(0, ∞) a check by explicit calculation is an instructive exercise for which some readers may want to join me now; all others are invited to jump to the two final paragraphs of this subsection. A change of the present rational parametrization is indicated, to shift the infinitely remote point in the complex b-plane to a finite location. To that end I propose to retrace the construction of the rational parametrization (11.6.89) of the matrices   AA T0 and to identify the particular T0AA = B˜ 1AA B1 associated with the Andreev– Altshuler saddle according to (11.6.87). It is sensible to represent B AA and B˜ AA as the limits B AA = limd→0 Bd and ˜B AA = limd→0 B˜ d with Bd ≡ P K −1 and B˜ d ≡ P K −1 , in terms of the diagonal d −d 2 × 2 matrices     00 1 0 P= , K ±d = ; (11.6.111) 01 0 ±d I may take the parameter d real such that the infinite point of the complex b-plane is approached along the real axis. As long as d is finite the corresponding 4 × 4 matrix T0 factorizes,  Td =

1 P K d−1 −1 P K −d 1



 =

K −d P P Kd



−1 K −d 0 0 K d−1

 .

(11.6.112)

Since the second factor in Td commutes with Λ I find the matrix identity lim Td ΛTd−1 = T0AA Λ(T0AA )−1 = RΛR −1 ≡ ΛAA , ⎛ ⎞ 1000   ⎜0 0 0 1⎟ K0 P ⎟ =⎜ R = R −1 = ⎝0 0 1 0⎠ . P K0 0100

d→0

(11.6.113)

The matrix R describes a permutation transposing the second and fourth entries, i. e., the Fermi–Fermi ones such that ΛAA = RΛR = R diag(1, 1, −1, −1)R = diag(1, −1, −1, 1) .

(11.6.114)

To find the part of the high-energy asymptotics connected with the Andreev– Altshuler saddle I go back to the starting point (11.6.88) and reparametrize the integration variables as T0 = RT0 . The Jacobian of the transformation T0 → T0 is unity and the integration domain is unchanged. After invoking the cyclic invariance of the supertrace I get

11.7

Universality of Spectral Fluctuations: Non-Gaussian Ensembles

 Z0 =



−1

dμ(T0 ) e 2 Str T0 ΛT0 i

R diag(e B ,e D ,−e A ,−eC )R

541

.

(11.6.115)

I now rationally parameterize T0 and am led to a superintegral differing from (11.6.92) by the interchange e D ↔ −eC . The Gaussian approximation to that integral therefore yields the announced result Z (2) = Z (1) |e D ↔−eC . 1 -corrections to the leading-order saddle-point corrections under The absence of |e| discussion is worthy of comment. We here find the RMT analogue of the exactness of the diagonal approximation of the semiclassical approximation for dynamics without time reversal invariance. As revealed by the Duistermaat-Heckman theorem [14], the present leading-order saddle-point analysis exhausts the full generating function since the GUE is distinguished among the Wigner–Dyson symmetry classes by affording a symplectic structure for the integration domain of the matrices T0 . In the semiclassical approximation of Chap. 10 the periodic-orbit expansion of the generating function had to be based on the assumption of large positive imaginary parts of the energy offsets. When the periodic-orbit expansion for Z (e A , e B , e D , eC ) was built with that assumption, the part Z (2) (e A , e B , eC , e D ) was exponentially suppressed and only Z (1) (e A , e B , eC , e D ) obtained; the part Z (2) (e A , e B , eC , e D ) was then established by invoking unitarity, in the form of the Riemann–Siegel lookalike. The present discussion of the RMT sigma model affords real offsets for which both Z (1) and Z (2) are of equal weight.

11.7 Universality of Spectral Fluctuations: Non-Gaussian Ensembles Hackenbroich and Weidenm¨uller presented an alternative derivation of the two-point function of ensembles of random matrices [15] which does not assume Gaussian statistics for the matrix elements12 Hi j . These ensembles are characterized by matrix densities of the form P(H ) ∝ exp[−N Tr V (H )] ,

(11.7.1)

constrained only such that moments exist. The appearance of the trace secures invariance of the ensemble w.r.t. the desired “canonical” transformations, be these unitary, orthogonal, or symplectic. Here, I shall confine myself to the unitary case of complex Hermitian matrices H . By choosing V (H ) = H 2 , we would be back to the Gaussian ensemble for which the mean density of levels is given by the semicircle law. For general V (H ), that latter law does not reign, but after rescaling the energy axis in the usual way, e = E N (E), we shall recover the same two-point function as for the GUE above. This will be no surprise for the reader who has studied level dynamics and appreciated, in particular, the arguments of Sect. 6.10. Universality 12

For recent progress on the universality problem within RMT see Ref. [16, 17]

542

11 Superanalysis for Random-Matrix Theory

of spectral fluctuations is at work here. Once we have ascertained the independence of the two-point function Y (e) of the function V (H ) and invoke the ergodicity a` la Pandey (see Sect. 4.14.2), we shall have gone a long way toward understanding why a single dynamical system with global chaos in its classical phase space has universal spectral spectral fluctuations. As a preparation to the reasoning of Hackenbroich and Weidenm¨uller, we need to familiarize ourselves with delta functions of Grassmann variables.

11.7.1 Delta Functions of Grassmann Variables For Grassmann as for ordinary variables, the delta function is defined by [18]  (11.7.2) f (η0 ) = dη δ(η − η0 ) f (η) . With f (η) = f 0 + f 1 η, we check the simple representations  1 δ(η − η0 ) = dσ eiσ (η−η0 ) = η − η0 . i

(11.7.3)

Neither representation suggests drawing anything peaked, but who would want to draw graphs for functions of Grassmann variables anyway. A more serious comment is that δ(aη) = aδ(η) for an arbitrary complex number a; in particular, δ(−η) = −δ(η). The latter two properties are well worth highlighting since they contrast to δ(ax) = |a|−1 δ(x) for real a. The first of the representations in (11.7.3) is obviously analogous to the familiar Fourier integral representation of the ordinary $ delta function, δ(x) = (2π)−1 dk exp ikx. Thus equipped, we can proceed to delta functions of supermatrices. Starting with the simplest case, 2 × 2 supermatrices of the form Q=

 ∗ aη , η ib

(11.7.4)

we define δ(Q − Q ) as the product of delta functions of the four matrix elements ∗

˜ = δ(a − a )δ(b − b )δ(σ ∗ − σ )δ(σ − σ ) δ(Q − Q)

(11.7.5)

and note the integral identities   1 ˆ exp iN Str Q(Q ˆ dQ − Q ) 2π   ˆ exp iN Str Q(Q ˆ − Q ) f (Q) ; f (Q ) = d Qd Q

δ(Q − Q ) =

(11.7.6)

all integration measures are defined here according to our convention d Q = (dadb/2π)dσ ∗ dσ ; note that the factor N in the exponent is sneaked in at no cost, for the sake of later convenience. When generalizing to 2n × 2n supermatrices,

11.7

Universality of Spectral Fluctuations: Non-Gaussian Ensembles

543

replacing a in (11.7.4) by an n ×n Bose√ block etc., we still accompany each bosonic differential increment with a factor 1/ 2π and therefore must replace the factor 2 (2π)−1 in the first of the identities (11.7.6) by (2π)−n to get the correct power of 2π required by the Fourier-integral representation of the ordinary delta function. The second of the identities (11.7.6) remains intact for any n.

11.7.2 Generating Function We return to the generating function (11.6.9) for the two-point function and average over the non-Gaussian ensemble (11.7.1),  ˆ = d H P(H )Z ( E) ˆ Z ( E)   ˆ . (11.7.7) = (−1) N dΦ ∗ dΦ d H P(H ) exp[i Φ † L(H − E)Φ] The admitted non-Gaussian character of P(H ) forbids us from resorting to the Hubbard–Stratonovich transformation employed in Sect. 11.6.1. We still enjoy the unitary invariance P(H ) = P(H ) with H = U HU † and U an arbitrary unitary N × N matrix. Due to that invariance, the integrand of the foregoing supervector integral is, after the ensemble average, a “level-space scalar”, i.e., it depends on the vector Φ and its adjoint only through N 

∗ ˜ αβ L β ; Φαi Φβi Lβ = N Q

(11.7.8)

i=1

˜ was already encountered in (11.6.11). note that the 4 × 4 supermatrix Q Instead of Hubbard–Stratonovich, now we use the second of the delta function identities (11.7.6) for 4 × 4 supermatrices and f (Q) = 1 to write13   N ∗ ˆ ˆ exp[iN Str Q(Q ˆ ˜ L)]exp[i Φ † L(H − E)Φ] ˆ Z ( E) = (−1) dΦ dΦ d Qd Q −Q   ˆ exp[iN Str Q Q] ˆ dΦ ∗ dΦexp[i Φ † L(H − Eˆ − Q)Φ] ˆ = (−1) N d Qd Q    ˆ exp(iN Str Q Q) ˆ Sdet L(H − Eˆ − Q) ˆ −1 = (−1) N d Qd Q    ˆ −1 ˆ exp(iN Str Q Q) ˆ Sdet (H − Eˆ − Q) = d Qd Q  ˆ exp(iN Str Q Q) ˆ exp Str Tr ln(H − Eˆ − Q) ˆ . = d Qd Q (11.7.9)

13 Our unscrupulous change of the order of integrations as well as the naive use of the saddle-point approximation for the N -fold energy integral later admittedly give a certain heuristic character to this section.

544

11 Superanalysis for Random-Matrix Theory

Even though the reader should be at peace by now with compact notation, it may be well to spell out that the superdeterminant above is 4N ×4N and that within it we must read H as the tensor product of the N × N random matrix H in level space with ˆ are 4 × 4 matrices the 4 × 4 unit matrix in superspace, whereas conversely Eˆ and Q in superspace and act like the N × N unit matrix in level space. In the last line of the foregoing equation, I have written the overall supertrace (which sums over 4N diagonal elements) as Str Tr, to reserve the symbol Str for the four-dimensional superspace, denoting by Tr the trace in level space. At this point, we may imagine the matrix H diagonalized in level space and do the ensemble average with the help of the joint density of levels P(E) = N exp[−N



V (E i )]

i

, (E i − E j )2

(11.7.10)

i< j

following from (11.7.1); the normalization constant N secures Then, the ensemble averaged exponential in (11.7.9) reads

$

d N E P(E) = 1.

ˆ exp Str Tr ln(H − Eˆ − Q) (11.7.11)       ˆ . = N d N E exp 2 ln |E i − E j | − N V (E i ) − Str ln (E i − Eˆ − Q) i< j

i

i

Inasmuch as we are interested in large values of N , the foregoing N -fold integral over the energies E i can be done in the saddle-point approximation. To that end, we simply observe that the third of the three terms in the exponent is smaller than the first two by one order in N such that it gives rise to a “prefactor” that can be evaluated at the saddle and then taken out of the integral. The remaining integral cancels against the normalization factor, N /N = 1, so that we are simply left with the saddle-point value of the prefactor,   sp ˆ = exp − ˆ ; (11.7.12) exp Str Tr ln(H − Eˆ − Q) Str ln (E i − Eˆ − Q) i sp

here E i denotes the saddle-point value of the ith integration variable, determined by the saddle-point equation  sp E j(=i) i

2 sp

sp − N V (E i ) = 0 . − Ej

(11.7.13)

We are done with the ensemble average and confront integrals over the supermaˆ for the average generating function trices Q, Q  ˆ = Z ( E)

  sp ˆ exp iN Str Q Q ˆ − ˆ . d Qd Q Str ln (E i − Eˆ − Q) i

(11.7.14)

11.7

Universality of Spectral Fluctuations: Non-Gaussian Ensembles

545

Now, it is indicated to split the logarithm in the foregoing exponent into a leadingorder term and a perturbation smaller by one order in N , as in the transition from (11.6.24) to (11.6.25), by again introducing the center energy E = (E 1 + E 2 )/2 and the diagonal matrix E˜ = (E − , E − , E + , E + ) as in (11.6.32), immediately taking E 1 = E 3 , E 2 = E 4 . The analogue of (11.6.25),  ˆ = Z ( E)

  sp ˆ exp iN Str Q Q ˆ − ˆ d Qd Q Str ln (E i − E˜ − Q) 

× exp −



Str



i sp Ei

ˆ − E˜ − Q

−1

˜ , ( Eˆ − E)

(11.7.15)

i

has its first exponential invariant under the hyperbolic transformations discussed in ˆ integral, we Sect. 11.6.2. When invoking the saddle-point approximation for the Q must treat the second exponential above as a prefactor and thus involve the saddlepoint equation iQ =

 1 d E (E ) 1  ˆ sp ) ; ≈ = G( E˜ + Q ˆ sp − E sp ˜ +Q ˆ sp − E

N i E˜ + Q E i

(11.7.16)

it is in line with the leading-order treatment in N when we approximate here the sum by an integral with the mean level density as a weight and so involve the mean Greens function G on the right-hand side; at any rate, the foregoing equation ˜ as a function of the matrices Q ˆ sp (Q, E) implicitly fixes the saddle-point value Q ˜ The Parisi–Sourlas–Efetov–Wegner theorem secures the integral over the and E. ˆ around that saddle at unity so that we are left with the integral over fluctuations δ Q the supermatrix Q  ˆ = Z ( E)

  sp ˆ sp − ˆ sp ) d Q exp iN Str Q Q Str ln (E i − E˜ − Q i





ˆ sp )( Eˆ − E) ˜ × exp − N Str G( E˜ + Q

(11.7.17)

which itself invites a final saddle-point approximation. Here again, the second exponential plays the role of a prefactor, and the saddle-point equation reads ˆ sp + iQ iQ

ˆ sp ˆ sp ∂Q ˆ sp ) ∂ Q = 0 . − G( E˜ + Q ∂Q ∂Q

(11.7.18)

From this and the previous saddle-point equation (11.7.16), we conclude that ˜ =0 ˆ sp (Q sp , E) Q

=⇒

˜ . iQ sp = G( E)

(11.7.19)

But due to the hyperbolic invariance of the first exponential in (11.7.17), a whole manifold of saddles arises. We insert the integration measure (11.6.41) and (11.6.42)

546

11 Superanalysis for Random-Matrix Theory

and get, in the fashion of (11.6.50),  ˆ = Z ( E)

 ˆ T0 G( E) ˜ T −1 . dμ(T0 ) exp Str( E˜ − E) 0

(11.7.20)

˜ is expressed in terms of the mean level density The mean Green function G( E) (E) as in (11.6.82), but the latter is of course ensemble specific, and in the present case is not given by the semicircle law. Thus recovering the generating function of the nonlinear sigma model, we see that the non-Gaussian random-matrix ensemble underS consideration does indeed have the same two-point function as the GUE, save for the different mean density of levels. Nevertheless, the universal cluster function arises once that density is scaled away, as described in the discussion of the generating function (11.6.78) and (11.6.79) at the end of the preceding section.

11.8 Universal Spectral Fluctuations of Sparse Matrices Another important ensemble of matrices, for which universality a` la Wigner and Dyson has been demonstrated, is formed by random Hermitian matrices with independent and identically distributed entries. I shall not enter into a detailed discussion but rather refer the reader to the important contributions of Khorunzhy, Khoruzhenko, and Pastur [19] and Anna and Zee [20]. The early work by Mirlin and Fyodorov is closest to the spirit of this chapter [21]. The latter authors pioneered a superanalytic approach based on a functional generalization of the Hubbard–Stratonovich transformation. They found that the most important characteristic of the ensembles with independent elements is the mean number p of non-zero elements per row or column. More precisely, if the density of non-zero entries is an even function with finite moments and if p grows with the matrix size, as p ∼ N α with 0 < α ≤ 1, both the mean and the higher cluster functions of the level density come out as for the Gaussian ensembles of the same symmetry (real symmetric, complex Hermitian, or quaternion real). The only “nontrivial” case thus appears to be one of very sparse matrices that have a finite mean number p of non-zero entries randomly placed among the about N − p zeros per row, even when N grows indefinitely. The mean density of levels then deviates from the semicircular form, but still, one has the truly remarkable property that, as long as p exceeds a certain threshold value, p > pc , the spectral correlations remain the same as for the Gaussian ensembles. Only when p is decreased below pc , a kind of Anderson localization occurs, and the eigenvalues become decorrelated [22]. The numerical value of the threshold pc is non universal, i.e., it depends on the density of non-zero entries and can be found by solving a certain integral equation. However, numerical experiments with sparse matrices show [23] that the value of pc is typically smaller than two. It is indeed an impressive manifestation of universality a` la Dyson and Wigner that two non-zero elements out of thousands or

11.9

Thick Wires, Banded Random Matrices, One-Dimensional Sigma Model

547

more vanishing ones suffice to produce the statistics familiar from the Gaussian (or circular) ensembles. What is crucial here is the arbitrary position of the non-zero entries within the rows or columns. In the subsequent section, we shall see that quite different statistics arise if the non-zero elements are bound to be concentrated near the principal diagonal of the matrix.

11.9 Thick Wires, Banded Random Matrices, One-Dimensional Sigma Model 11.9.1 Banded Matrices Modelling Thick Wires We have seen a link between quantum chaos and disordered systems in our discussion of quantum localization in Chap. 7. The disordered systems considered there were strictly one-dimensional and that restriction can now be eased in favor of the quasi one-dimensional behavior of a particle hopping from one lattice site to neighboring sites in a thick wire. I propose to return Anderson’s tight-binding Hamiltonian of (7.2.1) Hmn = Tm δmn + Wr δm,n+r ,

(11.9.1)

where the indices refer to sites, Tm is a single-site potential, and Wr is a hopping amplitude. Anderson’s model can account for an electron moving in a thick wire of length L and cross-section S in the following way. We think of the wire as divided in L/lel slices, whose length lel is the mean free path lel , i.e., the mean distance between scattering impurities. Each such slice provides b ≡ k 2F S transverse “channel states.” The moving electron can make intraslice transitions between different channel states and may also hop from one slice to one of the two neighboring sites. The site label in Anderson’s Hamiltonian may be chosen such that it increases by b within one slice before proceeding to doing the same for the neighboring slice to the right. Very schematically, we may associate r = ±b with an interslice hop and |r | < b with an intraslice transition. Thus, the total number of “sites” in the “lattice” is N = bL/lel = k 2F S L/lel , and the Hamiltonian matrix (11.9.1) has a band structure: Its elements Hmn are either strictly zero outside a band of width 2b around the diagonal, Hmn = 0 for |m − n| > b, or at least fall to zero sufficiently rapidly as the skewness |m − n|/b grows large. To introduce disorder, we take the elements Hmn as independent random numbers with the joint density

P(H ) = N

  |H |2  H2  , ij exp − ii exp − 2Jii i< j 2Ji j i=1

N ,

(11.9.2)

548

11 Superanalysis for Random-Matrix Theory

where the variances Ji j suitably express the assumed bandedness. The exponential or the box distribution are examples convenient to work with,  (λ2 /2b) exp(−|i − j|/b) Ji j = J (|i − j|) = (λ2 /2b)Θ(b − |i − j|)

,

(11.9.3)

where λ fixes the unit of energy; all subsequent explicit results will be specified for the exponential distribution. We face an ensemble of random Hermitian matrices not invariant under unitary transformations; nor do we allow for time-reversal invariance, inasmuch as the off-diagonal elements have equal variances for real and imaginary parts. Of course, the bandwidth b should be small, b/N → 0 as N → ∞, or else we would be back to the GUE. A great deal more physics is accessible through the quasi one-dimensional character of this model, compared to the strictly one-dimensional model of Chap. 7. We cannot even take for granted localization of all eigenfunctions since the effectively long-range random hops of range b = k 2F S ∼ S make for competition between diffusion and quantum coherence. Localization will prevail only if the length L of the sample is larger than the localization length l while for short samples, i.e., L < l, classical diffusion is not hindered by quantum coherence. It is well to support the foregoing argument by an order-of-magnitude estimate of the various quantities involved. As the particle hops about in the lattice, starting at some site, classical diffusion will cease to provide an appropriate description once its spatial uncertainty has reached the quantum localization length l. Such quantum cutoff of diffusion must surely happen if the time needed for diffusive exploration of the sample length L, the so-called Thouless time τc = L 2 /D, is large compared to the time scale /Δ on which the particle begins to resolve the level spacing Δ, i.e., to feel the discreteness of the spectrum. Assuming that the hopping particle is a fermion whose energy is near the Fermi energy E F = 2 k 2F /2m, we may estimate the level spacing as Δ = E F /(k 3F L S) since k 3F L S is the total number of singleparticle states with energies between 0 and E F . Similarly, the diffusion constant is roughly given by the elastic mean free path lel and the Fermi velocity k F /m as D = lel k F /m, such that the localization condition L 2 /D  /Δ yields L  lel (k 2F S) ≡ l. Conversely, if the particle can diffusively explore the whole sample before resolving the level spacing, one has the case of a short sample in the sense L  lel (k 2F S) ≡ l. The length scale dividing short and long is just the localization length l = lel (k 2F S) = blel .

(11.9.4)

In quasi one-dimensional wires, l  lel . To spell out the important conclusion once more, diffusion prevails if the wire is short, whereas localization becomes effective in long wires. The detailed behavior of the eigenvectors of the banded matrices under study turns out to be governed by a scaling parameter depending on N and b as

11.9

Thick Wires, Banded Random Matrices, One-Dimensional Sigma Model

x ∝ N /b2 ,

549

(11.9.5)

as was first found numerically Casati, Molinary, and Izrailev [24, 25] and later confirmed analytically by Fyodorov and Mirlin [26].

11.9.2 Inverse Participation Ratio and Localization Length Although in presenting Lloyd’s model in Chap. 7 we calculated the mean localization length l, we shall presently follow Fyodorov and Mirlin [26] and discuss quantum localization for the ensemble of banded random matrices (11.9.3) in terms of the so-called inverse participation ratio, a quantity intimately related to l. For the μth eigenvector of a matrix H , that quantity is defined as Pμ =

N 

|ψμn |4 .

(11.9.6)

n=1

√ If the eigenvector is “extended” rather than localized, |ψμn | ∼ 1/ N such that Pμ ∝ 1/N and Pμ → 0 as N → ∞, but a finite value of Pμ results for an exponen1 tially localized vector, ψμn = (2lμ )− 2 exp(−|n − n μ |/2lμ ). A one-line calculation indeed reveals that Pμ ≈ 1/4lμ for l  1 and N → ∞. To characterize a whole matrix H rather than one of its eigenvectors, we employ the spectral average 1 Pμ  = N (E)ΔE



E+ΔE/2

dE E−ΔE/2



 μ

   Pμ δ(E − E μ ) 

(11.9.7)

ΔE→0

and to make sure that we don’t pick up an exceptional matrix, we perform the disorder average, i.e., the average over the ensemble (11.9.3), P ≡ Pμ  =

1  Pμ δ(E − E μ ) . N (E) μ

(11.9.8)

The ensemble-averaged density of levels N (E) appearing here must be evaluated for the present matrix ensemble. I shall not devote separate space to that endeavor but refer the reader to Refs. [27–30] where it is shown that Wigner’s semicircle law holds if the bandwidth grows no more slowly than b2 ∝ N for N → ∞, the case of interest. In fact, we shall see below that the semicircle law must arise as a by-product of our discussion of localization. In Chap. 7, we extracted the localization length l from the real part of the Green function Tr (E − H )−1 ; the inverse participation ratio P, on the other hand, requires the of two matrix elements of the resolvent, G nn (E) = n|(E − H )−1 |n =  product μ 2 −1 μ |ψn | (E − E μ ) , one retarded and the other advanced,

550

11 Superanalysis for Random-Matrix Theory

N ρ(E)P = lim ↓0

  G nn (E − i)G nn (E + i) π n

= lim

  |ψnμ |2 |ψnν |2 π n,μ,ν (E − E μ − i)(E − E ν + i)

= lim

μ  μ   |ψn |4 = |ψn |4 δ(E − E μ ) . π n,μ (E − E μ )2 +  2 n,μ

↓0

↓0

(11.9.9)

The reader will appreciate that only diagonal terms in the foregoing double sum over eigenvectors survive the limit as  goes to zero. I should hurry to add that there is a deep reason for not settling on the average Green function but on average products thereof when attempting to understand transport properties of quasi one-dimensional wires in terms of random band matrices. It turns out that a full characterization requires not just one such quantity but a whole set of multilocal ones like  μ |ψn 1 |2q |ψnμ2 |2q . . . |ψnμk |2q δ(E − E μ ) (11.9.10) μ

with positive integer exponents q. The reason is that finite samples are not selfaveraging in their transport behavior, such that sample-to-sample fluctuations require a probability density for the inverse participation ratio rather than just a mean value. Even though the calculation of the whole set of moments is not much more difficult than that of just the lowest-order set [26], our P, I hold fast to pedagogic principles and present the simplest case first.

11.9.3 One-Dimensional Nonlinear Sigma Model The expression (11.9.9) for the inverse participation ratio closely resembles the average product of two Green functions from which we had previously extracted the two-point correlator of the level density. However, an important difference to the )G(E 2+ ) is that we no longer look at representation indetwo-point correlator G(E 1− pendent traces G(E ± ) = n G nn (E ± ) but at individual matrix elements G nn (E ± ) in the “position” representation where the index n refers to lattice sites. That difference is no obstacle for the superanalytic procedure from the ensemble average via Hubbard–Stratonovich to the sigma model. We just have to carry along the site index and shall thus be led to a separate supermatrix Q i for each site. The sigma model thus becomes one-dimensional, in contrast to the zero-dimensional one of the previous section. The zero-dimensional model will still be contained as the limit √ b  N. We can even obtain the product G nn (E − )G nn (E + ) from a generating function, in analogy to (11.6.5) and (11.6.6). Leaving the specification of an appropriate generating function to the reader as Problem 11.13, I proceed here to apply the moment-generating function (11.4.16) and write the superintegral

11.9

Thick Wires, Banded Random Matrices, One-Dimensional Sigma Model

G nn (E − )G nn (E + ) = (−1)?



  ∗ ∗ dΦ ∗ dΦ z 1n z 1n z 2n z 2n

  4 × exp i

α=1

551

(11.9.11) 

  Φα† L α H − E α + iΛα  Φα ,

not bothering about an overall sign since we know the final result must be positive. This differs from the generating function for the GUE (11.6.9) only by the appear∗ ∗ z 1n z 2n z 2n and in that one and only one energy argument ance of the prefactor z 1n E appears; the positive infinitesimal  remains on display here until eventually dealt with in the fashion of (11.9.9). Clearly, a (4N )-component supervector Φ like (11.6.8) is employed, but it is now convenient to proceed immediately to the “advanced-retarded” notation and order the components as ⎛ ⎞ ⎛ ⎞ Φ1 z1 ⎜η1⎟ ⎜Φ2⎟ ⎟ ⎜ ⎟ Φ=⎜ ⎝z 2⎠ = ⎝Φ3⎠ ; η2 Φ4

(11.9.12)

then the diagonal 4 × 4 matrices L = diag(1, 1, −1, 1) and Λ = diag(1, 1, −1, −1) immediately take the form familiar from (11.6.32). Were the reader inclined to ponder for a moment about the similarity and the reason for the slight differences between (11.6.9) and (11.9.11), that moment would be well spent. As in Sect. 11.6, we proceed to the Gaussian average. The disorder ensemble (11.9.3)and (11.9.4) now gives exp(i i j Hi j u i j ) = exp(− 12 i j Ji j u i∗j u i j ) where ∗ L α Φα j . It is precisely due to the dependence of the variance Ji j on u i j = α Φαi the site indices i, j that now we must deal with the “local” 4 × 4 supermatrix ∗ ˜ αβi = Φαi Φβi Q

(11.9.13)

 rather than with the “global” one in (11.6.11) and have exp(i i j Hi j u i j ) =  ˜ iLQ ˜ j L). Therefore, instead of the average generating function exp(− 12 i j Ji j Str Q (11.6.12), we get    ∗ ∗ (11.9.14) G nn (E − )G nn (E + ) = (−1)? dΦ ∗ dΦ z 1n z 1n z 2n z 2n     1 ˜ ˜ ˜ ˜ Ji j Q i L Q j L − iE × exp Str − Qi L −  Q i LΛ . ij i i 2 With the disorder average out of the way, we attack the integral over the supervector Φ which as usual enters the exponent quartically. The superanalytic Hubbard– Stratonovich transformation (11.6.21) must be modified to deal with the product ˜ j L; it is easy to see that we may use the generalization ˜ iLQ Q

552

11 Superanalysis for Random-Matrix Theory

  1 ˜ iLQ ˜ jL exp Str − Ji j Q (11.9.15) ij 2     1 ˜ iL d N Q exp Str − (J −1 )i j Q i L Q j L − i Qi L Q = (−1) N ij i 2 M4N in which N auxiliary supermatrices Q i appear as integration variables, each to be integrated over its own manifold M4 . As a little aside, we ought to convince ourselves of the validity of the foregoing identity. Really no more than the temporary introduction of an ordinary orthogonal N × N matrix is required to diagonalize the real symmetric variance ˜ i reads form in the Q matrix, O T J O = diag( j1 , j2 , . . . j N ). Thus, the quadratic   ˜ ˜ ˜ ˜ ˜ ˜ ˜ J Str Q L Q L = j Str q L q with q = O . But if Q L is pseuQ i j k L k ik i i ij ij k k i ˜ L such that we may use (11.6.21) N times, dounitarily diagonalizable, so is q k $     +  exp[− 12 k jk Str q˜ k L)2 = k (−1) N dqk exp − Str 21jk (qk L)2 − iqk L q˜ k L . Undoing the orthogonal transformation we confirm (11.9.15). With the help of the superanalytic Hubbard–Stratonovich transformation, (11.9.15), our averaged product of Green functions reads  G nn (E − )G nn (E + ) = (−1)?

 dN Q M4N

  ∗ ∗ dΦ ∗ dΦ z 1n z 1n z 2n z 2n

(11.9.16)

    1 −1 ˜ (J )i j Q i L Q j L − Str Q i L iE − Λ − iQ i L , × exp Str − ij i 2 

and now the Φ-integral has the desired Gaussian form. The superanalytic variant (11.4.17) of Wick’s theorem provides the following generalization of (11.6.22):  G nn

(E − )G

nn

(E + )

= (−1)

?

 BB d N Q gaa (n)grrB B (n) + garB B (n)graB B (n)

M4N

     1 × exp − (J −1 )i j Str Q i L Q j L − Str ln L iE − Λ − iQ i L ; ij i 2 (11.9.17) it may be well to spell out that each site but the nth contributes the usual factor ∗ ∗ z 1n z 2n z 2n Sdet−1 (iE − Λ − iQ i L) while for the nth, to which the prefactor z 1n pertains, Wick’s theorem (11.4.17) and the generating function (11.4.16) give two factors, i.e., the nth superdeterminant of the foregoing form and the combination of matrix elements of g(n) = (E − Q n + iΛ)−1 → (E − Q n )−1

(11.9.18)

in the prefactor; the imaginary infinitesimal will turn out to be dispensable in the 4 × 4 supermatrix g(n) since Q i will become effective with a finite imaginary part. Note that only bosonic elements of g(n) arise in (11.9.17) due to the purely bosonic

11.9

Thick Wires, Banded Random Matrices, One-Dimensional Sigma Model

553

∗ ∗ character of the quantity z 1n z 1n z 2n z 2n “averaged” over; the indices “1” and “3” do indeed refer to the bosonic entries of the supervector (11.9.12) in the advancedretarded notation now employed. The same minor embellishments as in our previous processing of (11.6.22) are in order now, i.e., to use SdetL = −1 and employ Q i L in place of Q i as integration variables. At the same time, I choose to expand the logarithm to first order in the positive infinitesimal  and thus get

 G nn (E − )G nn (E + ) = (−1)?

 d N Q exp − A(Q)

(M4 L) N

× f n exp[−i



(11.9.19)

Str (E − Q i )−1 Λ]

i

where the “action” and the prefactor f n are defined by A(Q) =

 Q2    1  −1 i (J )i j − J0−1 δi j Str Q i Q j + Str + ln E − Q i 2 ij 2J0 i

BB (n)grrB B (n) + garB B (n)graB B (n) f n = gaa

(11.9.20)

and the abbreviation J0 =

N  j=1

Ji j =

λ2 1 + e−1/b ≈ λ2 . 2b 1 − e−1/b

(11.9.21)

Even though there is no explicit large factor decorating the action A(Q), the summation over the N sites suggests that A is proportional to N . The double sum in the first term also shares that property, as becomes obvious when we consider the exponential form of Ji j given in (11.9.4) and its well known inverse [31]. We easily check 2bλ−2  (1 + e−2/b )δi j − e−1/b (δi, j+1 + δi, j−1 ) , 1 − e−2/b  B (11.9.22) 2δi j − δi, j+1 − δi, j−1 , (J −1 )i j − J0−1 δi j = 2 4bλ−2 e−1/b 2b2 B= ≈ . 1 − e−2/b λ2 (J −1 )i j =

At any rate, inasmuch as we are concerned with b  1, a saddle-point approximation appears in order for the Q-integrals. To prepare for that step, we write Q i = Q is + δ Q i and expand the action to second order in the increment δ Q i , A(Q) = A(Q s ) + δ A + δ 2 A + . . .     (J −1 )i j − J0−1 δi j Q sj , δ Q i J0−1 Q is − (E − Q is )−1 + δ A = Str i

j

554

11 Superanalysis for Random-Matrix Theory

1   −1 (J )i j − J0−1 δi j δ Q i δ Q j , δ 2 A = Str 2 ij 1  + Str J0−1 δ Q i2 − (E − Q is )−1 δ Q i (E − Q is )−1 δ Q i . 2

(11.9.23)

Obviously, the saddle-point equation δ A = 0 possesses homogeneous solutions Q is = Q s determined by the familiar equation J0−1 Q s = (E − Q s )−1 ,

(11.9.24)

and by reasoning precisely as in Sect. 11.6, we establish the relevant diagonal solution. Notational differences apart, we recover (11.6.28) and, for the manifold of nondiagonal saddles, (11.6.30). However, even though the diagonal saddle is homogeneous, we must allow for a separate pseudounitary transformation Ti at each site such that the manifold of saddles associated with the ith site reads Q is = E/2 − iπ (E)J0 Ti ΛTi−1 ,

(11.9.25)

where (E) = (2π J0 )−1 (4J0 − E 2 )1/2 is the semicircular mean density of levels which, the reader will recall, was extracted from the analogue (11.5.11) of the homogeneous saddle-point equation (11.9.25) in Sect. 11.5. It is appropriate to pause for a moment and appreciate that the homogeneous solution (11.9.24) can hardly be expected to be reliable when boundary effects come into play for b < ∼ N . Thus, we should expect 1  b  N as the range of validity of our procedure from this point on. As in (11.6.34), we employ pseudounitary matrices T0i to block diagonalize the supermatrices Q i , Q i = T0i

  Q ai 0 T −1 0 Q ri 0i

(11.9.26)

and from (11.6.41), we take the integration measure d Q = N

N , i=1

≡

N ,

d Q ai d Q ri dμ(T0i )

(qaBi − qrBi )2 (qaFi − qrFi )2 (qaBi − qrFi )2 (qaFi − qrBi )2

d Q ai d Q ri dμ(T0i ) F(Q ai , Q ri )

(11.9.27)

i=1

where dμ(T0i ) is as in (11.6.42). The 2 × 2 blocks Q ai , Q ri at each site contain the hard modes whose action increasing fluctuations can be integrated out by the saddle-point approximation. Thus, in analogy to (11.6.50), we get

11.9

Thick Wires, Banded Random Matrices, One-Dimensional Sigma Model

G nn (E − )G nn (E + ) =

 ,

 dμ(T0i ) f n

555

(11.9.28)

i

3 4  Qs 1  −1 × exp − i (J )i j − J0−1 δi j Str Q is Q sj Str i Λ − J0 2 ij i where according to (11.9.18) and (11.9.24), we could replace g(n) by Q sn /J0 . Each of the T0i integrals is over the coset space U (1, 1/2)/U (1/1)U (1/1). A slightly more suggestive form of the foregoing expression arises when we realize from (11.9.25) that the supermatrices Q is depend on the block diagonalizing matrices T0i only through Q(T0i ) ≡ T0i ΛT0i−1

(11.9.29)

and that in the exponent in (11.9.28), we may replace Q is by −iπ(E)J0 Q(T0i ). As already remarked around (11.6.30), these latter dimensionless 4 × 4 supermatrices are the basic constituents of the nonlinear supermatrix sigma model obeying Q(T0i )2 = 1 ,

Q(T0i )† = L Q(T0i )L ,

Str Q(T0i ) = Str Λ = 0 .

(11.9.30)

Together with (11.9.22) they allow us to write G nn (E − )G nn (E + ) =

 ,

 dμ(T0i ) f n e−S({T0i }) ,

(11.9.31)

i

 γ  Str Q(T0i )Q(T0,i+1 ) + π Str Q(T0i )Λ 2 i i  2 γ   =− Str Q(T0i ) − Q(T0,i+1 ) + π Str Q(T0i )Λ , 4 i i

S({T0i }) =

γ = [π (E)J0 ]2 B ∝ 2b2 . The dimensionless (but energy dependent!) parameter γ ∝ 2b2  1 defines a correlation length for the supermatrices Q(T0i ), as becomes especially transparent if we pass to a continuum description through Q(T0i ) − Q(T0,i+1 ) → −∂ Q(x)/∂ x $L N and i=1 → 0 d x. Continuum description or discrete, we find ourselves up to our ears immersed in field theory when we have to implement the one-dimensional sigma model set up in the foregoing expression for the average product of two Green functions, a task a bit more involved than for the zero-dimensional model (cf. Sect. 11.6.4).

11.9.4 Implementing the One-Dimensional Sigma Model Efetov and Larkin first showed that the continuum version of the one-dimensional model describes thick disordered wires [32]. These authors also demonstrated the

556

11 Superanalysis for Random-Matrix Theory

localization of eigenfunctions in the limit of long wires and found the localization length l = 2γ lel . Here I shall keep following Ref. [26] where the full statistics of localization was treated. Let us put our goal in view. The mean inverse participation ratio is given by (11.9.9) and (11.9.31) as N 1  Pn , N n=1  Pn = lim G nn (E − i)G nn (E + i) →0 π   ,  dμ(T0i ) f n e−S({T0i }) . = lim →0 π  i

P=

(11.9.32)

The matrix g(n) whose elements enter the prefactor f n according to (11.9.20) is determined by (11.9.18), (11.9.25), and (11.9.29) as g(n) =

E − iπ (E)Q(T0n ) . 2J0

(11.9.33)

We proceed to do the superintegral in (11.9.32) recursively, working our way inward site per site. We start at the edge, i = 1, and imagine that j − 1 ≥ 1 steps were already taken and yielded Y j (T0 j ) ≡

j−1  ,

dμ(T0i ) e Str



− γ2 Q(T0i )Q(T0,i+1 )−ηQ(T0i )Λ



,

i=1

where

η = π  .

(11.9.34)

We can continue until after (n − 1) steps we arrive with Yn (T0n ) at the site whose contribution Pn to the inverse participation ratio P we focus on; that site requires extra care since it attaches the extra factor f n to the integrand. Along the way we may rewrite the above integral as the recursion relation  Y j (T0 ) = dμ(T0 ) E(T0 , T0 ) Y j−1 (T0 ) ,  γ E(T0 , T0 ) = exp − Str Q(T0 )Q(T0 ) − ηStr Q(T0 )Λ , (11.9.35) 2 Y1 = 1 . But we may equally well work our way inward starting from the other end, at i = N with Y N = 1, and arrive after N − n steps with the contribution Y N −n (T0n ) to the final integral over T0n . Combining all of the pieces, that last integral reads  f n −ηStr Q(T0)Λ e . (11.9.36) Pn = lim η dμ(T0 ) Yn (T0 )Y N −n (T0 ) η→0 (π )2

11.9

Thick Wires, Banded Random Matrices, One-Dimensional Sigma Model

557

Now, the parametrization (11.6.39) and (11.6.40) of the matrix T0 must be invoked, together with the integration measure (11.6.42). The recursion relation (11.9.35) then appears as no small hurdle to jump since the function Y j might depend on all eight independent variables entering T0 , i.e., the Grassmannians ρ, ρ ∗ , σ, σ ∗ and the two complex eigenvalues t B , t F . A decisive step ahead of us is to show that only the two moduli |t B |, |t F | of these eight variables actually do enter. Fearlessly launching ourselves into the fight for that simplification, we scrutinize the kernel E(T0 , T0 ) in (11.9.35). Simply to check is Str Q(T0 )Λ = 4(|t B |2 + |t F |2 ). The evaluation of Str Q(T0 )Q(T0 ) is a little harder but still quite straightforward. Gruesome expressions to walk away from with displeasure can be avoided by replacing the moduli of the eigenvalues t B = |t B |eiϕ B , t F = |t F |eiϕ F of T0 with λ B = 1 + 2|t B |2 λ F = 1 − 2|t F |2

ranging in ranging in

1 ≤ λB < ∞ , − 1 ≤ λF ≤ 1

(11.9.37)

and temporarily employing the abbreviations μB =

%

λ2B − 1 ,

μF =

%

1 − λ2F ,

1 ∗ (ρ ρ − ρρ ∗ + σ σ ∗ − σ σ ∗ ) 2 ρ˜ = ρ − ρ, ρ˜ ∗ = ρ ∗ − ρ ∗ , σ˜ = σ − σ,

iδϕ =

ϕ˜ B = ϕ B − ϕ B − δϕ,

(11.9.38) σ˜ ∗ = σ ∗ − σ ∗ ;

ϕ˜ F = ϕ F − ϕ F − δϕ ;

here primed quantities refer to T0 and thus to integration variables in the recursion relation (11.9.35). Thus equipped, we get Str Q(T0 )Λ = 2(λ B − λ F )

(11.9.39)

= 2(λ B λ B − λ F λ F ) + (λ B − λ F )(λ B − λ F )(ρ˜ ρ˜ ∗ − σ˜ σ˜ ∗ )    −2 1 − 14 ρ˜ ρ˜ ∗ σ˜ σ˜ ∗ μ B μ B cos ϕ˜ B + μ F μ F cos ϕ˜ F   −(ρ˜ ρ˜ ∗ − σ˜ σ˜ ∗ ) μ B μ B cos ϕ˜ B − μ F μ F cos ϕ˜ F   +iσ˜ ρ˜ ∗ μ B μ F e−iϕ˜ B + μ F μ B eiϕ˜ F e−i(ϕ B −ϕ F )   −ρi ˜ σ˜ ∗ μ B μ F eiϕ˜ B + μ F μ B e−iϕ˜ F ei(ϕ B −ϕ F ) .

(11.9.40)

and Str Q(T0 )Q(T0 )

Now, we may seek a solution of the recursion relation (11.9.35) in the form Yk (λ B , λ F ). Accounting for the transformation (11.9.37), we confront

558

11 Superanalysis for Random-Matrix Theory

 dϕ B dϕ F

dρ ∗ dρdiσ ∗ diσ

Y j (λ B , λ F ) = (2π )2 1 0  × exp −2η(λ B −λ F )−(γ /2)Str Q(T0 )Q(T0 ) Y j−1 (λ B , λ F ) . (11.9.41) 

∞

dλ B



1

dλ F −1

1 (λ B − λ F )2



2π

We immediately conclude that the nilpotent phase shifts δϕ in ϕ˜ B , ϕ˜ F within the kernel can be eliminated by simply shifting the integration variables ϕ B , ϕ F

to ϕ˜ B , ϕ˜ F . Thereafter, the four primed Grassmannian integration variables appear only in the differences ρ˜ = ρ − ρ, etc., and may be replaced by the latter; due to d ρ˜ = dρ , etc., the integration measure remains unchanged. The pole in the integration measure at λ B = λ F = 1 deserves comment. It corresponds to the pole at t B = t F = 0 for the original parametrization (11.6.42) and needs to be cancelled if the above integral is to exist. Searching for such cancellation, we inspect the phase-averaged exponential 

 d ϕ˜ B d ϕ˜ F

exp − (γ /2)Str Q(T0 )Q(T0 ) (11.9.42) 2 (2π) 0

 ˜ σ˜ ∗ − E 1∗ iσ˜ ρ˜ ∗ + E 2 (ρ˜ ρ˜ ∗ − σ˜ σ˜ ∗ ) + E 3 ρ˜ ρ˜ ∗ σ˜ σ˜ ∗ = e−γ (λ B λ B −λ F λ F ) E 0 + E 1 ρi 2π

where the four quantities E i are functions of λ B , λ F , λ B , λ F and read E 0 = I0 (γ μ B μ B )I0 (γ μ F μ F ) ,  γ

E1 = μ B μ F I1 (γ μ B μ B )I0 (γ μ F μ F ) + (B ↔ F) e−i(ϕ B −ϕ F ) , 2 γ3 E2 = − (λ B − λ F )(λ B − λ F )E 0 2  4 + μ B μ B I1 (γ μ B μ B )I0 (γ μ F μ F ) − (B ↔ F) , (11.9.43) E3 =

3 γ2 (λ B − λ F )(λ B − λ F ) (λ B λ B + λ F λ F )E 0 2  4 − μ B μ B I1 (γ μ B μ B )I0 (γ μ F μ F ) − (B ↔ F)

≡ (λ B − λ F )(λ B − λ F )L(λ B , λ F |λ B , λ F ) . Cancellation indeed! Due to the definition of μ B , μ F in (11.9.38) and the smallargument behavior of the Bessel function I1 , the three functions E 1 , E 2 , E 3 all approach zero linearly as λ B , λ F → 1 and thus give rise to converging integrals over λ B , λ F in (11.9.41). We can wave good-bye for good to the terms with E 1 , E 2 since these are annulled by the subsequent fourfold Grassmann integral. The term with E 3 does survive$ since it comes with the maximal Grassmann monomial which inte˜ σ˜ ∗ d σ˜ ρ˜ ρ˜ ∗ σ˜ σ˜ ∗ = 1. Finally, the term with E 0 does not vangrates to unity, d ρ˜ ∗ d ρd ish at the boundary λ B = λ F = 1 and thus engenders $ a diverging bosonic integral; it is multiplied by the vanishing Grassmann integral d ρ˜ ∗ dρdσ ∗ dσ = 0, however,

11.9

Thick Wires, Banded Random Matrices, One-Dimensional Sigma Model

559

and the formal product ∞ × 0 is assigned the value exp[−γ (λ B − λ F )]Y j−1 (1, 1) by the Parisi–Sourlas–Efetov–Wegner theorem of Sect. 11.4.5. Thus, the recursion relation takes the form (11.9.44) Y j (λ B , λ F ) = e−γ (λ B −λ F ) Y j−1 (1, 1)  1  ∞ λB − λF



+ dλ B dλ F

L(λ B , λ F |λ B , λ F ) λ − λ 1 −1 B F



× e−2η(λ B −λ F ) Y j−1 (λ B , λ F ) . Before trying to solve for Y j , it is well to pull the expression (11.9.36) for the inverse participation ratio Pn to the level reached for the function Y j , i.e., to do the phase and Grassmann integrals. To that end, we should write the quantity f n /(π ρ)2 more explicitly. Recalling (11.9.20) and (11.9.18), we easily find f n /(π ρ)2 = −2ρ ∗ ρσ ∗ σ λ B (λ B − λ F ) + . . . ,

(11.9.45)

where the dots refer to submaximal Grassmann monomials annulled by the subsequent integration, analogous to the fate of the functions E 1 , E 2 in the recursion relation. The desired reformulation of (11.9.36) is Pn = lim 2η

 ∞

η→0

1

dλ B dλ F λ B −2η(λ B −λ F ) e Y N −n (λ B , λ F)Yn (λ B , λ F) −1 λ B − λ F 1

(11.9.46)

and suggests the next step ahead by its very appearance: The integral must be proportional to 1/η for the limit η → 0 to exist. Such singular behavior can be contributed only by large values of λ B . But in that range, λ B − λ F → λ B , and we are led to suspect that Yn (λ B , λ F ) becomes independent of λ F . The ansatz Yn (λ B , λ F )|λ B λ F = yn (2ηλ B ) ≡ yn (z) ,

y1 = 1

(11.9.47)

will in fact turn out to be consistent and immediately embellishes Pn to 

∞

dze−z y N −n (z)yn (z) .

Pn = 2

(11.9.48)

0

Now the recursion relation (11.9.44) must be updated to the ansatz (11.9.47) in the large-λ B limit. As a first simplification, we can dispense with the exponentially small boundary term, and thus  y j (2ηλ B ) = 1



∞

dλ B

λ B e−2ηλ B λ B



1

dλ F L(λ B , λ F |λ B , λ F ) y j−1 (2ηλ B ) .

(11.9.49)

−1

The integral over λ F becomes explicitly doable once we appreciate that μ B μ B ≈ λ

λ B λ B − 12 ( λ BB + λλ B )  1 and replace the Bessel functions with large arguments B

560

11 Superanalysis for Random-Matrix Theory

by their asymptotic forms expansions In (z) = (2π z)−1/2 ez (1 − 4n8z−1 . . .). Then, the kernel L(λ B , λ F |λ B , λ F ) takes the asymptotic form 2

 γ  λ

γ 3/2 λB  B

exp − + λ + γ λ L(λ B , λ F |λ B , λ F ) = & F F 2 λB λ B 2 2π λ B λ B 3 1 λ B  1  λ B × λ F λ F + (11.9.50) +

+ 2γ 2 λB λB 4 ×I0 (γ μ F μ F ) + μ F μ F I1 (γ μ F μ F ) which we can subject to the λ F -integration required in (11.9.49). The integral identity (see Problem 11.14) 

√  &  sinh a 2 + b2 2 d xe I0 b 1 − x = 2 √ a 2 + b2 −1 1

ax

(11.9.51)

and its first derivatives w. r. t. the real parameters a, b yield 

:

 ∞ z −z  z 

dz e Lγ duu −3/2 e−uz L γ (u)y j−1 (uz) , y j−1 (z ) = y j (z) = z z 3 0 0 : γ − γ (u+ 1 )  1 1 sinh γ L γ (u) = e 2 u cosh γ − + u+ sinh γ . (11.9.52) 2π 2γ 2 u ∞



One arrow remains in our quiver and will be shot presently, the largeness of the parameter γ ∝ 2b2  1. The kernel L γ (u) therefore has a sharp peak of width √ 1/ γ  1 at u = 1. Promising to demonstrate consistency later, I assume that the function y j (uz) varies slowly in u across the width of the peak of L γ (u), whatever the value of z may be. Then, it is advisable to shift and rescale the integration variable as ξ u =1+ √ γ

(11.9.53)

√ such that the following expansions in powers of 1/ γ become available: :   1 γ γ − γ (u+ 1 ) 1 1 1 L γ (u) = e 2 u 1− + u+ 2 2π 2γ 2 u :   γ − ξ2 ξ2 ξ3 ξ4 ξ6 1 = e 2 1− + + √ − + + O(γ −3/2 ) , 2π 4γ 4γ 2 γ 2γ 8γ 3ξ 15ξ 2 u −3/2 = 1 − √ + + O(γ −3/2 ) , 2 γ 8γ

11.9

Thick Wires, Banded Random Matrices, One-Dimensional Sigma Model −zu

e

−z

561

 z2ξ 2 zξ −3/2 ) , + O(γ 1− √ + γ 2γ



=e

zξ z 2 ξ 2

y j−1 (uz) = y j−1 (z) + √ y j−1 (z) + y (z) + O(γ −3/2 ) . (11.9.54) γ 2γ j−1 Upon doing the ξ -integrals, we transform the recursive integral equation (11.9.52) into the recursive differential equation14 ez y j (z) = y j−1 (z) +

 z2  y j−1 (z) − 2y j−1 (z) + y

j−1 (z) . 2γ

(11.9.55)

We should appreciate that the unit steps of the site index j are minute compared to the correlation length γ ; thus, a continuum approximation according to j → 2γ τ is indicated, with τ ranging in the interval [0, N /(2γ ) ≡ x]. Finally, we anticipate that y j (z) will decay with growing z on a scale ∝ 1/γ and introduce the rescaled independent variable y = 2γ z, N ]. 2γ (11.9.56) Such rescalings yield e−z = e−y/2γ → 1, reveal the first two terms in the curly bracket in (11.9.55) as negligible, and turn the recursion relation into the partial differential equation y j (z) = y2γ τ (y/2γ ) ≡ Y (y, τ )

∂ Y (y, τ ) = ∂τ

 y2

with

y ∈ [0, ∞) ,

τ ∈ [0, x =

 ∂2 ˆ (y, τ ) − y Y (y, τ ) ≡ LY ∂ y2

(11.9.57)

and the “initial” condition in (11.9.35) and (11.9.47) into Y (y, τ = 0) = 1 .

(11.9.58)

The very absence of all parameters from the foregoing initial-value problem demonstrates the consistency of the scaling assumptions made above, provided, of course, that a solution exists. The (local) inverse participation ratio Pn last given in (11.9.48) can also be expressed in terms of the function Y (y, τ ), 1 Pn → γ



∞

dy Y (y, x − τ )Y (y, τ ) .

0

Upon averaging over sites according to (11.9.32), we express the mean participation ratio as 14

Equation 72 on p. 3818 of Ref. [26] contains typos and should be read as our Eq. (11.9.55).

562

11 Superanalysis for Random-Matrix Theory

P=

 x  N 1  2 ∞ Pn = dy dτ Y (y, x − τ )Y (y, τ ) . N n=1 N 0 0

(11.9.59)

This is already an important asymptotic result, worthy of highlighting. We may read the prefactor PGUE ≡ 2/N of the foregoing integral as the inverse participation ratio of the GUE and refer our P to that unit. The result, P PGUE

 =



∞

x

dy 0

dτ Y (y, x − τ )Y (y, τ ) ≡ β(x) ,

(11.9.60)

0

depends only on the scaling parameter x = N /2γ ∝ N /b2 .

(11.9.61)

As already noted above, Casati, Molinari, and Izrailev [24] established such scaling through numerical work and even conjectured that β(x) = 1 + x/3. Such surprisingly simple behavior was indeed borne out by Fyodorov’s and Mirlin’s analysis of Ref. [26] which we are spreading out here. It will be convenient to go for the final goal via the Laplace transform 

∞

˜ p) = β(

d x e− px β(x) =

0



∞

dy Y˜ (y, p)2 .

(11.9.62)

0

Solutions of the above partial differential equation may be sought as composid2 tions of eigenfunctions of the operator Lˆ = y 2 dy 2 − y. Such eigenfunctions which decay to zero for y → ∞, are related to the modified Bessel functions K r (t) as √ √ fr (y) = 2 y K r (2 y) and come with the eigenvalues (r 2 − 1)/4. For imaginary indices r = iν, ν ∈ (0, ∞), the functions f iν (y) are mutually orthogonal and can be used as a basis in the sense of the Lebedev–Kontorovich transformation [33]. That ˜ transformation relates a function F(x) to its transform F(ν) as 

∞

˜ dν K iν (x) F(ν) ,  ∞ dx 2 ˜ K iν (x)F(x) , F(ν) = ν sinh π ν π x 0

F(x) =

(11.9.63)

0

provided (1) that F(x) is piecewise differentiable in (0, ∞) and (2) thet there is some positive number  such that 

 0

d x x −1 |F(x) ln x| < ∞

 and 

∞

d x x −1/2 |F(x)| < ∞ .

(11.9.64)

A little thought shows that we cannot naively invoke that transformation to repν 2 +1 √ √ resent Y (y, τ ) as a superposition of eigenfunctions ei 4 τ 2 y K iν (2 y) since the

11.9

Thick Wires, Banded Random Matrices, One-Dimensional Sigma Model

563

initial condition Y (y, 0) = 1 would yield the function F(x) = 1/x which does not qualify as the member of a Lebedev–Kontorovich pair. There is a way out found by Fyodorov and Mirlin and easily followed. The differential operator Lˆ has one additional eigenfunction which decays to zero for y → ∞ and pertains to the √ √ eigenvalue 0; indeed, Lˆ f 1 (y) = 0 where f 1 = 2 y K 1 (2 y). By simply including that eigenfunction, we represent the function Y (y, τ ) in search as √ √ Y (y, τ ) = b1 2 y K 1 (2 y) +



∞

dν b(ν) e−

ν 2 +1 4 τ

√ √ 2 y K iν (2 y)

(11.9.65)

0

and determine the expansion coefficients b1 and b(ν) from the initial condition Y (y, 0) = 1. That condition reads √ √  ∞ 1 − b1 2 y K 1 (2 y) √ √ dν b(ν) 2 y K iν (2 y) = √ 2 y 0

(11.9.66)

and reveals b(ν) as the would-be Lebedev–Kontorovich transform of the function F(x) = x1 − b1 K 1 (x). Due to the small-argument behavior of the Bessel function, K 1 (x) → x1 + x2 ln x2 for x → 0, F(x) → (1−b1 ) x1 )−b1 x2 ln x2 , and we conclude that the coefficient b1 is uniquely determined as b1 = 1. No other choice would meet the first of the conditions (11.9.64); the second of these is also met since for b1 = 1, the large-x behavior F(x) → x1 . Therefore, our function Y (y, τ ) is determined up to quadratures, namely, 2 b(ν) = ν sinh π ν π



∞ 0

1 dx 2 ν sinh πν 2 K iν (x) − K 1 (x) = x x π 1 + ν2

(11.9.67)

and the composition (11.9.65) 

2 √ Y (y, τ ) = 2 y K 1 (2 y) + π √



 2 ν sinh πν √ − 1+ν τ 2 4 dν e K iν (2 y) . (11.9.68) 1 + ν2

∞ 0

The final expression for the expansion coefficient in (11.9.65) as well as the following Laplace transform  (μ + 1) + u 2 (μ − 1) 1 ∞ √ ˜ du u Jμ−1 (2 yu) Y (y, p) = 2 2 p 0 (1 + u )  1 ∞ √ ≡ du F(u) Jμ−1 (2 yu) , p 0 & μ = 4p + 1

(11.9.69)

were found in Ref. [26] by some ingenious juggling with integrals involving trigonometric, hyperbolic, and Bessel functions. I shall sketch these interesting calculations below but propose to proceed to the final result first.

564

11 Superanalysis for Random-Matrix Theory

˜ p) of (11.9.61), Inserting the Laplace transform Y˜ (y, p) into the scaling function β( we obtain 



∞

˜ p) = p 2 β(

∞

dy 0

2

√ du F(u) Jμ−1 (2 yu)

.

(11.9.70)

0

√ integration variable y to t = 2 y and the orthogonality $ ∞A simple change of the 1 0 dt t Jν (tu)Jν (tv) = u δ(u − v) of the Bessel functions produce 1 p β( p) = 2



∞

2˜

du u −1 F(u)2 = p +

0

1 . 3

(11.9.71)

Reverting to the Laplace transform gives the celebrated scaling function β(x) = 1 + x/3 .

(11.9.72)

It is customary to reexpress that result in terms of the localization length l(N , γ ) ∝ 1/P. Remembering that x = N /2γ , we find l(N , γ ) as the geometric mean of its limiting values for complete delocalization and full localization 1 1 1 = + . l(N , γ ) l(N , ∞) l(∞, γ )

(11.9.73)

Clearly, then, there is no threshold or critical value of x for the onset of localization but rather a continuous transition from the delocalization typical of GUE matrices to well pronounced localization, as the scaling parameter x grows from O( N1 ) to O(N ).

11.10 Problems 11.1 Let ρ and σ be M × M matrices with anticommuting entries. Using (ρi∗j )∗ = σ = −σ˜ ρ, ˜ and (ρσ )† = −σ † ρ † . −ρi j , show that ρ †† = −ρ, ρ 11.2 A common factor x in a bosonic row or column of a superdeterminant can be pulled out as the factor x multiplying the superdeterminant; in contrast, a factor y in a fermionic row or column as 1/y. True? 11.3 Let ρ and ρ ∗ with (ρ ∗ )∗ = −ρ be a pair of Grassmannians and c an ordinary complex number. Define a new pair η = cρ and η = cρ ∗ = (c/c∗ )η∗ . Show that (η) = −η such that one may call η a complex conjugate of η.  † aρ with real a and imaginary b, ρ b even though not Hermitian, is still diagonalized by (11.3.31) and (11.3.32); give a 11.4 Show that the 2 × 2 supermatrix H =

11.10

Problems

565

redefinition of the complex conjugation of Grassmann variables that restores unitarity of the matrix U in (11.3.32). Convince yourself that no integral over the Grassmannians η and η∗ is affected by such a redefinition of the complex conjugation. 11.5 Use the generating function (11.4.5) to prove Wick’s theorem Si1 . . . Sin S ∗j1 . . . S ∗jn  =



(a −1 )i1 ,P j1 . . . (a −1 )in ,P jn

P

where the sum is over all n! permutations of the { j} and Si S ∗j  = (a −1 )i j . ˆ 0) from (11.5.7) invariant under unitary transforma11.6 Why is the action A( Q, ˆ tions of the 2 × 2 matrix Q? $11.7 Show that the saddle-point approximation gives the Grassmann integral dη1 dη2 exp(−N η2 η1 ) exactly. Generalize to arbitrary Gaussian integrals, Grassmann, ordinary, and super. ∗

11.8 Convince yourself of G(E + ) = G(E − ) . 11.9 Show that pseudounitary N × N matrices form a group. ˆ 11.10 Show that pseudounitary matrices leave the scalar product (Φ, LΦ) invariant ˆ in the sense (T Φ, Lˆ T Φ) = (Φ, LΦ) .    hB 0 a ρ† U −1 be a Hermitian 2 × 2 supermatrix, = U 11.11 Let H = 0 hF ρ b $ and imagine a superintegral d H f (H ) where d H = dadbdρ ∗ dρ. Change integration variables to the eigenvalues h B , h F and the Grassmannian “angles” η∗ , η in the diagonalizing ∗matrix U given in (11.3.31) and (11.3.32). Show that the Berezinian ,ρ ) = (h B − h F )−2 . reads J ( h a,b,ρ ∗ B ,h F ,η ,η 

11.12 Convince yourself of the equivalence of the delta functions (11.6.1) and (11.6.3) for 2 × 2 supermatrices. Argue that the two i’s in the integral representation could be replaced by any complex number. Why does the factor (2π)−1 in (11.6.1) have to be dropped for 4 × 4 supermatrices? 11.13 Which generating function would have the product of two matrix elements of the resolvent as derivatives? Which point of Chap. 7 provides a clue? Write the generating function as a superintegral and give the expression provided by the onedimensional nonlinear sigma model. 11.14 Even though possibly not aware of it, you know the integral identity (11.9.51). Consider the$Fourier transform of $ ∞a spherically symmetric function F(|x|) in three dimensions, d 3 xeik·x F(|x|) = 0 drr 2 F(r ) f (kr ) where

566

11 Superanalysis for Random-Matrix Theory





2π

f (ρ) =

dϕ 0

0

π

dθ sin θ eiρ cos θ = 4π

sin ρ ρ

To save labor the polar axis was chosen parallel to the vector k here. Now choose axes less wisely and enjoy 

1 iax

dx e −1

√  &  sin a 2 + b2 2 J0 b 1 − x = 2 √ . a 2 + b2

as the fruit of lesser wisdom. Analytic continuation to imaginary a, b produces (11.9.51).

References 1. K.B. Efetov: Adv. Phys. 32, 53 (1983) 2. J.J.M. Verbaarschot, H.A. Weidenm¨uller, M.R. Zirnbauer: Phys. Rep. 129, 367 (1985) 3. K.B. Efetov: Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997) 4. T. Guhr, A. M¨uller-Groeling, H.A. Weidenm¨uller: Phys. Rep. 299, 192 (1998) 5. H. Flanders: Differential Forms with Applications to the Physical Sciences (Dover, New York, 1989) 6. F.A. Berezin: Introduction to Superanalysis (Reidel, Dordrecht, 1987) 7. F.A. Berezin: Method of Second Quantization (Academic, New York, 1966) 8. G. Parisi, N. Sourlas: Phys. Rev. Lett. 43, 744 (1979) 9. A.J. McKane: Phys. Lett. A 76, 33 (1980) 10. F. Wegner: Z. Physik 49, 297 (1983) 11. F. Constantinescu, F. de Groote: J. Math. Phys. 30, 981 (1989) 12. Y.V. Fyodorov: In E. Akkermans, G. Montambaux, J.-L. Pichard, J. Zinn-Justin (eds) Les Houches, Session LXI, 1994, Mesoscopic Quantum Physics (Elsevier, Amsterdam, 1995) 13. L. Sch¨afer, F. Wegner: Z. Phys. B 38, 113 (1968) 14. J.J. Duistermaat, G.J. Heckman: Invent. Math. 69, 259 (1982) 15. G. Hackenbroich, H.A. Weidenm¨uller: Phys. Rev. Lett. 74, 4118 (1995) 16. L. Erd˜os, J. Ramirez, B. Schlein, T. Tao, V. Vu, H. Yan: arXiv:0960.44v1[math.PR](24 Jun 2009) 17. M. Kieburg, T. Guhr: to be published 18. N. Lehmann, D. Saher, V.V. Sokolov, H.J. Sommers: Nucl. Phys. A 582, 223 (1995) 19. A.M. Khoruzhy, B.A. Khoruzhenko, L.A. Pastur: J. Phys. A 28, L31 (1995) 20. J.D. Anna, A. Zee: Phys. Rev. E 53, 1399 (1996) 21. A.D. Mirlin, Y.V. Fyodorov: J. Phys. A 24, 2273 (1991) 22. Y.V. Fyodorov, A.D. Mirlin: Phys. Rev. Lett. 67, 2049 (1991) 23. S.N. Evangelou: J.Stat. Phys. 69, 361 (1992) 24. G. Casati, L. Molinari, F. Izrailev: Phys. Rev. Lett. 64, 16 (1990); J. Phys. A24, 4755 (1991) 25. S.N. Evangelou, E.N. Economou: Phys. Lett. A151, 345 (1991) 26. Y.V. Fyodorov, A.D. Mirlin: Int. J. Mod. Phys. B8, 3795 (1994) 27. M. Ku´s, M. Lewenstein, F. Haake: Phys. Rev. A44, 2800 (1991) 28. L. Bogachev, S. Molchanov, L.A. Pastur: Mat. Zametki 50, 31 (1991); S. Molchanov, L.A. Pastur, A. Khorunzhy: Theor. Math. Phys. 73, 1094 (1992) 29. M. Feingold: Europhys. Lett. 71, 97 (1992) 30. G. Casati, V. Girko: Rand. Oper. Stoch. Eq. 1, 1 (1993)

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Index

A Accelerated decoherence, 285, 288–291 Action and angle variables, 146 Anderson model, 248, 251, 547 Andreev gap, 38, 472 Andreev scattering, 38 Antidot structures, 475 Antilinearity, 16, 44, 315 Antiunitarity, 16, 17, 44, 315 Antiunitary symmetries, 4, 15–45, 315, 319, 321 Asymptotic level spacing distributions, 86 Avoided crossing, 48, 49, 188 B Baker map, 355, 475 Band matrices, 247, 250, 547–564 BBGKY hierarchy, 236 BCS-Hamiltonian, 38 Berezinian, 99, 501, 520, 525 Berry conjecture, 79, 222 Berry’s diabolo, 58 Berry–Robnik conjecture, 472 Bifurcation, 473 Billiard, 79, 215, 222 on surface of negative curvature, 192 Bloch functions, 249 Bogolyubov-deGennes Hamiltonian, 39 Bohigas–Giannoni–Schmit conjecture, 8, 61, 170 Born approximation, 281, 330, 338 Bose–Einstein condensate, 2, 475 Break time, 8, 10, 254, 264, 332, 333, 335 Brownian motion, 9, 170, 224, 243 C Calogero–Moser dynamics, 176 Canonical ensemble, 195, 208 Canonical transformation, 145, 342

Canonical transformations of Hamiltonians or Floquet operators orthogonal, 21, 28, 34 symplectic, 27, 35 unitary, 19, 24, 34 Cardioid billiard, 364 Cat map, 192, 355, 396, 405 Caustic, 146, 393, 407, 414, 476 p-space, 147, 412 q-space, 147, 412 Central limit theorem, 105 Chaotic resonator, 475 Chapman–Kolmogorov equation, 226 Charge conjugation, 39, 43 Chiral symmetry, 43 Chirikov’s standard map, 253, 326 Circular ensembles of random matrices, 84, 208, 290, 337 Cluster function, 83, 101, 103, 110, 112, 119–121, 137, 142, 167, 236 Cocycle decomposition, 351 Codimension of a level crossing, 50–53, 68, 85, 149, 180, 188, 193 Coherence, 285 Coherent states, 286, 337 Coherent states of an angular momentum, 268, 336 Collapse and revival, 2, 12, 273 Collision time, 224 Complex (quasi) energy, 10, 85, 279, 296–303 Complex conjugation of Grassmannians, 491, 497, 514, 564 Conductance, 213 Conjugate points, 362, 391, 408 Conventional time reversal, 17, 27 Cooper pair, 38 Coulomb gas model, 230, 236–243 Coupling agents, 282

F. Haake, Quantum Signatures of Chaos, Springer Series in Synergetics, 3rd ed., C Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-05428-0, 

569

570 Covariance under time reversal of Floquet operators, 33 Cubic repulsion, 300, 316, 319, 321 Cumulants, 113, 119, 120 Cycle expansion, 474 Cycle structure of permutations, 374 D Damped rotator, 326–335 Degree of level repulsion, 5, 12, 36, 50, 52 Delta function of a Grassmann variable, 542 Density–density correlation function, 82, 85, 161, 165 Detailed balance, 230, 321–326 Determinants as Gaussian integrals, 96–101, 220, 304 Diagonal approximation, 437 Diffusion, 8, 225, 247, 253, 328, 333 Disordered systems, 248, 261, 482, 547 Dyadic vector, 289, 291 Dyson–Pastur equation, 238 E Ehrenfest time, 106, 406, 473 Eigenvector distributions, 76 Einstein–Brillouin–Keller approximation, 147, 183 Energy-dependent propagator, 398 Equilibration of spectra, 224–234 Ergodicity, 106, 146, 188, 351, 355 of level dynamics, 188 of random-matrix ensembles, 62, 81, 121, 124, 209, 225 Euler–MacLaurin summation formula, 313 Exponential instability of initial-value problem, 348 Exponential proliferation of periodic orbits, 13, 357, 389, 406, 426, 434, 436 Exponential stability of boundary value problem, 362 F Fermi’s golden rule, 284, 287, 336 Fictitious-particle dynamics, 9, 174, 176, 209 Fidelity decay, 4 Floquet operator, 9, 10, 12, 29, 32, 33, 79, 170, 251 Flow, 15 Fluctuation-dissipation theorem, 282 Focal points, 393 Fokker–Planck equation, 227, 228, 236 Form factor, 101–103, 110, 112, 124, 137 Fresnel integral, 101, 386 Frobenius–Perron operator, 359, 474

Index Functional equation, 118, 406 Furstenberg’s theorem, 250, 254 G Gap probability, 88, 151, 195 Gaussian ensembles of random matrices, 61 Gaussian ensembles of random matrices, 83, 209, 212, 228, 243, 301, 303, 314 Gaussian integral, multiple, 96, 260, 277, 304 Generating function, 343, 432 Ghost orbits, 473 Ginibre’s ensemble, 303 Graphs, 405 Grassmann variables, 96, 304, 337 complex conjugation of, 484, 496, 497 Green function, 73, 216, 258, 423, 482, 508, 510, 546 advanced, 429, 517 retarded, 429, 517 Gross–Pitaevski equation, 2, 475 H Hadamard-Gutzwiller model, 364 Hamilton–Cayley theorem, 112, 179, 398, 406 Hamilton–Jacobi equation, 2, 57 Hamiltonian embedding of dissipative dynamics, 279 Hard modes, 520, 554 Hartwig–Fischer theorem, 127 Heat bath, 279, 329, 336 Heisenberg time, 13, 106, 473 symplectic case, 112 HOdA sum rule, 355, 361, 426, 438, 443 Hole probability, 310–314 Hubbard–Stratonovich transformation, 484, 499, 506, 546 Hydrogen, 28, 475 Hyperbolicity, 355 I Inverse participation ratio, 549 J Jacobi equation, 391 Jacobi identity, 174, 243 Jacobi matrix and determinant, 344 Jacobian for Grassmann variables, see Berezinian Joint distribution of eigenvalues of Gaussian ensembles, 66, 68, 303 K Kick, 30

Index Kicked rotator, 7–9, 12, 31, 33, 247, 326, 355, 482 experiment, 248 limit of top, 274 vs Anderson model, 251 Kicked top, 2–5, 11, 12, 19, 31, 33, 35, 48, 79, 135, 248, 264, 274, 299, 355 classical phase space of, 184 damped, 297 Kramers’ degeneracy, 4, 5, 22, 34, 50, 57, 64, 102, 111, 188 L Lagrangian manifold, 407–413, 476 in the energy shell, 410 Langevin equation, 328 Lax form, 178, 243 Legendre transformation, 399 Leibnitz product rule, 174 Level clustering, 4 Level repulsion, 75 Level width, 321 Level clustering, 12, 75, 145, 249 Level crossing, 149 intermultiplet, 180 intramultiplet, 181 Level curvature, 212–220 Level repulsion, 4, 10, 12, 47, 247, 249 complex levels, 299 Level spacing distribution complex levels, 309 Level staircase, 75 Level velocity, 221–224 Level width, 10, 12 Lie algebra so(N ), 40 sp(N ), 43 u(N ), 20 Lie group O(N ), 21 S O(N ), 21 Sp(N ), 27 U (N ), 20 Liouville picture, 346 Liouville–von Neumann equation, 280 Lloyd’s model, 249, 258, 549 Local and global energy scales, 74, 234–242 Local energy average, 429 Local stretching rate, 350 Localization, 246–276, 482, 546–564 destruction of, 279, 326, 332 Localization length, 10, 12, 140, 189, 191, 249, 258, 263, 548, 549

571 Lyapunov exponent, 1, 6, 13, 250, 348, 388 of endless trajectories, 351 of periodic orbits, 351, 353 M Map, 29 Markovian processes, 10, 226, 280, 281, 330, 338 Maslov index, 147, 394 Master equation, 10, 280–288, 329, 335, 336 Microcanonical ensemble, 179, 192, 194, 195 Microreversibility, 321 Microwave resonators, 56, 213 Minimality vs extremality of the action, 390 Mixed phase space, 355, 405, 472–475 Mixing, 355 Moment-generating function, 499 Monodromy matrix, 353, 356, 395, 419, 424 Morse index, 387–391, 395, 399, 417 Morse theorem, 391 Multiplet, 47, 150, 169, 180 N Nearly degenerate perturbation theory, 48, 188, 319 Newton’s formulae, 113, 394, 398 Nilpotent, 491, 497, 502, 514 Nonconventional time reversal, 27, 44 Nonseparability, 2, 57 Nonstandard symmetry classes, 36, 54, 68, 472 Normal form, 473 codimension of, 473 Numerical variable, 491, 494, 497, 514 O Ornstein–Uhlenbeck process, 227 Orthogonal polynomials, 120 Overdamped pendulum, 285 P Parisi–Sourlas–Efetov–Wegner theorem, 504, 522, 533, 545 Particle-hole symmetry, 39 Partner orbit, 363, 368 action difference, 369 Path integral, 385 Pauli master equation, 284, 330 Pauli matrices, 18, 25 Pechukas–Yukawa gas, 9, 180, 209, 243 confined, 209 Periodic orbits, 13, 80, 140, 146, 153, 192 bunches of, 363 hierarchies, 380 Periodically kicked systems, 33

572 Pfaffian, 98, 109, 230 Planck cell, 269, 472 Poincar´e invariant, 347, 371 Poincar´e map, 353, 406 Poincar´e surface of section, 352 Poisson brackets, 15, 145, 174, 242, 266, 342 from commutators, 184 Poisson summation formula, 153, 159 Poissonian ensemble, 233 Poissonian process, 4, 13, 150, 167, 301, 302, 337 Pollicott–Ruelle resonances, 474 Porter–Thomas distribution, 79 Primitive periodic orbit, 394 Pseudo-orbit, 365, 425 Pseudo-vector, 28 Pseudounitary symmetry, 512 Q Quantum chaos criterion, 57, 79, 279 Quantum map, 30, 102, 250, 252, 254, 289, 296, 308, 314, 332 Quantum measurement process, 288 Quantum well, 475 Quasi-energy, 4, 30, see Complex (quasi) energy Quasi-periodicity, 1, 251, 264, 270, 272, 279, 293 Quaternion real matrices, 26, 27, 66 R Random potential, see Disordered systems Random unitary matrices, 61 Random-phase approximation, 252, 263, 276, 328 Rate equation, 284, 324, 330 Recurrence, 1, 2, 11, 270, 273, 288 Replica trick, 261 Resummation of periodic orbits, 406 Riemann zeta function, 96, 118 Riemann–Siegel lookalike, 117, 422, 427, 434, 541 Ruelle-Pollicott resonances, 359 S Saddle-point integration, 487 Scars, 80, 474 Schr¨odinger cat state, 10, 286 long-lived, 287 Second variation, 386, 508, 521 Secular coefficients, 112, 129, 139 Self-averaging, 53, 189, 290, 291 Self-correlation terms, 397 Self-encounter, 13, 363

Index 2-encounter, 363, 366 l-encounter, 373 duration, 368 Self-inversiveness, 117, 398 Semi-Poissonian distribution, 167 Semicircle law, 73, 75, 83, 141, 215, 228, 234, 490, 546 Semiclassical limit, 144–167, 383–475 Sensitivity to initial data, 3 to perturbations, 4, 254, 273 Shadowing theorem, 365 Sieber–Richter pair, 364 Sigma model, 453, 482, 515 advanced-retarded notation, 518 Bose-Fermi notation, 518 one-dimensional, 547–564 rational parametrization, 533 semiclassical construction orthogonal class, 461 unitary class, 449 Soft modes, 520 Sparse matrices, 546 Spectral determinant, see Secular polynomial Spin relaxation, 281 Stability matrix, see Monodromy matrix Stable manifold, 411 Stationary-phase approximation, 155, 156, 160, 386 Strange attractor, 290, 292, 327 Stretching factor, 350 Stroboscopic description, 9, 29, 354 Sum rule of Hanny and Ozorio de Almeida, see HOdA sum rule Superfluorescence, 283 Superintegral, 483 Supermatrix, 491 Superposition of spectra, 151, 302, 303 Supertrace, 492 Supervector, 491 Sutherland–Moser dynamics, 176 Symmetry breaking, 224–234 Symplectic structure, 347 T Taylor series for level spacing distributions, 95 Tenfold way, see Nonstandard symmetry classes Tetrad, 317 Thomas–Fermi distribution, 154, 158, 159, 168, 405 Thouless formula, 260 Thouless time, 548

Index Time ordering, 29, 32 Time reversal, 15–45 Time-scale separation, 235–242, 247, 279, 281, 285 Toeplitz determinant, 104, 125, 127 Torus quantization, 7, 147 Trace formula, 13 flows, 404 maps, 396 Transition from clustering to repulsion, 170, 244 Transitions between universality classes, 9, 170, 233–242, 244 Trotter’s product formula, 384 Truncated exponential, 309 Turning point, 399 Two-body collisions, 187 U Unfolding inhomogeneous spectra, 75, 158, 166, 239, 297, 300 Universality of spectral fluctuations, 9, 47, 61, 170, 383

573 V Van Vleck propagator, 383, 387, 393, 413, 475 Vandermonde determinant, 86, 107, 231, 304 Von Neumann–Wigner theorem, 48 W Wave chaos, 2, 56, 475 Wedge product, 347, 498, 528 Weyl symmetry, 531 Weyl’s law, 74, 154, 159, 190, 404–406, 423 Wick’s theorem, 499, 552, 565 for Grassmann variables, 500 superanalytic, 502 Wigner distributions, 70, 75, 86, 141, 302, 334 Wigner function, Husimi function, 474 WKB, 183, 415 Wojciechowski dynamics, 176 Z Zaslavsky’s map, 326, 331 Zeta function, 360, 422

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