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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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<ν
μ=1
with β = 1, 2, or 4. Clearly, the Gaussian factor in the foregoing density is nothing but the joint distribution (4.2.20) of the matrix elements expressed in the eigenrepresentation of H ; the product of eigenvalue differences, on the other hand, is the energy dependent factor in the Jacobian for the transformation from matrix elements to eigenvalues and the “angles” in the diagonalizing matrix. The factorization of that Jacobian into two factors, one energy dependent and the other a function of the angles mentioned, signals statistical independence of eigenvalues and eigenvectors of the Gaussian ensembles under discussion.
4.4 Eigenvalue Distributions for Nonstandard Symmetry Classes Having discussed the level-repusion phenomena for the new nonstandard symmetry classes introduced by M. Zirnbauer and A. Altland in Sect. 3.5 I can now make the obvious guess for the eigenvalue distribution,
4.5
Level Spacing Distributions
69
⎛ ⎞ -1 ... N . -1 ... N . N , , α β E 2 − E 2 E μ exp ⎝−A P(E) = const E μ2 ⎠ . μ ν μ<ν
μ
(4.4.1)
μ=1
The guess concerns only the preexponential factors relevant for level repulsion. The Gaussian factor has the same origin as for the Dyson classes. For each class, the Hamiltonian matrices H and the measure d H must be restricted by the pertinent symmetries. The Gaussian density exp −ATrH 2 for (the independent elements of) H then automatically yields the above Gaussian in the eigenrepresentation of H . Of course, the matrix density exp −ATrH 2 also yields the power law prefactors as the Jacobian for the transformation from the independent matrix elements in H to eigenvalues and “angles” in the matrix diagonalizing H , but the evaluation of that Jacobian I dare replacing by the “guess”. As would be hard to imagine otherwise, the guess turns out correct [6]. I recall the level repulsion exponents from Sect. 3.5, class D : β class C : β class DIII : β class CI : β
=2 =2 =4 =1
α α α α
=0 =2 =1 =1,
(4.4.2)
and spell out the intuitive interpretation suggested by the symmetry of all spectra about zero energy: The factor |E μ |α means repulsion of the level E μ from its mirβ ror image −E μ ; similarly, the factor (E μ + E ν )β in E μ2 − E ν2 reflects repulsion between E μ and the image −E ν of E ν . At energies much exceeding the mean level spacing the interaction of levels with their distant images at negative energy is expected to become negligible. In that limit the level statistics must reduce to Wigner-Dyson for the given exponent β. Significant peculiarities arise only for energies of the order of the mean level spacing. In particular, the mean density of levels there comes out as ρ(E) = Tr δ(E − H) ∼ |E|α .
(4.4.3)
The latter power-law behavior and the exponent α must be expected to be universal, shared by all systems of a universality class, since they result from the common symmetry. Of course, far away from the power-law dip of the level density at zero energy, the mean density will follow the nonuniversal semicircle law. The interested reader will find a more comprehensive discussion of the level statistics of the new ensembles in [6]. For the exponents α, β of the chiral symmetry classes see [5].
4.5 Level Spacing Distributions There is no problem in calculating the distributions of the level spacings for the three ensembles of random matrices, provided that the consideration is again restricted to the smallest number of dimensions, such that only a single pair of levels occurs.
70
4 Random-Matrix Theory
Taking the joint distribution (4.3.15) for N = 2, P(S) = const
+∞
−∞
d E+
+∞ −∞
d E − δ (S − |E + − E − |)
× |E + − E − |n−1 e−A(E+ +E− ) . 2
2
(4.5.1)
With the unit of energy set such that the mean spacing is unity, $ $ ∞the elementary integrals yield S = d SS P(S) = 1, and with the normalization 0 d S P(S) = 1, ⎧ −S 2 π/4 ⎪ ⎨(Sπ/2)e 2 P(S) = (S 2 32/π 2 )e−S 4/π ⎪ ⎩ 4 18 6 3 −S 2 64/9π (S 2 /3 π )e
orthogonal unitary symplectic.
(4.5.2)
These distributions show the expected degree of level repulsion in their behavior for S → 0 and share a Gaussian fall-off at large S. They are plotted, together with $S their integrals I (S) = 0 d S P(S ), in Fig. 4.1. The P(S) in (4.5.2) will be referred to as the Wigner surmises. The spacing distributions can also be worked out for higher dimensions. The limit N → ∞ which is of special interest will be dealt with in Sect. 4.12. Quite surprisingly, the asymptotic distributions differ only little from the Wigner distributions valid for N = 2, as is obvious from Fig. 4.2.
$S Fig. 4.1 Distributions of nearest neighbor spacings P(S) and their integrals I (S) = 0 d S P(S ) for the Poissonian random process (p) the orthogonal (o), unitary (u), and symplectic (s) ensembles (Gaussian or circular) of random matrices
4.6
Invariance of the Integration Measure
71
o u s
Fig. 4.2 (a) Deviations of the asymptotic (N → ∞) level spacing distributions of the orthogonal (o), unitary (u), symplectic (s) (Gaussian or circular) ensembles from the respective Wigner surmises. (b) Numerical results for the difference P(S) − P Wigner (S) for a kicked top pertaining to the orthogonal universality class. Histogram obtained from 101 Floquet matrices of the form (7.6.2) with dimension 1001, α = π/2, and λ ranging from 8.50 to 9.50 in steps of 0.01. Smooth curve as in (a). For the derivation of the asymptotic distributions, see Sect. 4.12
4.6 Invariance of the Integration Measure When introducing the three Gaussian ensembles of random Hermitian matrices in 4.2, I used the differential volume elements dH consisting of the products of differentials of all independent matrix elements. These are ⎧1...N 1, ... N ⎪ ⎪, ⎪ d H d Hi j ⎪ ii ⎪ ⎪ ⎪ i i< j ⎪ ⎪ ⎪ 1, ... N ⎨1...N , d Hii d 2 Hi j dH = ⎪ ⎪ i i< j ⎪ ⎪ ⎪ 1, ... N 0, 1...N ... 3 ⎪ , ⎪ ⎪ (0) ⎪ dh dh (μ) ⎪ nn nm ⎩ n
n<m
orthogonal unitary
(4.6.1)
symplectic
μ
where for the complex elements in the unitary case, I introduced the convention d 2 Hi j = dRe Hi j dIm Hi j ; equivalently, one may treat Hi j and Hi∗j , rather than ReHi j and ImHi j , as independent and thus interpret d 2 Hi j as d Hi∗j d Hi j /(2i). It is about time to reveal the invariance of dH under the pertinent canonical transformation, thus completing the invariance proof for the Gaussian ensembles. To that end, I must show that a canonical transformation H → H comes with a Jacobian equal to unity, det(∂ H /∂ H ) = 1. For the sake of transparency, I shall write out the reasoning for the smallest nontrivial matrix dimension, N = 2; the validity for arbitrary N will become clear as well. First, I turn to the GUE and once more employ the unitary matrix (4.3.6) to
transform the Hermitian matrix H as H = U HU † or
72
4 Random-Matrix Theory
H11
H22
H12
H12∗
∗ = c2 H11 + s 2 H22 − c s (e∗ H12 + e H12 ) ∗ 2 2 ∗ = s H11 + c H22 + c s (e H12 + e H12 ) ∗ = c s e (H11 − H22 ) + c2 H12 − s 2 e2 H12 ∗ = c s e (H11 − H22 ) − s 2 e∗2 H12 + c2 H12
(4.6.2)
where I have abbreviated c = cos Θ, s = sin Θ, and e = eiφ . The very appearance
of (4.6.2) suggests reading the matrices H and H as four-component vectors h and
h related by a 4 × 4 matrix U as h = Uh. The latter matrix is unitary with respect to the scalar product (a, b) = Tr A† B of complex four-component vectors a and b related to 4×4 matrices A and B in the manner just introduced; the unitarity follows trivially from (Ua, Ub) = Tr (U AU † )† U BU † = TrA† B = (a, b). The Jacobian of the transformation (4.6.2), simply the determinant det U, is thus unimodular; in fact, it is equal to unity since that is manifestly so for the infinitesimal version of (4.6.2), as is immediately checked with c → 1, se → Θ with infinitesimal Θ. With det U = 1, we have the desired invariance dH = dH , at least for the GUE of complex Hermitian 2 × 2 matrices. But the argument is immediately extended to N × N matrices by letting a, b, h, . . . be vectors with N 2 complex components and U an N 2 × N 2 matrix. The GOE is dealt with quite analogously. We have to consider the orthogonal ˜ between real symmetric matrices. Again securing nice transformation H = O H O looking formulae by choosing N = 2 and using the orthogonal matrix O from (4.3.2), we get the transformation
H11 = c2 H11 + s 2 H22 − 2c s H12
H22 = s 2 H11 + c2 H22 + 2c s H12
H12 = c s (H11 − H22 ) + (c2 − s 2 )H12
(4.6.3)
with c = cos Θ, s = sin Θ as above. To reveal the Jacobian of this transformation as equal to unity, a little gymnastics is necessary since the 3 × 3 matrix in (4.6.3) is not orthogonal: Neither rows nor columns form pairwise orthogonal three-component vectors. The gymnastics will hardly produce sweat though since it amounts to no more than rewriting the 3 × 3 Jacobian as a 4 × 4 determinant, 2 2 c s −cs −cs 2 2 2 2 c s −2cs 2 2 cs cs = s c s c 2cs (4.6.4) cs −cs c2 −s 2 . cs −cs c2 − s 2 cs −cs −s 2 c2 The equality is easily seen by first adding the fourth to the third column in the 4 × 4 determinant and then subtracting the third from the fourth row. It is not by chance that the foregoing 4 × 4 determinant results from the 4 × 4 Jacobian of the transformation (4.6.2) upon specializing as e → 1; that specialization carries the transformation (4.6.2) into a slightly blown-up version of (4.6.3) which encompasses the latter for the at present real symmetric matrix H . But the 4 × 4 version of the transformation (4.6.3) has already been shown above to have a Jacobian equal
4.7
Average Level Density
73
to unity and due to the equality (4.6.4) the Jacobian for (4.6.3) equals unity as well. I trust the reader has refrained from indulging in blindly evaluating the determinants in (4.6.4). The faithless who may have arrived at det(∂ H /∂ H ) = 1 in so deviant a manner must now work hard to generalize from N = 2 to arbitrary N . We others just proceed as in the unitary case and elevate the 3 × 3 Jacobian to an N (N + 1)/2 × N (N + 1)/2 one which equals its blown-up N 2 × N 2 version and thus unity.
4.7 Average Level Density The joint distribution of energy levels (4.3.15) not only implies the distribution P(S) of level spacings but also the ensemble-averaged level density (E) ¯ =
d E 2 . . . d E N P (E, E 2 , . . . , E N ) .
(4.7.1)
To secure a well-defined limit for the foregoing (N −1)-fold integral as N → ∞, one sets the energy scale such that the second-order moments take the values for the GUE,
Hi j H ji = Hii2 =
1 . 4N
(4.7.2)
Then, the density of levels has the semicircular form √ (2/π ) 1 − E 2 (E) ¯ = 0
for |E| < 1 for |E| > 1
(4.7.3)
known as Wigner’s semicircle law [11]. That law is also valid for the Gaussian orthogonal and symplectic ensembles, even with the same value unity of the radius, provided that the energy scale is appropriately chosen. On the other hand, if the chosen independently of N , the width of the Gaussian distribution for the Hi j is √ radius of the semicircle becomes proportional to N , and the mean level spacing √ proportional to 1/ N . The most elegant and up-to-date method of deriving the semicircle law involves the representation of determinants by Gaussian integrals over anticommuting variables. Readers with indomitable curiosity about that derivation may jump to Sect. 4.13 below where Grassmann integration is introduced and then immediately on to Chap. 11. There we shall obtain the average level density from the average Green function
74
4 Random-Matrix Theory
G(E ± i0+ ) = lim
N →∞
1 1 i& E Tr ∓ 1 − (E/2λ)2 = N E ± i0+ − H 2λ2 λ
(4.7.4)
as (E) ¯ = ± π1 ImG(E ∓ i0+ ), where the energy scale is set such that |Hi j |2 = λ2 /N instead of (4.7.2). In contrast to the local fluctuations in the spectrum, the average level density (4.7.3) has no universal validity; rather it is a specific property of the Gaussian matrix ensembles. It differs, in particular, from the semiclassical average level density (Weyl’s law, also known as the Thomas–Fermi distribution) for Hamiltonians of systems with f classical degrees of freedom, 1 ¯ SC (E) = f h
d f pd f qδ (E − H ( p, q)) ,
(4.7.5)
which estimates the number of energy levels in the interval [E, E + ΔE] as the number of Planck cells contained in the phase-space volume of that energy shell. Weyl’s law will be derived in Sect. 5.7 for classically integrable dynamics and in Sect. 10.3.3 for the nonintegrable case.
4.8 Unfolding Spectra It may be well to digress from random-matrix theory for a moment and to discuss the problem of how to extract an average level density from a given sequence of measured or calculated levels. The problem is analogous to that of finding a smoothed density of particles for a gas or a liquid. Easiest to cope with are spectra (or systems of particles) that appear homogeneous on energy (or length) scales exceeding a certain minimal range ΔE min : the number ΔN of levels (degenerate ones counted with their degree of degeneracy) per energy interval ΔE yields the average density ¯ = ΔN /(ΔE N ) provided ΔE ≥ E min . More often, however, one encounters “macroscopically” inhomogeneous spectra (analogous to compressible fluids with density gradients) and then faces the problem of having to distinguish between local fluctuations in the level sequence and a systematic global energy dependence of the average density. When defining (E) ¯ =
ΔN 1 ΔE N
(4.8.1)
one must take an energy interval ΔE comprising many levels, ΔN 1, so that local fluctuations do not prevent the ratio ΔN /ΔE from being smooth in E. On the other hand, ΔE must appear infinitesimal with respect to the scale on which (E) ¯ varies systematically. For the obvious conflict to be reasonably resolvable, a self-consistent separation of energy scales must be possible. By requiring that the average density (E) ¯ should vary little over an interval of the order of the local mean level spacing 1/(E)N ¯ , one obtains the self-consistency condition
4.8
Unfolding Spectra
75
2 ¯ (E) N (E) ¯ .
(4.8.2)
In practice, the following procedure has proven viable. One considers the convolution of the density (E) = (1/N ) iN= 1 δ(E − E i ) with the Gaussian g(E) =
Δ (E) =
1 2 2 √ e−E /Δ , Δ π
(4.8.3)
N 1 g(E − E i ) N i =1
(4.8.4)
and chooses the width Δ by minimizing the mean square deviation between the level staircase σ (E) =
1 Θ(E − E i ) N i
(4.8.5)
and the integral of Δ (E), σΔ (E) =
E
−∞
d E Δ (E ) .
(4.8.6)
If an optimal value of Δ can be found in this way, one may make the identification Δ (E) = (E) ¯ and identify σΔ (E) with the average level staircase σ¯ (E). Fortunately, the mean square deviation between σ (E) and σΔ (E) often depends weakly on Δ for Δ in the range of several typical level spacings; in such cases the self-consistency condition (4.8.2) is well obeyed. The concept of level spacing fluctuations (and of other measures of local fluctuations in the spectrum) requires revision when the average density (E) is energydependent. For instance, inasmuch as the terms “level clustering” and “level repulsion” are meant to describe local fluctuations, it would be quite inappropriate to describe a spectral region with high ¯ as one with less repulsion than a region with smaller . ¯ Indeed, before it even begins to make sense to compare local fluctuations from two spectral regions with different average densities, one must secure uniformity of the average spacing throughout the spectrum by local changes of the unit of energy. In stating, at the end of Sect. 4.5, that the Wigner distributions (4.5.2) hold for the three Gaussian ensembles of random matrices not only for the smallest dimensions possible but also, to a satisfactory degree of accuracy, in the limit N → ∞, it was tacitly assumed that the levels rescale to a uniform average density with the help of Wigner’s semicircle law. A natural way of “unfolding” a spectrumto uniform density can be found as follows: One looks for a function f (E) such that the rescaled levels
76
4 Random-Matrix Theory
ei = f (E i )
(4.8.7)
have unit mean spacing, the mean evaluated with respect to energy intervals [E − ΔE/2 , E + ΔE/2]. The corresponding rescaled interval goes from f (E − ΔE/2) to f (E + ΔE/2). The requirement 1=
Δe ΔN
=
=
ΔE ΔN
f (E) = f (E)/(E)N ¯
1 ΔN
[ f (E + ΔE/2) − f (E − ΔE/2)]
(4.8.8)
yields, upon integration, the function f (E) as N times the average level staircase σ¯ (E) =
E
−∞
d E (E ¯ ) .
(4.8.9)
The rescaled levels are thus ei = N σ¯ (E i ) .
(4.8.10)
The unfolding (4.8.10) is certainly natural, but it is not the only one possible. For many applications, another unfolding, ei = E i N (E i ) ,
(4.8.11)
is convenient. The two rescalings (4.8.10) and (4.8.11) are strictly equivalent only in the (uninteresting) case of homogeneous spectra [(E) ¯ = const]; but with respect to spectral regions in which the energy dependence of (E) is sufficiently weak to be negligible, there is practically no difference between (4.8.10) and (4.8.11).
4.9 Eigenvector Distributions 4.9.1 Single-Vector Density For several applications, the distribution of the components of the eigenvectors of random matrices are of interest [12–15]. Considering first the Gaussian unitary ensemble of N × N matrices, one faces unit-norm eigenvectors with N in general complex components cn = cn + icn
. For a given operator H , every eigenvector can be unitarily transformed into an arbitrary vector of unit norm. The only invariant
4.9
Eigenvector Distributions
77
characteristic of eigenvectors is therefore the norm itself, and the joint probability for the N complex components of an eigenvector must read PGUE ({cn }) = const δ 1 −
N
|cn |
,
2
(4.9.1)
n=1
where the constant is fixed by normalization. Evidently, PGUE ({cn }) is non-zero only on the surface of a d-dimensional unit sphere with d = 2N . A convenient quantity to compare with numerical data is the reduced density of, say, |c1 |2 , i.e., of the probability of finding the first basis state populated in an eigenstate of H, PGUE (y) =
d 2 c1 . . . d 2 c N δ y − |c1 |2 PGUE ({c}) ,
(4.9.2)
which we shall calculate presently. For the Gaussian orthogonal ensemble of N × N matrices, the eigenvectors can, without loss of generality, be assumed real. The joint probability density for the N components cn is therefore concentrated on the surface of a d-dimensional unit sphere with d = N , PGOE ({c}) = const δ 1 −
N
cn2
,
(4.9.3)
n=1
and the reduced density of c12 reads PGOE (y) =
dc1 . . . dc N δ y − c12 PGOE ({c}) .
(4.9.4)
The Gaussian symplectic ensemble of random N ×N Hamiltonians, finally, again has unit-norm eigenvectors with N complex components, and N is necessarily even. The joint density of the components cn , PGSE ({c}), is given by (4.9.1); the reason is analogous to that given above for the unitary case: Every eigenvector of H can be symplectically transformed into an arbitrarily prescribed unit-norm vector. However, now the natural reduced density to compare to numerical data is slightly different from those above. Recalling that in the symplectic case there is no loss of generality in choosing a basis of the form, |1 , T |1 , |2 , T |2 , . . . |N /2 , T |N /2 ,
(4.9.5)
the two eigenvectors of H pertaining to one given eigenvalue can be written as |e1 = c1 |1 + c˜ 1 T |1 + c2 |2 + c˜ 2 T |2 . . . T |e1 = −˜c1∗ |1 + c1∗ T |1 − c˜ 2∗ |2 + c˜ 2∗ T |2 + . . . .
(4.9.6)
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4 Random-Matrix Theory
However, any linear combination of |e1 and T |e1 together with an orthogonal combination such as α|e1 + βT |e1 , −β|e1 + αT |e1
(4.9.7)
with |α|2 + |β|2 = 1 is also a pair of unit-norm eigenvectors of H pertaining to the eigenvalue under consideration. A diagonalization code for H starting with the matrix H in the basis (4.9.5) may yield the two eigenvectors in the form (4.9.7) with any choice of α and β [rather than with α = 1, β = 0 as was the case in (4.9.6)]. What remains invariant, however, under the transition from (4.9.6) to (4.9.7) are the occupation probabilities for the subspaces spanned by |i and T |i. Thus, one is led to compare the reduced density for, say, |c1 |2 + |˜c1 |2 , PGSE (y) =
d 2 c1 d 2 c˜ 1 d 2 c2 d 2 c˜ 2 . . . d 2 c N /2 d 2 c˜ N /2 × δ y − |c1 |2 − |˜c1 |2 P ({c, c˜ })
(4.9.8)
with the corresponding distribution obtained by diagonalizing H numerically. In the basis set of (4.9.5), the joint distribution of all components {c, c˜ } reads PGSE ({c, c˜ }) = const δ 1 −
N /2
|cn | + |˜cn | 2
2
.
(4.9.9)
n=1
For all three Gaussian matrix ensembles, the joint distribution of all components of eigenvectors are uniformly concentrated on d-dimensional unit spheres, with d suitably related to the matrix dimension N (d = N , 2N , 2N for the orthogonal, unitary, and symplectic cases, respectively). In a unified notation involving d real variables x1 . . . xd , the properly normalized joint distribution reads P
(d)
(x1 , . . . , xd ) = π
−d/2
d d 2 Γ xn . δ 1− 2 n=1
(4.9.10)
By integrating out d − l of the variables x, one obtains the reduced density (d−l−2)/2 l Γ (d/2) xn2 . 1− P (d,l) (x1 , . . . , xl ) = π −l/2 Γ ((d − l)/2) n=1
(4.9.11)
The reduction from P (d) to P (d,l) is conveniently carried out by using a Fourier integral representation for the delta function in (4.9.10) so as to make the (d −l)-fold integral over xl+1 , . . . xd the (d − l)th power of a single integral. Different choices for d and l now yield the reduced densities PGOE (y), PGUE (y), and PGSE (y) defined above.
4.9
Eigenvector Distributions
79
For the orthogonal case [12, 13], PGOE (y) =
d x P (N ,1) (x)δ(y − x 2 )
Γ (N /2) 1 (1 − y)(N −3)/2 = √ . √ y π Γ [(N − 1)/2]
(4.9.12)
The analogous result for the unitary case is PGUE (y) =
d x1 d x2 P (2N ,2) (x1 , x2 )δ y − x12 − x22
= (N − 1)(1 − y) N −2 ,
(4.9.13)
and for the symplectic case PGSE (y) =
d x1 d x2 d x3 d x4 P (2N ,4) (x1 . . . x4 )
×δ y − x12 − x22 − x32 − x42 = (N − 1)(N − 2)y(1 − y) N −3 .
(4.9.14)
Considering that the mean y is of the order 1/N , it is convenient to go over to the variable η = y N and to perform the limit N → ∞. After proper normalization, this yields the densities 1 e−η/2 , PGOE (η) = √2πη PGUE (η) = e−η , PGSE (η) = ηe−η .
(4.9.15)
The first of these densities is known as the Porter–Thomas distribution [16, 17]. Interestingly, the densities P(y) or P(η) for the three ensembles are quite different in their y (or η) dependence, as illustrated in Fig. 4.3. This figure also compares these functions with data obtained by diagonalizing the Floquet operators of kicked tops with the appropriate symmetries and fully chaotic classical limits. The reliability of random-matrix theory is indeed impressive. Figure 4.3 also reveals, incidentally, that the eigenvector statistics yield a quantum mechanical chaos criterion. Nowhere in this section has the Gaussian nature of the matrix ensembles been used. In fact, all results hold true unchanged for the eigenvectors of unitary matrices from Dyson’s circular ensembles to which we shall turn in Sect. 4.11. Quite different in status even though similar in content is Berry’s conjecture for wave functions of billiards: Amplitudes sampled at independent points make up a Gaussian distribution; independent means further apart than a typical wavelength [18, 19]. That conjecture is well obeyed except for wavefunctions localized near
80
4 Random-Matrix Theory
1 I(η)
0
r
o
0
u
s
η
0.5
Fig. 4.3 Eigenvector statistics for the orthogonal (o), unitary (u), and symplectic (s) (Gaussian or circular) ensembles of random matrices (smooth curves), compared with numerically obtained statistics for kicked tops $ η of the appropriate universality classes. Plotted are the integrated distributions, i.e., I (η) = 0 dξ P(ξ ). The curve labelled (r ) refers to a kicked top under conditions of regular motion in the classical limit
periodic orbits. For a comprehensive discussion of these matters, including the socalled scars, the reader is referred to St¨ockmann’s book [20].
4.9.2 Joint Density of Eigenvectors In concluding Sect. 4.3, we saw that eigenvectors and eigenvalues of matrices from the Gaussian ensembles are statistically independent. That observation allows us to write the joint densities of the eigenvectors. For a GUE matrix, the ith component of the μth eigenvector is a complex number to be denoted by ciμ . The joint density in question must express normalization and mutual orthogonality of all N eigenvectors while abstaining from any further prejudice, ⎡
PGU E
⎤. N N 1...N N , , 2 ∗ ⎣ ⎦ = δ |ciμ | − 1 δ ciμ ciν . μ=1
i=1
μ<ν
(4.9.16)
i=1
The product of delta functions is not in conflict with the angle-dependent part of the Jacobian discussed in Sect. 4.3. Those two densities differ only in the number and the character of the variables admitted: The Jacobian of Sect. 4.3 has normalization and orthogonality already worked in and parametrizes the eigenvectors in terms of the remaining variables, referred to as “angles” there. The foregoing density applies to the circular unitary ensemble (CUE) to be met with in Sect. 4.11 as well, simply because all relevant properties are common to Hermitian and unitary matrices, i.e., normalizability and mutual orthogonality of eigenvectors as well as statistical independence of eigenvectors and eigenvalues.
4.10
Ergodicity of the Level Density
81
The modifications appropriate for the orthogonal and symplectic classes are obvious and will not be written here.
4.10 Ergodicity of the Level Density When describing in Sect. 4.8 how a spectrum determined either experimentally or by computation can be unfolded to uniform mean density, it was necessary to define a mean level density by a local spectral average. Similar spectral averages, local or global, must be performed when extracting measures of fluctuations from a given spectrum. For instance, to construct the level spacing distribution, the distances between neighboring levels from the whole level sequence must be gathered. Random-matrix theory, on the other hand, provides ensemble averages of quantities like the level density and the spacing distribution. From empirical and numerical evidence, one knows that for almost all individual Hamiltonians many observables related to the spectrum have spectral means that agree very well with their averages over the appropriate ensemble of random matrices, provided only that the number of levels is large (N 1 or N → ∞). Such (overwhelmingly likely) accurate agreement of spectral and ensemble averages is quite analogous to the equivalence of time and ensemble averages of certain observables in many-particle systems; in both cases one speaks of ergodic behavior of the observables in question. Proving ergodicity is often a hard task. Now, I shall sketch the proof of “ergodic” behavior for the density of levels, (E) =
N 1 δ(E − E i ) , N i =1
(4.10.1)
whose spectral average is (E) =
1 ΔE
E+ΔE/2
d E (E )
(4.10.2)
E−ΔE/2
and the ensemble mean reads (E) ¯ =
d N Eδ(E − E 1 )P (E 1 , E 2 , . . . , E N ) .
(4.10.3)
Ergodic behavior means that there is a limit, to be identified, in which the ensemble variance of (E) vanishes, and the ensemble mean of (E) is (E), ¯ (E) → (E) ¯
2 Var (E) = (E)2 − (E) → 0 .
(4.10.4)
82
4 Random-Matrix Theory
Indeed, when the fluctuations of (E) within the ensemble vanish, one may identify (E) = (E). ¯ Following the arguments of Pandey [21], I shall now show that the limit in question amounts to letting the number N of levels go to infinity, the energy interval ΔE used for the spectral average shrink to zero, and the mean number of levels ΔN contained in ΔE grow indefinitely, N → ∞,
ΔE → 0 ,
ΔN → ∞ .
(4.10.5)
This limit formally resolves the conflict discussed in Sect. 4.8: The interval ΔE must be sufficiently large that it comprises a large number of levels ΔN but small enough for (E) ≈ ΔN /(ΔE N ) and (E) ¯ to be constant within it. By writing ΔN = N (E)ΔE, ¯ one sees that there is no contradiction between the three requirements (4.10.5); one may imagine ΔN ∼ N ε ,
ΔE ∼ N ε−1 ,
0<ε<1.
(4.10.6)
The first of the properties (4.10.4) becomes obvious when the ensemble-averaged density (E) ¯ is represented by a Taylor series,
(E) =
1 ΔE
E+ΔE/2
d E (E ¯ )
E−ΔE/2
= (E) ¯ +
1
¯ (E)(ΔE)2 + . . . 24
→ (E) ¯
for ΔE → 0 .
(4.10.7)
A little more work is needed for the ensemble variance Var (E) =
1 ΔE
2
E+ΔE/2
d E d E
(E )(E
)
E−ΔE/2
−(E ¯ )(E ¯ ) .
(4.10.8)
One encounters here the density–density correlation function for the matrix ensemble [7, 22–24], S(E , E
) = (E )(E
) − (E ¯ )(E ¯
) .
(4.10.9)
By inserting (4.10.1), S(E , E
) =
1 δ(E − E i )δ(E
− E j ) − (E ¯ )(E ¯
) N2 ij
(4.10.10)
4.10
Ergodicity of the Level Density
83
and by separating diagonal from off-diagonal terms in the double sum, S(E , E
) =
1 ¯ ) − (E ¯ )(E ¯
) δ(E − E
)(E N 1 δ(E − E i )δ(E
− E j ) . + 2 N i= j
(4.10.11)
The last term in (4.10.11) could be evaluated with the help of the reduced distribution function for two eigenvalues; it obviously has the meaning of the probability density for finding a level at E and another one at E
, up to the normalization factor (N − 1)/N . An interesting consequence of the inhomogeneity of the Gaussian ensembles with respect to energy is manifest in the correlation function S(E , E
) at this point. Inasmuch as the ensemble-averaged density (E) ¯ is energy-dependent according to Wigner’s semicircle law, the correlation function S depends on both E and E
separately, rather than on the difference between them. Before one can hope for stationarity in E, one must renormalize the correlation function ¯ )(E ¯
) = δ(E − E
)/N (E ¯ ) − 1 S(E , E
)/(E δ(E − E i )δ(E
− E j )/(E ¯ )(E ¯
)N 2 +
(4.10.12)
i= j
and rescale all energies by referring them to the local mean spacing 1/N (E), ¯ e = E N (E) ¯ ,
(4.10.13)
so as to have ¯ )(E ¯
) = δ(e − e
) − Y (e , e
) , S(E , E
)/(E
Y (e , e
) = 1 −
δ(e − ei )δ(e
− e j ) ,
(4.10.14)
i= j
where Y (e , e
) is known as Dyson’s two-level cluster function [7, 22–24]. Indeed, this function is stationary, i.e., it depends only on the difference e − e
for the three Gaussian ensembles and reads YGOE (e) = s(e)2 − J (e)D(e) YGUE (e) = s(e)2 YGSE (e) = s(2e)2 − I (2e)D(2e)
(4.10.15)
84
4 Random-Matrix Theory
with sin πe πe ∂s D(e) = ∂e $e I (e) = 0 de s(e )
s(e) =
=
1 π
(4.10.16)
Si (π e)
J (e) = I (e) − 12 sgn (e) . A derivation of (4.10.15) will be presented in Sect. 4.14 and for the GUE in Chap. 11. By inserting (4.10.15) in (4.10.8) and recalling that the averaging interval ΔE is chosen small enough to forbid any variation of (E) ¯ within, we can attack the double-energy integral of (4.10.8), starting with
(E) ¯ Var(E) = ΔN
2
ΔN
de
de δ(e
− e ) − Y (e
− e ) .
(4.10.17)
0
Neglecting relative corrections of order 1/ΔN ,
(E) ¯ Var (E) = ΔN
2
⎧2 2 [ln (2π ΔN ) + γ + 1 − π8 ] ⎪ ⎪ ⎨ π2 × π12 [ln (2π ΔN ) + γ + 1] ⎪ ⎪ 2 ⎩ 1 [ln (4π ΔN ) + γ + 1 + π8 ] 2π 2
GOE GUE
(4.10.18)
GSE
where γ is Euler’s constant. For all three cases, the ensemble variance of (E) vanishes as (ln ΔN )/(ΔN )2 as ΔN → ∞. The ergodicity of the level density, (E) → (E), ¯ is thus established. A rougher argument leads to the same conclusion more rapidly. We just have to realize that the auxiliary functions constituting the cluster function are bounded everywhere and fall off with increasing e like s(e) ∝ D(e) ∝ J (e) ∝ 1/e. It follows that the cluster functions themselves are bounded and enjoy the asymptotic large-e behavior YGOE (e) ∝ YGUE (e) ∝ 1/e2 and YGSE (e) ∝ 1/e. Like the delta function δ(e
− e ) in the integrand of (4.10.17), the cluster function Y (e
− e ) can therefore contribute no worse than a term of the order ΔN to the double integral in the orthogonal and unitary cases and no more than a term ∝ ΔN lnΔN in the symplectic case. Such growth is obviously outdone by the common prefactor ∝ (ΔN )−2 . I shall come back to ergodicity in Sect. 4.17.
4.11 Dyson’s Circular Ensembles Without mentioning possible applications to the Floquet operators of periodically driven systems but certainly with scattering matrices in mind, Dyson introduced three so-called circular ensembles of random unitary matrices [7, 25]. In view of
4.11
Dyson’s Circular Ensembles
85
the respective canonical transformations of their member matrices, these ensembles carry the names orthogonal (COE), unitary (CUE), and symplectic (CSE). The joint densities of matrix elements are easily written; they are in fact identical in appearance to the joint densities of eigenvectors for Hermitian or unitary matrices of the appropriate class; in particular, for the CUE, the density in question is nothing but (4.9.16). Most importantly, defining the unimodular eigenvalues of unitary matrices by their eigenphases φ j , F|φ j = e−iφ j |φ j , j = 1, 2, . . . N ,
(4.11.1)
the joint distribution for the eigenphases is P ({φ}) = (1/Nβ )
, e−iφi − e−iφ j β i< j
, φi − φ j β = (1/Nβ ) 2 sin 2 i< j
(4.11.2)
where β = n −1 denotes the degree of level repulsion which we have seen is smaller by 1 than the codimension n of a level crossing, i.e., ⎧ ⎪ ⎨1 β= 2 ⎪ ⎩ 4
COE CUE CSE
(4.11.3)
and Nβ is a normalization constant. Since P({φ}) depends on the differences φi −φ j only, the circular ensembles, in contrast to the Gaussian ones, are homogeneous. Indeed, the reduced distribution of a single eigenphase φ is a constant, (φ) ¯ =
d N φ P ({φ}) δ(φ − φ1 ) =
1 . 2π
(4.11.4)
Therefore, no unfolding is necessary in a spectrum of quasi-energies. The foregoing distribution is, of course, just the ensemble mean of the level density, the quasi-energy analogue of (4.7.1) and (4.10.1), (φ) =
N 1 δ(φ − φi ) . N i=1
(4.11.5)
Similarly, the density–density correlation function, (φ)(φ ) = S(φ − φ ) +
1 2π
2 ,
(4.11.6)
86
4 Random-Matrix Theory
the quasi-energy analogue of (4.10.10), is now a function of the difference φ − φ
without any prior unfolding. When dealing with the quantum spectra of classically integrable systems with at least two degrees of freedom in the next chapter, we shall meet with Poissonian fluctuations, i.e., that register no correlations between neighboring levels. Unitary matrices with such spectra are nicely modelled with the so-called Poissonian ensemble (PE) whose joint distribution of eigenphases is just the constant P({φ}) =
1 2π
N PE ;
(4.11.7)
this fits into the scheme of (4.11.2) and (4.11.3) with a vanishing degree of level repulsion, β = n − 1 = 0; clustering as opposed to repulsion of levels is indeed an important quantum or wave signature of integrability.
4.12 Asymptotic Level Spacing Distributions The asymptotic (N → ∞) level spacing distributions differ from the Wigner distributions (4.5.2) which are strictly valid for N = 2. Since it is known [7] that the Gaussian and circular ensembles have the same asymptotic level spacing distributions, it will be convenient here to take Dyson’s circular ensembles as a starting point for a quantitative analysis of the limit N → ∞. Our first goal will be to derive Taylor series for the spacing distributions P(S) in powers of S; then, connecting these with asymptotic results for large S by suitable Pad´e approximants yields P(S) for 0 ≤ S < ∞ with any desired accuracy. Easiest to treat is the case of the CUE which I propose to deal with in now. The product of eigenvalue differences in (4.11.2) can be related to a Vandermonde determinant ([7], Chap. 12), ⎞ 1 z 1 z 12 . . . z 1N −1 1, ... N ⎜1 z 2 z 2 . . . z N −1 ⎟ 2 2 ⎟ ⎜ (z i − z j ) = det z ik−1 = det ⎜ . ⎟ , .. ⎠ ⎝ .. . i< j N −1 2 1 zN zN . . . zN ⎛
(4.12.1)
where the abbreviation z = e−iφ has been used. The reader may easily check the correctness of (4.12.1) by realizing that both the product and the determinant are polynomials in z N of order N − 1 with roots z 1 , z 2 , . . . , z N −1 ; a corresponding statement holds true for z N −1 , z N −2 , . . . . Then, the squared modulus in (4.11.2) for n − 1 = 2 is also given by a determinant, 1, ... N i< j
N i−1 ∗k−1 z i − z j 2 = det zl zl , l =1
(4.12.2)
4.12
Asymptotic Level Spacing Distributions
87
whereupon the eigenvalue distribution for the CUE takes the form N2−1 det
P ({φ}) =
N
zli−1 zl∗k−1
.
(4.12.3)
l =1
For the discussion that follows, a little lemma about the mean of a symmetric function f ({φ}) of all N phases φ with the weight (4.12.3) will be helpful, f ({φ}) = N2−1 N !
d N φ f ({φ}) det z ii−1 z i∗k−1 .
2π 0
(4.12.4)
To prove this identity,2 we begin with f ({φ}) =
N2−1
2π
d φ f ({φ}) det N
0
N
zli−1 zl∗k−1
l =1
and take out of the determinant the sums over l from all elements in its first row, f ({φ}) =
N2−1
2π
d N φ f ({φ})
0
⎛ ×
zl∗
1
N
zl∗2
N N ⎜ N ∗ ∗2 z det ⎜ j =1 z jz j j =1 z jz j ⎝ j =1 j .. .. .. l =1 . . .
...
⎞
⎟ . . .⎟ . ⎠ .. .
Due to the symmetry of the integrand, the sum in question can be replaced by a factor N and the index l set to 1 in the first row. By subtracting from the lth row z l−1 times the first row, 1 f ({φ}) =
N2−1 N ⎛
⎜ ×det ⎝
2π
d N φ f ({φ})
0
N
1
j =2
.. .
zj
N
z 1∗
j =2
.. .
z j z ∗j
z 1∗2 ∗2 j =2 z jz j .. .
N
⎞ ... . . .⎟ ⎠ . .. .
The foregoing argument is now repeated: The sum over j in the second row is replaced by a factor (N − 1) in front of the determinant and the index j set equal to 2 in the second row. One so proceeds row by row until (4.12.4) is established.
2
In Sect. 10.8, an alternative method for dealing with problems of this kind will be presented.
88
4 Random-Matrix Theory
In a first application of (4.12.4), the normalization factor N2 in (4.12.3) can be determined as N2 = N !(2π ) N since after setting f ({φ}) = 1 in (4.12.4), the integral over the ith phase can be carried out in each element of the ith row:
1 = N2−1 N ! det =
2π 0
dφi z ii−1 z i∗k−1
N2−1 N ! det (2π δik )
= N2−1 N !(2π ) N .
(4.12.5)
In the next step toward constructing the level spacing distribution, we now consider the probability E(S) that a phase interval of length S (in units of the mean spacing 2π/N ) is empty of levels, E(S) =
N ,
π −π S/N
i = 1 −π +π S/N
dφi
P ({φ}) .
(4.12.6)
It should be noted that the homogeneity of the distribution P({φ}) justifies the particular location of the integration range in (4.12.6). The spacing distribution P(S) equals the second derivative of the “gap probability” E(S),
P(S) =
∂2 E . ∂ S2
(4.12.7)
Indeed, let S = x +y, and consider the difference E(x +y)− E(x +y+Δx) which is the probability that x + y is empty and the neighboring interval Δx not empty. For small Δx, the probability that Δx contains two or more levels is of second or higher order in Δx. Therefore, −[∂ E(x + y)/∂ x]Δx is the probability that x + y is empty and Δx contains one level. Reasoning analogously, [∂ 2 E(x + y)/∂ x∂ y]ΔxΔy is the probability that x + y is empty and the surrounding intervals Δx and Δy each contain one level. The latter probability, on the other hand, may be expressed as the product of the “single-event” probability Δx and the conditional probability P(S)Δy with S = x + y; note that in the units used, the mean density of levels is unity. The identity (4.12.7), together with (4.12.6), will find further applications in Chaps. 5 and 6. Proceeding with the evaluation of E(S) for the CUE, we observe that the integral in (4.12.6) affects all N phases symmetrically, whereupon the identity (4.12.4) is once more applicable. As in (4.12.5), the integral over the lth phase can be carried out by integrating all elements of the lth row in the determinant,
4.12
Asymptotic Level Spacing Distributions
89
π −π S/N
dNφ det zll−1 zl∗k−1 N (2π) −π +π S/N π −π S/N dφl iφl (l−k) = det e −π +π S/N 2π sin [π (k − l)S/N ] = det δkl − . π(k − l)
E(S) =
(4.12.8)
Now, this determinant and its second derivative P(S) will be Taylor expanded, E(S) =
∞
El Sl ,
P(S) =
l =0
∞
(l + 1)(l + 2)El+2 S l ,
(4.12.9)
l =0
and a manageable expression for the expansion coefficient El be constructed . The starting point for the expansion in question is the usual representation of a determinant as a sum of products of N elements, 1, ... N
sin [π (k − Pk)S/N ] P (−1) , δk,Pk − E(S) = π(k − Pk) k P
(4.12.10)
where P denotes a permutation of the set of indices 1, 2, 3, . . . , k, . . . N and Pk the index associated with k by the permutation P. Second, the Taylor series of sin x in powers of x must be employed for x = π (k − Pk)S/N , 0 ... ∞ (π S/N )2l+1 1 sin [π (k − Pk)S/N ] (−1)l = (k − Pk)2l . π (k − Pk) π l (2l + 1)!
(4.12.11)
Finally, by inserting the binomial expansion of (k − Pk)2l , we arrive at E(S) =
P
(−1)
P
1, ... N
0 ... ∞ (π S/N )2l+1 1 (−1)l+1 π l (2l + 1)! 0 ... 2l 2l 2l−t t × (−k) (Pk) . t t
δk,Pk +
k
(4.12.12)
The zeroth-order Taylor coefficient may be read off as E 0 = E(0) = 1. That value is in fact immediately obvious from the definition of E(S) : An interval of zero length contains no eigenvalue with probability unity. To obtain the El with l > 0, consider the case where n out of the N factors in a summand in (4.12.12) are of nonvanishing order in S, and let these orders be 2l1 + 1, 2l2 + 1, . . . , 2ln + 1. The
90
4 Random-Matrix Theory
remaining (N − n) factors are of the type δk,Pk . The contribution to E(S) of such terms with fixed n and l1 . . . ln reads3 T (S, n, {li }) 1 ... N
=
2l1 +1 1 (π S/N )2ln +1 l1 + ... +ln +n (π S/N ) (−1) . . . π n n! k ... k (2l1 + 1)! (2ln + 1)! 1
×
0 ... 2l1
n
...
0 ... 2ln
t1
×
tn
2l1 t1
...
2ln (−k1 )2l1 −t1 . . . (−kn )2ln −tn tn
˜ ˜ t1 ˜ n tn . (−1)P Pk . . . Pk 1
(4.12.13)
P˜
The remaining n! permutations P˜ reshuffle the n indices on the n summation variables k1 . . . kn ; they are the remnants of the N ! original permutations P in (4.12.12) after requiring k = Pk for N −n of the original N factors in the product in (4.12.12). Since each P˜ is nothing but a P with the N −n constraints just mentioned, ˜ the respective parities are identical, (−1)P = (−1)P ; the latter property of the P˜ has already been used in writing (4.12.13). The sum over the P˜ may be recognized as t the n × n determinant det (ki j ), whereupon T (S) takes the form T (S, n, {li }) (−1)l1 +l2 + ... +ln +n (π S/N )2l1 +1 (π S/N )2ln +1 = . . . π n n! (2l1 + 1)! (2ln + 1)! 0 ... 2ln 0 ... 2l1 2l1 2ln × ... (−1)t1 +t2 + ... +tn ... t tn 1 t t 1
×
1 ... N
n
2l −t +t det ki i i j .
(4.12.14)
{k}
Clearly, E(S) − 1 results when T is summed first over all l1 . . . ln from 0 to ∞ and then over n from 1 to ∞. At this point, it is well to realize that the n-fold sum over k1 . . . kn draws nonzero contributions only when all n summation variables ki are different from one another in their values and, more importantly,
Actually, (4.12.13) requires l1 = l2 = . . . = ln . When some of the li coincide such that k n i = n, different values are assumed, 1 ≤ k ≤ n, with multiplicities n 1 , n 2 , . . . , n k and the factor 1/n! must be replaced by 1/(n 1 !n 2 ! . . . n k !). For the Taylor coefficients El , given in (4.12.18 or 4.12.21), the distinction in question is irrelevant, due to the appearance of another factor n 1 ! . . . n k !/n! accompanying the summation over the configurations {li }. 3
4.12
Asymptotic Level Spacing Distributions
91
all ti are different from one another, and all 2li − ti are different from one another.
(4.12.15)
Indeed, when two or more ki or ti coincide, the determinant has at least two identical rows and thus vanishes; when (2li − ti ) = (2l j − t j ), on the other hand, it is obvious from (4.12.3) that there is a pairwise cancelling of determinants in the sums over ki and k j . The constraint (4.12.15) yields an upper bound for the number n in T (S, n, {li }) for a fixed value of l=
n n (2li + 1) = 2 li + n , i =1
(4.12.16)
i =1
the order of T (S, n, {li }) in S. The inequality in question, n2 ≤ l ,
(4.12.17)
will turn out to amount to a strong restriction of the number of terms T (S, n, {li }) contributing to the Taylor coefficient El . Postponing the proof of (4.12.17) until after its exploitation, we concentrate now on the Taylor coefficient4 El =
π l N ×
2 n ≤l
n=1,2, ...
n 0,1,2, ... l −n (−1)(l+n)/2 δ li , π n n! 2 l ... l i =1 1
n
... 2ln 1 ... 2l1 1 1 1 2l1 2ln (−1)t1 + ... +tn ... ... (2l1 + 1)! (2ln + 1)! t t tn 1 t 1
×
1 ... N
2l −t +t det ki i i j .
n
(4.12.18)
k1 ... kn
The reader will appreciate the power of (4.12.17) in restricting the allowable values of n in (4.12.18) when observing that due to the definition (4.12.16) even (odd) values of l admit only even (odd) values of n. It follows that 16 (25) is the smallest even (odd) value of l for which two different values of n are allowed (n = 2, 4, and n = 3, 5, respectively). Note that n = 1 can contribute only to l = 1 : For n = 1, no permutation P in (4.12.12) or P˜ in (4.12.13) except the identity can contribute.
For the sake of typographical convenience, δ(m, n) here denotes the Kronecker symbol usually written as δmn .
4
92
4 Random-Matrix Theory
At this point, the asymptotic large-N behavior must be discussed. To that end, we observe that, to within corrections of relative order 1/N , each of the n sums over a ki may be replaced by an integral to yield N
N x+1 1 1+O for x > 0 . x +1 N
kx =
k =1
(4.12.19)
Thus, the n-fold sum over the ki in (4.12.18) gives
= det
k
1 ... N
2l −t +t det ki i i j
k1 ... kn
2li −ti +t j
1 = N det 2li − ti + t j + 1
l
k
(4.12.20)
whereupon the Taylor coefficient El takes the final form El(UE)
=
2 n ≤l
π
l−n (−1)
n!
n=1,2, ...
×
0 ... 2l1 t1
×
1, ... n k
...
(l+n)/2 0,1,2, ...
0 ... 2ln
det
tn
l1 ... ln
n
l −n δ li , 2 i =1
1 2li − ti + t j + 1
2lk 1 (−1)tk . (2lk + 1)! tk
(4.12.21)
The Taylor coefficients so determined indicate impressively rapid convergence of the Taylor series for P(S) throughout the range of S from which the first few ¯ S 2 , . . . draw important contributions, say 0 ≤ S < 3. moments S, ∼ On the other hand, for S > ∼ 1.7 Dyson’s asymptotic result [27] (UE) (S) = E as
B=
π −1/4 2 1 24
e2B S −1/4 e−(π
ln 2 + 32 ζ (−1) ,
2
/8)S 2
(4.12.22)
holds with excellent accuracy. In particular, Dyson’s asymptotics and the cheaply accessible Taylor series have an overlapping range of validity in the right wing of P(S) such that Pad´e interpolations between the two approximations offer themselves for numerical evaluations of P(S) with any desired accuracy [26]. Finally, we turn to the proof of the inequality (4.12.17) from the constraints (4.12.15) and the definition (4.12.16). For the sake of clarity, let us restate the task in question, reformulated slightly as a counting problem. With a fixed integer n,
4.12
Asymptotic Level Spacing Distributions
93
we wish to determine the configurations of two sets of integers, t1 , t2 , . . . , tn and l1 , l2 , . . . , ln , that minimize the sum
l=
n
(2li + 1) = n + 2
i =1
n
li
!
= min
(4.12.23)
i =1
under the constraints (a) 0 ≤ ti ≤ 2li , (b) li = 0, 1, 2, . . . , (c)
t1 = t2 = . . . = tn ,
(d) 2l1 − t1 = 2l2 − t2 = . . . = 2ln − tn . As a “warm-up” we consider the “primitive ladder” configurations li = i − 1 for i = 1, 2, . . . , n. Then, the ti are uniquely fixed by the above constraints: t1 = 0 and 2l1 − t1 = 0 by (a), t2 = 1 and 2l2 − t2 = 1 by constraints (c,d), and so forth, ti = 2li − ti = i − 1. The number l in this case takes the value l = n 2 which already corresponds to the minimum asserted. The temptation may arise to search for lower values of l by cutting the primitive ladder at a lower height, say li = i − 1 for i = 1, . . . , n − ν + 1 and occupying one of its steps ν times. It is easy to see, however, that a violation of constraints (c,d) thus occurs. A conflict between (c,d) and a multiple occupancy of some integer by several li can be avoided only by leaving some lower integers unoccupied by any li . The extreme of such a configuration is that of “complete degeneracy”, li = L for i = 1, 2, . . . , n. Due to constraint (a), this L must obey 2L + 1 ≥ n, which implies l = n(2L + 1) ≥ n 2 . For odd n, the smallest integer thus allowed is L min = (n − 1)/2 and in that case the totally degenerate configuration has the same l as the primitive ladder, l = n 2 . Moreover, the {ti } and {2li − ti } = {2L − ti } are equally densely packed in both configurations, {ti } = {2L − ti } = {0, 1, 2, . . . , n − 1}. If n is even, L min = n/2, and therefore l is larger than n 2 for the totally degenerate configuration. Now, it is clear how to show the impossibility that l will assume a value below n 2 . Consider a general trial configuration l1 = l2 = . . . = ln 1 ≡ L 1 ln 1 +1 = . . . = ln 1 +n 2 ≡ L 2 .. . ln 1 +n 2 + ... +n k−1 +1 = . . . = ln 1 +n 2 + ... +n k = L k
(4.12.24)
94
4 Random-Matrix Theory
with n1 + n2 + . . . + nk = n and L1 < L2 < . . . < Lk . Minimizing l means minimizing all of the L i . By the foregoing argument for the fully degenerate configuration, L 1 is minimal when n 1 is chosen odd and then L 1,min = (n 1 −1)/2, {t1 , t2 , . . . , tn 1 } = {2L 1 −t1 , . . . , 2L 1 −tn 1 } = {0, 1, . . . , n 1 − 1}. Having chosen n 1 as odd and n 1 < n but otherwise arbitrary, we turn to the n 2 -fold degenerate value L 2 . The corresponding n 2 integers tn 1 +1 , . . . , tn 1 +n 2 must obey ti ≥ n 1 and 2L 2 −ti ≥ n 1 . Just as before, the minimum value of L 2 for fixed n 2 is found by assigning to the n 2 integers ti in question the smallest values consistent with constraints (c,d), i.e., tn 1 +1 , . . . , tn 1 +n 2 = 2L 2 − tn 1 +1 , . . . , 2L 2 − tn 1 +n 2 = {n 1 , n 1 + 1, . . . , n 1 + n 2 − 1} . To accommodate these values of the ti , the number L 2 must obey 2L 2 ≥ 2n 1 + n 2 −1. By requiring n 2 to be odd, the minimal value of L 2 is L 2,min = n 1 +(n 2 −1)/2. Now, the argument can be continued to yield L i,min =
i−1
nj +
j =1
1 (n i − 1) , i = 1, 2, . . . , k , 2
(4.12.25)
provided that all n i are chosen odd. The corresponding value of l is l=
k i =1
n i 2L i,min + 1 =
k
2 ni
= n2 ,
(4.12.26)
i =1
independent of the n i , provided only that the n i are all odd. A larger value of l results when one or more n i are chosen even, analogously to what was found for the totally degenerate configuration. The reader will realize that a special case of the configuration (4.12.24) with all n i odd is the primitive ladder, for which n i = 1 for i = 1, 2, . . . , n. Now, it is obvious that the lower bound lmin = n 2 cannot be beaten. It is realized in different configurations {li } which all have in common the closest packing of the {ti } and {2li − ti } as {t1 , . . . , tn } = {2l1 − t1 , . . . , 2ln − tn } = {0, 1, . . . , n − 1} .
(4.12.27)
4.12
Asymptotic Level Spacing Distributions
95
This completes our consideration of the spacing distribution for the CUE. The circular orthogonal and circular symplectic ensembles can be treated analogously [26]. The corresponding probabilities E(S) that an interval 2π S/N will be empty are represented as determinants [7] similar to (4.12.8), and these determinants are expanded in powers of S. In the limit N → ∞, the following Taylor coefficients arise: El(OE)
n π l−n (−1)(l+n)/2 1,2, ... l −n = δ li , 2 n! 2 l ... l n = 1,2, ... i =1 2 l≥2n −n
×
0 ... l1
...
t1
×
0 ... ln tn
1
n
1 det 2li − 2ti + 2t j + 1
2lk 1 2tk (2lk + 1)!
1, ... n k
(4.12.28)
for the orthogonal ensemble, and El(SE)
n (l+n)/2 1,2, ... l −n (−1) = π l−n δ li , n! 2 l ... l n = 1,2, ... i =1 2 l≥2n −n
×
0 ... l1 t1
×
...
1, ... n k
0 ... ln tn
1
n
1 det 2li − 2ti + 2t j + 1
4tk 2lk + 1 1 1− 2tk (2lk + 1)! 2lk + 1
(4.12.29)
for the symplectic ensemble. Here, as in (4.12.21), the remaining determinants are n × n. It may be worth noting that the restriction on n in (4.12.28) and (4.12.29), l ≥ 2n 2 − n, is even more stringent than was l ≥ n 2 in (4.12.21): The range 1 ≤ n ≤ 4 suffices to determine the El of the orthogonal and symplectic ensembles up to order l = 44; with the help of (4.12.9), the first 42 Taylor coefficients of the spacing distribution are thus accessible; they have been tabulated in [26]. The interested reader may also consult the reference for Pad´e interpolations between the small-S behavior inherent in the above Taylor expansions and the asymptotics at large S [27]. In concluding this section, it is appropriate to point out that, for most practical purposes, the Wigner distributions (4.5.2) are quite adequate. The rms deviations within a pair of competing distributions turn out to be
∞ 0
d S P(S) − PWigner (S)
2
1/2
⎧ −4 ⎪ ⎨1.6 × 10 = 0.4 × 10−4 ⎪ ⎩ 1.5 × 10−5
OE UE SE .
(4.12.30)
96
4 Random-Matrix Theory
The differences P(S) − PWigner (S) for the three ensembles are plotted in Fig. 4.2. Spacing histograms drawn from experimentally or numerically determined (quasi) energy spectra are notoriously too rugged to allow for a distinction as small as that indicated in (4.12.30) or Fig. 4.2. However, the kicked top represented in Fig. 4.2b does provide evidence for the inaccuracy of the Wigner surmise. Incidentally, the most urgent need for precise values of the spacing distribution arises in a context whose connection to quantum chaos has been a painfully open problem for quite a while. The nonreal zeros of Riemann’s zeta function all have, according to Riemann’s own hypothesis [28], one and the same real part 1/2. While the hypothesis remains unproven to this day [29], many millions of zeros have been computed numerically and do abide by the claim. The imaginary parts of the known zeros have a spacing distribution in fantastic agreement with that of the GUE [30, 31]. The GUE statistics of the Riemann zeros suggests that they might be related to the energy spectrum of some dynamic system with a fully chaotic classical limit. Search for a Hamiltonian is ongoing [32].
4.13 Determinants as Gaussian Grassmann Integrals Before proceeding with random matrices, we had better equip ourselves with a tool that will greatly facilitate certain types of averages. I aim at representing N × N determinants as Gaussian integrals over anticommuting variables [33–36], det A =
⎛
1, ... N
⎝
j
⎞ dη∗j dη j ⎠ exp
−
1 ... N
ηi∗ Aik ηk
.
(4.13.1)
ik
To explain this integral representation, the anticommuting (Grassmann) variables ηi and their conjugates ηi∗ must be introduced through their algebra 5 ηi ηk∗ + ηk∗ ηi = 0 ηi ηk + ηk ηi = 0
(4.13.2)
ηi∗ ηk∗ + ηk∗ ηi∗ = 0 . It follows that ηi2 = ηi∗2 = 0. Thus, the most general function of a single such variable is linear, F(η1 ) = a + bη1 with a and b ordinary commuting numbers. Similarly, the most general function of two Grassmann variables is bilinear, F(η1 , η2 ) = a + bη1 + cη2 + η1 η2 d .
(4.13.3)
5 For the purposes of the present chapter, the asterisk on N of the Grassmannians may be taken to just distinguish these variables as independent from those without an asterisk; only in Chap. 11 will the “complex conjugation” relation between the two sets of N variables be needed and explained
4.13
Determinants as Gaussian Grassmann Integrals
97
Differentiation with respect to a variable η is defined analogously to differentiation with respect to usual commuting variables, (∂/∂η1 )F(η1 ) = b, or ∂ ∂ F(η1 , η2 ) = b + η2 d , F(η1 , η2 ) = c − η1 d ∂η1 ∂η2 ∂ ∂η1
∂ ∂ ∂ F(η1 , η2 ) = − F(η1 , η2 ) = −d . ∂η2 ∂η2 ∂η1
(4.13.4)
The sign of mixed derivatives is found by arranging the product of η’s in F in the order inverse to that of the derivatives; obviously, ∂/∂ηi and ∂/∂η j anticommute like ηi and η j and (∂/∂ηi )η j = −η j (∂/∂ηi ) for i = j. It also follows that ∂ 2 /∂ηi2 = 0. When mixed derivatives with respect to η and η∗ arise, all of the above rules hold analogously, i.e., (∂/∂ηi )∂/∂η∗j = −(∂/∂η∗j )∂/∂ηi . Like differentiation, integration is defined as a linear operation. In addition, it is required that
dηi ηi =
dηi∗ ηi∗
=1,
dη1 1 =
dηi∗ 1 = 0
(4.13.5)
and that the differentials dηi and dηi∗ anticommute just like the ηi and ηi∗ themselves. Coming back to N = 2 as in (4.13.3) and (4.13.4), from the definition just given, $ $ $
dη1 F(η1 , η2 ) = b + η2 d dη2 F(η1 , η2 ) = c − η1 d $ dη2 dη1 F(η1 , η2 ) = − dη1 dη2 F(η1 , η2 ) = d .
(4.13.6)
For arbitrary N , we may write the most general function as the series
f (m 1 , . . . , m N )η1m 1 η2m 2 . . . ηmN N
(4.13.7)
dη N . . . dη1 F(η1 , . . . , η N ) = f (1, 1, . . . , 1) .
(4.13.8)
F(η1 , . . . , η N ) =
{m i =0,1}
and then
In short, differentiation and integration are identical operations for Grassmann variables. Grassmann integrals enjoy the privilege of not requiring boundaries to become definite. Now, the integral representation (4.13.1) of a determinant is easily checked to be a consequence of the above definitions. Only the N th order term of the Taylor
98
4 Random-Matrix Theory
expansion of the exponential on the r.h.s. in (4.13.1) can make a non-zero contribution to the 2N -fold integral, ⎞
⎛
,
⎝
D≡
dη∗j dη j ⎠ exp (−η∗ Aη) =
⎛
⎞
⎝
dη∗j dη j ⎠
1, ... N
j
=
⎛ ⎝
j
1, ... N
(−1) N ∗ (η Aη) N N!
⎞ dη j dη∗j ⎠
j
1 ∗ (η Aη) N . N!
(4.13.9)
For the integral to collect a non-zero contribution, the 2N -fold sum in (η∗ Aη) N must be restricted so as to involve precisely all the N different η and their conjugates η∗ . There are (N !)2 such additive terms, according to the permutations of the N naturals 1, 2, . . . , N as labels on the N factors η and N factors η∗ , D=
⎛ ⎝
⎞
1, ... N
dη j dη∗j ⎠
j
1 ∗ η η j η∗ η j . . . ηi∗N η jN N ! {i} { j} i1 1 i2 2
× Ai1 j1 Ai2 j2 . . . Ai N jN .
(4.13.10)
Each of the N ! permutations {i} yields one and the same contribution to D, once the N ! permutations { j} are summed over. Thus, the factor 1/N ! may be dropped and only one permutation {i} admitted, D=
⎛ ⎝
1, ... N
⎞
dη j dη∗j ⎠
η1∗ ηP1 η2∗ ηP2 . . . η∗N ηP N A1,P1 . . . A N ,P N .
(4.13.11)
P
j
At this point, the 2N -fold integral may be evaluated with the help of (4.13.5). The result is +1 or −1, depending on whether the permutation P1, P2, . . . P N is even or odd with respect to the natural sequence 1 2 . . . N ; I shall denote such a sign factor by (−1)P . The final result, D=
(−1)P A1,P1 A2,P2 . . . A N ,P N ≡ det A ,
(4.13.12)
P
is in fact the familiar representation for the determinant of the matrix A. We shall also encounter the so-called Pfaffian of real antisymmetric 2N × 2N matrices a = −a˜ which is partially, i.e., up to the sign, defined by Pf (a) 2 = det a .
(4.13.13)
4.13
Determinants as Gaussian Grassmann Integrals
99
A complete definition employs the Gaussian Grassmann integral
Pf (a) =
dη2N dη2N −1
1 ... 2N 1 . . . dη1 exp ηi aik ηk 2 ik
.
(4.13.14)
The proof of this integral representation provides an opportunity to reveal the interesting behavior of Grassmann integrals under transformations of the integration variables. It suffices to consider the homogeneous linear transformation ηi =
L ik ξk
(4.13.15)
k
and to invoke (4.13.8) to write the integral over all η’s as
η dξ N . . . ξ1 J ( )F(η1 ({ξ }), . . . , η N ({ξ })) ξ = J d n ξ F({η({ξ })})
f (1, . . . 1) =
= J f (1, . . . 1)
d ξ N
L 1,i1 L 2,i2 . . . L N ,i N ξi1 ξi2 . . . ξi N
i 1 ... i N
= J f (1, . . . 1)
1 ... N
dNξ
= J f (1, . . . 1) det L .
L 1,P1 . . . L N ,P N ξP1 . . . ξP N
P
(4.13.16)
The foregoing chain of equations should be mostly self-explanatory provided that the reader has cautiously gone through the derivation of the integral representation (4.13.1) of det A above. Nonetheless, a few remarks may be helpful. First, I have taken the Jacobian J in search out of the integral assuming that it is independent of the ξ ’s; that assumption will be borne out self-consistently and is of course due to the linearity of the transformation under consideration. Next, only those index configurations i 1 , i 2 , . . . i N can contribute to the integral that are permutations of the naturals 1, 2, . . . n since any Grassmann variable squares to zero. Finally, a sign factor (−1)P arises from the identity ξP1 ξP2 . . . ξP N = (−1)P ξ1 ξ2 . . . ξ N , whereupon the definition (4.13.12) of a determinant is recognized. Comparing the first and last member of the chain, we arrive at the Jacobian alias Berezinian 1 1 η = J( ) = ξ det L det ∂η ∂ξ
(4.13.17)
100
4 Random-Matrix Theory
which is interesting indeed: While one is used to the Jacobian as the determinant of the derivatives of “old” with respect to “new” commuting variables, here we meet with the inverse of that determinant in the case of anticommuting variables. Equipped with the Jacobian (4.13.17), now we can prove the integral representation (4.13.14) for a Pfaffian. To that end, we introduce a real orthogonal transformation from the 2N η’s to 2N ξ ’s and choose the orthogonal 2N × 2N matrix L such that (1) det L = 1 and (2) so as to bring the antisymmetric matrix a to block diagonal form with 2 × 2 blocks
0 ai −ai 0
,
ai = ai∗
i = 1 ... N ,
(4.13.18)
along the diagonal. Such an orthogonal transformation can be found (see Problem 4.10), and the ±iai are the (not necessarily different) eigenvalues. Now, the Grassmann integral in (4.13.14) becomes a product of twofold integrals, N ,
dξ2 dξ1 exp(ai ξ1 ξ2 ) =
i=1
N ,
ai =
√
det a ,
(4.13.19)
i=1
and we have reached our goal. An equivalent definition of the Pfaffian is Pf (ai j ) =
1 (−1)P aP1,P2 aP3,P4 . . . aP2N −1,P2N (N )! P
(4.13.20)
where the summation is over all permutations of 1 2 . . . 2N restricted by P1 < P2, P3 < P4, . . . . It can be established by expanding the exponential in the original integral representation (4.13.14) and arguing analogously to our procedure from (4.13.9) to (4.13.10); readers with no prior familiarity with Grassmann algebra will want to exercise their new skills by solving Problem 4.11. The reader is kindly invited to appreciate the difference between the Grassmann integrals presented above and the more familiar Gaussian integrals over commuting variables. The correspondent of (4.13.1) involves N pairs of complex conjugate variables z i , z i∗ , i = 1 . . . N and runs over all N complex planes; it reads , N d 2 zi i=1
π
⎛ exp ⎝−
ij
⎞ z i∗ Ai j z j ⎠ =
1 det A
(4.13.21)
where the differential volume element is to be interpreted according to d 2 z = dRe zd Imz = dz ∗ dz/(2i). The real part of the symmetric matrix A must be positive definite for the Gaussian integral to converge. Note that now an inverse determinant
4.14
Two-Point Correlations of the Level Density
101
is given by a Gaussian integral. On the other hand, an inverse square root of a symmetric determinant can be written as the “real” Gaussian integral
+∞ −∞
⎛ ⎞ N , d xi 1 1 exp ⎝− , xi ai j x j ⎠ = √ √ 2 2π det a ij i=1
(4.13.22)
the counterpart of the Grassmann integral (4.13.14). Now, the matrix a is symmetric, may have even or odd dimension, and is restricted to have a positive definite real part to ensure convergence of the integral.6 Note that no convergence precautions are necessary for the previous Gaussian Grassmann integrals. Simple proofs of the “bosonic” integral representations (4.13.21) and (4.13.22) involve transformations of the integration variables diagonalizing the respective matrices A and a, just as in the “fermionic” cases. The foregoing integral representations can be employed profitably when the matrices involved have random elements and when powers of the determinants with exponents ±1 or ±1/2 are to be averaged. Several such cases will arise in the sequel. The much in vogue art of writing ratios of determinants as mixed Gaussian integrals over both commuting (“bosonic”) and anticommuting (“fermionic”) variables, mostly referred to as “The Supersymmetry Method”, will be discussed in Chap. 11.
4.14 Two-Point Correlations of the Level Density 4.14.1 Two-Point Correlator and Form Factor We first encountered the two-point correlation function of the level density in Sect. 4.11, when dealing with the ergodicity of the mean level density in Gaussian ensembles. Far more important is the use of the two-point function as a measure of spectral fluctuations, since in contrast to the semicircular mean density, these fluctuations can serve as models for the largely universal fluctuations we find in the spectra of dynamic systems. In view of such applications, the two-point function is worth special consideration. The cluster functions (4.10.15) of the level density come out the same for the Gaussian and circular ensembles; they are, however, most easily calculated for the latter and that task will be taken up here.7 We shall take advantage of a slightly rewritten version of the density of eigenphases: Representing the 2π -periodic delta function by a Fourier series,
6
Analytic continuation to imaginary ai j yields the Fresnel integrals to be met in Chap. 10.
7
The two-point function of the Gaussian unitary ensemble will be treated in Chap. 11.
102
4 Random-Matrix Theory
(φ) =
N ∞ ∞ 1 in(φ−φi ) 1 e = tn einφ , 2π N i=1 n=−∞ 2π N n=−∞
(4.14.1)
where the nth Fourier coefficient is determined by
tn =
N
−inφi
e
=
i=1
Tr F n 1 Tr F n 2
for for
β = 0, 1, 2 , β=4
(4.14.2)
i.e., the trace of the nth power of the unitary matrix F whose spectrum is at issue. For short, I shall refer to tn as the nth trace. The factor 1/2 in tn = 12 TrF n for the symplectic case accounts for Kramers’ degeneracy; it is indeed customary for that case to count every one of the doubly degenerate levels just once in the density (φ) and its Fourier coefficient tn and to use the said factor 1/2. Incidentally, inasmuch as unitary matrices from the circular ensembles model Floquet matrices, the exponent n may be associated with a dimensionless time counting the number of iterations of the “quantum map” F. Now, the density–density correlation function (4.11.6) reads (φ)(φ ) = =
1 2π N 1 2π N
2
|tn |2 ein (φ−φ )
n
2 n
|tn |2 exp i
2π n(e − e ) ; N
(4.14.3)
here I have accounted for the homogeneity of the circular ensembles as tn tn = δn,−n |tn |2 and rescaled the phase variable by referring to the mean spacing, e = (N /2π)φ, analogously to (4.10.13). The mean squared-in-modulus traces |tn |2 , the so-called form factor, are now revealed as the Fourier coefficients of the density– density correlation function. Obviously, that form factor is even in n and obeys |t0 |2 = N 2 . Proceeding toward Dyson’s two-point cluster function Y (e), according to (4.10.14), we subtract the squared mean density (1/2π)2 to get the “connected two-point correlator”, rescale by dividing by the squared mean density, subtract the delta peak arising at zero spacing, change the overall sign, take a short breath, and then rejoice in
Y (e) = δ(e) −
∞ 2π n 2 2 |t | cos e . n N 2 n=1 N
(4.14.4)
It remains to calculate the form factor |tn |2 for the various circular ensembles. The ensemble averages to be performed are easier for the characteristic function
4.14
Two-Point Correlations of the Level Density
103
i β (ke−inφi + k ∗ e+inφi ) P˜ n N (|k|) = exp − 2 i =
N ,
exp − i|k| cos nφi
(4.14.5)
i=1
than for the form factor itself. The extra convenience gained is due to the multiplicative structure of the quantity to be averaged: Every phase φi enters through its exclusive exponential. As anticipated in writing out (4.14.5), the homogeneity of β the joint density of eigenphases of the circular ensembles allows P˜ n N to depend on ∗ the pair of complex conjugate variables k, k only through the modulus |k|. Thus, β no error is incurred by assuming k = k ∗ = |k| and simply writing P˜ n N (k) for the characteristic function. The form factor is eventually obtained from the second β derivative of P˜ n N (k) at k = 0, i.e., the special case m = 1 of |tn |2m = (−4)m
2m m
−1
d 2m ˜ β P (k) n N dk 2m
.
(4.14.6)
k=0
4.14.2 Form Factor for the Poissonian Ensemble Upon invoking the Poissonian joint distribution (4.11.7), we immediately get the characteristic function in terms of the Bessel function J0 (k), P˜ n0N (k) = J0 (k) N
(4.14.7)
and thus the form factor, connected correlator, and cluster function as |tn |2 = N for n > 0 , 1 2 SPE (e) = δ(e) 2π YPE (e) = 0 .
|t0 |2 = N 2 (4.14.8)
Needless to say, these simple signal the complete absence of correlations results N between levels; the trace tn = i=1 e−iφi results from N steps of a random walk in a complex plane with each step of unit length but arbitrary direction.
4.14.3 Form Factor for the CUE Now, I employ the joint eigenphase density in its determinantal form (4.12.3), N 1 e−iφi (l−m) PCUE ({φ}) = det (2π) N N ! i=1
l,m=1, ...,N
(4.14.9)
104
4 Random-Matrix Theory
together with the lemma (4.12.4) for the CUE mean of a symmetric function of all N eigenphases. Then, the desired average reads P˜ n2N (k) =
d N φ −ik i cos nφi e det e−iφl (l−m) . (2π) N
(4.14.10)
When the factor exp (−ik cos(nφl )) is pulled into the lth row of the determinant, the lth phase appears only in that row. The average can thus be done row by row, whereupon the characteristic function takes the form of a Toeplitz determinant,
dφ −i(l−m)φ −ik cos (nφ) e e 2π 0 -s=+∞ . = det J|s| (k)(−i)|s| δl,m−ns .
P˜ n2N (k) = det
2π
(4.14.11)
s=−∞
Readers wanting to see how the Bessel functions J|s| (k) with nonnegative integral orders |s| = 1, 2, . . . arise are invited to change the integration variable in the middle member of the foregoing chain of equations as nφ → φ and to split the resulting integral into n additive pieces each of which again runs between the limits 0 and 2π and equals any other one up to a phase factor. The sum of the n phase factors equals n if (l − m)/n is an integer and vanishes otherwise. Due to J|s| (0) = δs,0 , the characteristic function reflects proper normalization, P˜ n (0) = 1. For n ≥ N , the determinant in (4.14.11) becomes diagonal and yields P˜ n2N (k) = J0 (k) N =⇒ |tn |2 = N
for
N ≤ n.
(4.14.12)
To find the form factor in the interval 1 ≤ n ≤ N , we expand in powers of k, det M(k) = exp Tr lnM(k) (4.14.13) 1 = exp Tr ln 1 + k M (0) + k 2 M
(0) + O(k 3 ) 2 1 = exp kTrM (0) + k 2 Tr M
(0) − M (0)2 + O(k 3 ) . 2
The derivatives of the Bessel functions at zero argument, J|s| (0) = 12 δ|s|,1 and
(0) = 14 δ|s|,2 − 12 δs,0 yield TrM (0) = 0, TrM (0)2 = − 12 (N −n), and TrM
(0) = J|s| 2 3 N − 2 . We thus find det M(k) = e−nk /4+O(k ) and get the CUE form factor as
|tn
|2
= |t−n
|2
=
N 2 δn,0 + n N
for 0 ≤ n ≤ N for N ≤ n .
(4.14.14)
Using this in the general expression (4.14.4) for the cluster function, we obtain
4.14
Two-Point Correlations of the Level Density
YCUE (e) =
N 2π n 1 2 (N − n) cos + 2 e . N N n=1 N
105
(4.14.15)
Finally, on replacing the sum by an integral over τ = n/N in the limit N → ∞, we get the result anticipated in (4.10.15),8 1 δ(τ ) + |τ | for |τ | ≤ 1 |tn |2 → K CUE (τ ) = N 1 for |τ | > 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
k
= N ! (1/N1 )
2π>φ1 >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 , k
|e−iφk − e−iφl | =
, k
=i =i
φk − φl ) 2 det eimφ1 , . . . , eimφ N |m=(N −1)/2, ... ,−(N −1)/2 (4.14.18) det eimφl .
2 sin(
N (N −1) 2 N (N −1) 2
9 By dynamic rather than random-matrix arguments, a case could be construed for a possibly competing scale, the “Ehrenfest” time τ E = N −1 ln N alias n E = ln N , i.e., the time scale on which a perturbation of the Floquet operator F becomes noticeable in expectation values of observables, given global chaos in the classical limit.
4.14
Two-Point Correlations of the Level Density
107
Three remarks on the foregoing identity are in order. First, the modulus sign could be dropped since the phase ordering makes all sines positive; second, the underlined row index m in the determinants runs in unit steps from (N − 1)/2 to −(N −1)/2 whereas indices not underlined run as usual through the naturals 1 . . . N ; finally, the reader may check the last equality by patiently going backward from the product of sine functions to the product of eigenvalue differences and then on to the familiar Vandermonde determinant, without ever invoking a modulus operation to chase phase factors, as I sketch without further comment here: ,
φk − φl 2 sin 2 k
=i
N (N −1) 2
,
e−i(φk −φl )/2 − ei(φk −φl )/2
k
=i
N (N −1) 2
,
ei(φk +φl )/2 (e−iφk − e−iφl )
k
=i
N (N −1) 2
,
ei(N −1)φi /2 det e−i(k−1)φl
i
=i
N (N −1) 2
det (eimφ1 , . . . , eimφ N ) .
(4.14.19)
Importing (4.14.18) into the characteristic function (4.14.17) and assuming, for simplicity, even N , I fix the order of integrations to do first those over the phases with even indices and pull the (2k)th such integral into the (2k)th row of the determinant for k = 1, . . . , N /2; whereas the (2k)th integral first appears over the interval φ2k+1 < φ2k < φ2k−1 , we may replace the lower limit with zero, simply by adding the N th column of the determinant to the (N − 2)th, the resulting (N − 2)th to the (N − 4)th, and so forth and thus obtain P˜ n1N (k) = (1/N1 )N ! × det (e
dφ1 dφ3 . . . dφ N −1 i N (N −1)/2
φ1 >φ3 >...>φ N −1
i(mφ1 −k cos nφ1 )
φ1
,
dφ ei(mφ−k cos nφ) ,
0
ei(mφ3 −k cos nφ3 ) ,
φ3
dφ ei(mφ−k cos nφ) , . . . ) . (4.14.20)
0
The remaining integral over φ1 , φ3 , . . . , φ N −1 has a symmetric integrand so that we may extend the integration range to the (N /2) dimensional hypercube of edge length 2π and make up by the factor 1/(N /2)!. At this point, it is convenient to represent the N × N determinant by a Gaussian Grassmann integral, P˜ n1N (k) = (−1)?
N! (N /2)! N1
2π 0
dφ1 dφ3 . . . dφ N −1
108
4 Random-Matrix Theory
×
1...N , l
⎞ ⎛ ⎞ ⎛ (N −1) ...− (N −1 ) 2 , 2 dηl∗ ⎝ dηm ⎠ exp ⎝− ηl∗ Ml,m ηm ⎠ , (4.14.21) m
l,m
where M is the matrix whose determinant is under work. Note that I have lumped powers of i and −1 into an undetermined sign factor (−1)? ; this maneuver saves labor since the overall sign may change with every step of the following calculation and can be determined rather simply in the final result. As if retracing the proof of the Grassmann integral representation in Sect. 4.13, we replace the exponential by the N th term of its Taylor expansion and realize that each of the N powers of the quadratic form η∗ Mη must provide a different η∗ and that there are N ! different ways of so drawing the product η1 η2 . . . η N all of which contribute equally. Thus, the exponential in (4.14.21) is replaced by the N -fold sum over all the m of η1∗ ηm 1 η2∗ ηm 2 . . . η∗N ηm N φ1 i(m 1 φ1 −k cos nφ1 ) ×e dφei(m 2 φ−k cos nφ) 0 φ3 i(m 3 φ3 −k cos nφ3 ) ×e dφei(m 4 φ−k cos nφ) ............
0
×ei(m N −1 φ N −1 −k cos nφ N −1 )
(4.14.22)
φ N −1
dφ ei(m N φ−k cos nφ) .
0
When the N -fold integral over the η∗ ’s is now done, it leaves no trace outside the sign factor (−1)? . To prepare for the remaining integral over the η’s, we must take a closer look at the pairs ηm 2i−1 ηm 2i and their accompanying double phase integrals. The latter may be antisymmetrizedin the indices m 2i−1 , m 2i since their symmetric parts do not contribute to the sum m,m ηm ηm Am,m with
A
m,m
2π φ 1
(k) = dφ dφ ei(mφ−k cos nφ+m φ −k cos nφ ) antisym 2π i 0 0 2π 2π 1
= dφ dφ sign(φ − φ )ei(mφ−k cos nφ+m φ −k cos nφ ) 4π i 0 0
(−i)|s|+|s | = J|s| (k)J|s | (k) (4.14.23) δm,−m −n(s+s ) m + ns s,s
manifestly antisymmetric. The Bessel functions in Am,m arise when the phase integral is done after expanding the integrand in powers of k. Now, the characteristic function reads
4.14
Two-Point Correlations of the Level Density
109
⎞ ⎛ N , 2 N ! (2π ) ⎝ dηm ⎠ P˜ n1N (k) = (−1)? (4.14.24) (N /2)! N1 m {m} × ηm 1 ηm 2 Am 1 m 2 ηm 3 ηm 4 Am 3 m 4 . . . ηm N −1 ηm N Am N −1 m N . It is worth recalling the representation of the Pfaffian by a Gaussian Grassmann integral from Sect. 4.13: We are confronting such a beast since the inteN N grand of the foregoing integral may be rewritten as 2 2 ( 21 m,m ηm Amm (k)ηm ) 2 = N ( N2 )! 2 2 exp ( 21 m,m ηm Amm (k)ηm ) such that -
P˜ n1N
N
N ! (4π ) 2 = (−1)? N1
.
⎡
⎤ 1 d N η exp ⎣ ηm Amm (k)ηm ⎦ 2 m,m
= (”) Pf (A(k)) & = (”) det A(k) 1 = (”) exp Tr ln A(k) . 2
(4.14.25)
Recalling that the form factor is given by the second derivative of the characteristic function at k = 0, according to (4.14.6), we may expand the matrix A in powers of k up to second order, & P˜ n1N = (”) det A(0) exp Tr
×
(4.14.26)
1 −1 1 2 −1
−1
2 . k A (0) A (0) + k A (0) A (0) − ( A (0) A (0)) 2 4
must equal unity; with the help of J|s| (0) = δs,0 , we find √ Here, the square? bracket det A(0) = (−1) 2 N /2 /(N − 1)!! ; the final sign factor (−1)? thus turns out as +1 and the normalization factor N1 of the eigenphase density of the COE as N1 = (
N N )! π 2 22N . 2
(4.14.27)
Had it not been for the secret desire to catch that normalization factor, I could of course have left an overall proportionality factor undetermined throughout the above calculation, as I have done with the overall sign factor. At any rate, the main goal of the present subsection is also reached quickly once we use the derivatives of the Bessel functions (given after (4.14.14)) to establish (i) Tr A−1 (0) A (0) = 0 for n = 0, N [m + n − (N + 1)/2]−1 , (ii) Tr A−1 (0) A
(0) = −2N + n m=1
110
4 Random-Matrix Theory
(iii) Tr[A−1 (0) A (0)]2 = 0 for n ≥ N , and (iv) Tr{−A−1 (0) A
(0) + [ A−1 (0)A (0)]2 } = 2n − n nm=1 [m + (N − 1)/2]−1 for 1 ≤ n ≤ N. The COE form factor thus comes out as ⎧ ⎪ N2 ⎪ ⎨ |tn |2 = |t−n |2 = 2n − n nm=1 m+(N1−1)/2 ⎪ ⎪ ⎩2N − n N 1
m=1 m+n−(N +1)/2
for n = 0 for 1 ≤ n ≤ N
(4.14.28)
for N ≤ n .
After approximating sums by integrals in the limit N → ∞, ⎧ ⎨δ(τ ) + 2|τ | − |τ | ln(1 + 2|τ |) for |τ | ≤ 1
1 |tn |2 → K COE (τ ) = ⎩2 − |τ | ln 2|τ |+1 N 2|τ |−1
for |τ | > 1 ,
YCOE (e) = s(e)2 − J (e)D(e)
(4.14.29)
with D(e) = s (e) and J (e) an integral of s(e), as defined in (4.10.16). Plots of K COE (τ ) and YCOE (τ ) are shown in Fig. 4.4. For future reference, I note the behavior of the cluster function at small and large e, ⎧ ⎨1 −
YCOE (e) =
⎩
1
π 2 e2
π2 |e| 6
−
+
π4 |e|3 60
3+cos 2πe 2π 4 e4
−
π4 4 e 135
+ . . . , |e| 1
+ ... ,
|e| 1 .
(4.14.30)
The nonoscillatory part of the cluster function arises, as for the CUE, from the corner of 2|τ | in the form factor while the jump of the third derivative of K COE at the Heisenberg time yields the oscillatory part of YCOE (e); the respective leading terms are displayed in the large-e expansion; see Problem 4.13.
4.14.5 Form Factor for the CSE Our starting point is the CSE analogue of (4.14.17) for the characteristic function, P˜ n4N (k) ∝
2π>φ1 >φ2 ... φ N >0
×
,
1...N , −iφk −iφl 4 e dφ1 . . . dφ N −e
e−ik cos(nφi ) ,
k
(4.14.31)
i
which differs from (4.14.17) only in the value of the level repulsion exponent. An overall factor was dropped and will be restored at the end by invoking normalization.
4.14
Two-Point Correlations of the Level Density
111
(The interested reader may take up the extra labor of evaluating the normalization constant N4 of the joint density of eigenphases; see Problem (4.12).) Now, it is convenient to extend the N -fold integral to a (2N )-fold one, P˜ n4N (k) ∝
2π>φ1 >φ2 ... φ2N >0
×
N ,
e−ik cos(nφi )
i=1
dφ1 . . . dφ2N
1...2N , k
φk − φl sin 2
∂ δ(φ2i−1 − φ2i − ) , ∂φ2i−1
(4.14.32)
where is to be sent to zero from above. To check the equivalence of the two foregoing expressions, one may integrate by parts with respect to the phases with odd labels in the latter. Only those terms survive in the limit → 0 for which the N differentiations have turned precisely those N sines into cosines which are assigned vanishing arguments by the delta functions; the remaining sines then group 4 . Incidentally, the delta functions to quadruples as, symbolically, s13 s14 s23 s24 → s24 in (4.14.31) reflect Kramers’ degeneracy. To do the (2N )-fold phase integral, we recall the identity (4.14.19) as extended to 2N ordered phases and integrate over every second phase to chase the delta functions. Renaming the remaining N integration variables as φ2k → φk and letting → 0, P˜ n4N ∝
2π
d N φ det (m eimφ1 , eimφ1 , m eimφ2 , eimφ2 , . . .) , × e−ik cos(nφi ) 0
(4.14.33)
i
where the underlined row index m again goes in unit steps through half-integers, m ≤ (2N − 1)/2. Note that the integration range again is the hypercube. As in (4.14.20), we encounter a determinant that contains every integration variable in a pair of columns. In fact, the further reasoning precisely parallels the above one for the COE, starting with the Grassmann integral representation of the determinant as in (4.14.21). The antisymmetric matrix A whose Pfaffian is incurred is now 2N × 2N and takes the actually simpler form Am,m (k) = (m − m )
∞
J|s| (k)(−i)|s| δ(m + m + ns) .
(4.14.34)
s=−∞
Again, we get the form factor from the k-expansion (4.14.26) as ⎧ 2 ⎪ ⎨N n CSE 2 CSE 2 |tn | = |t−n | = 12 n + 14 n m=1 ⎪ ⎩ N
1 N + 12 −m
for n = 0 for 1 ≤ n ≤ 2N for 2N ≤ n .
(4.14.35)
112
4 Random-Matrix Theory
Finally, by employing τ = n/N and invoking the limit N → ∞, we arrive at 1 CSE 2 δ(τ ) + |tn | → K CSE (τ ) = N 1
|τ | 2
−
|τ | 4
ln |1 − |τ || for |τ | ≤ 2 for |τ | > 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.
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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,
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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.
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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)
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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)
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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
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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
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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
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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)
166
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)
174
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.
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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
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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
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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
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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
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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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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
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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
