Quantum Signatures of Chaos (eBook)

Preface to the Third Edition 8
Preface to the Second Edition 10
Preface to the First Edition 12
Foreword to the First Edition 16
Contents 18
1 Introduction 25
References 37
2 Time Reversal and Unitary Symmetries 39
2.1 Autonomous Classical Flows 39
2.2 Spinless Quanta 40
2.3 Spin-1/2 Quanta 41
2.4 Hamiltonians Without T Invariance 43
2.5 T Invariant Hamiltonians, T2 = 1 45
2.6 Kramers' Degeneracy 46
2.7 Kramers' Degeneracy and Geometric Symmetries 46
2.8 Kramers' Degeneracy Without Geometric Symmetries 49
2.9 Nonconventional Time Reversal 51
2.10 Stroboscopic Maps for Periodically Driven Systems 53
2.11 Time Reversal for Maps 55
2.12 Canonical Transformations for Floquet Operators 57
2.13 Beyond Dyson's Threefold Way 60
2.13.1 Normal-Superconducting Hybrid Structures 62
2.13.2 Systems with Chiral Symmetry 67
2.14 Problems 68
References 69
3 Level Repulsion 71
3.1 Preliminaries 71
3.2 Symmetric Versus Nonsymmetric H or F 72
3.3 Kramers' Degeneracy 74
3.4 Universality Classes of Level Repulsion 76
3.5 Nonstandard Symmetry Classes 78
3.6 Experimental Observation of Level Repulsion 80
3.7 Problems 81
References 82
4 Random-Matrix Theory 84
4.1 Preliminaries 84
4.2 Gaussian Ensembles of Hermitian Matrices 85
4.3 Eigenvalue Distributions for Dyson's Ensembles 89
4.4 Eigenvalue Distributions for Nonstandard Symmetry Classes 91
4.5 Level Spacing Distributions 92
4.6 Invariance of the Integration Measure 94
4.7 Average Level Density 96
4.8 Unfolding Spectra 97
4.9 Eigenvector Distributions 99
4.9.1 Single-Vector Density 99
4.9.2 Joint Density of Eigenvectors 103
4.10 Ergodicity of the Level Density 104
4.11 Dyson's Circular Ensembles 107
4.12 Asymptotic Level Spacing Distributions 109
4.13 Determinants as Gaussian Grassmann Integrals 119
4.14 Two-Point Correlations of the Level Density 124
4.14.1 Two-Point Correlator and Form Factor 124
4.14.2 Form Factor for the Poissonian Ensemble 126
4.14.3 Form Factor for the CUE 126
4.14.4 Form Factor for the COE 129
4.14.5 Form Factor for the CSE 133
4.15 Newton's Relations 135
4.15.1 Traces Versus Secular Coefficients 135
4.15.2 Solving Newton's Relations 138
4.16 Selfinversiveness and Riemann--Siegel Lookalike 140
4.17 Higher Correlations of the Level Density 141
4.17.1 Correlation and Cumulant Functions 141
4.17.2 Ergodicity of the Two-Point Correlator 144
4.17.3 Ergodicity of the Form Factor 147
4.17.4 Joint Density of Traces of Large CUE Matrices 150
4.18 Correlations of Secular Coefficients 152
4.19 Fidelity of Kicked Tops to Random-Matrix Theory 158
4.20 Problems 164
References 165
5 Level Clustering 167
5.1 Preliminaries 167
5.2 Invariant Tori of Classically Integrable Systems 167
5.3 Einstein--Brillouin--Keller Approximation 169
5.4 Level Crossings for Integrable Systems 171
5.5 Poissonian Level Sequences 172
5.6 Superposition of Independent Spectra 173
5.7 Periodic Orbits and the Semiclassical Density of Levels 175
5.8 Level Density Fluctuations for Integrable Systems 180
5.9 Exponential Spacing Distribution for Integrable Systems 187
5.10 Equivalence of Different Unfoldings 188
5.11 Problems 189
References 190
6 Level Dynamics 191
6.1 Preliminaries 191
6.2 Fictitious Particles (Pechukas-Yukawa Gas) 192
6.3 Conservation Laws 199
6.4 Intermultiplet Crossings 202
6.5 Level Dynamics for Classically Integrable Dynamics 203
6.6 Two-Body Collisions 209
6.7 Ergodicity of Level Dynamics and Universality of Spectral Fluctuations 210
6.7.1 Ergodicity 210
6.7.2 Collision Time 211
6.7.3 Universality 212
6.8 Equilibrium Statistics 214
6.9 Random-Matrix Theory as Equilibrium Statistical Mechanics 217
6.9.1 General Strategy 217
6.9.2 A Typical Coordinate Integral 221
6.9.3 Influence of a Typical Constant of the Motion 227
6.9.4 The General Coordinate Integral 228
6.9.5 Concluding Remarks 230
6.10 Dynamics of Rescaled Energy Levels 231
6.11 Level Curvature Statistics 234
6.12 Level Velocity Statistics 243
6.13 Dyson's Brownian-Motion Model 246
6.14 Local and Global Equilibrium in Spectra 256
6.15 Problems 264
References 266
7 Quantum Localization 268
7.1 Preliminaries 268
7.2 Localization in Anderson's Hopping Model 269
7.3 The Kicked Rotator as a Variant of Anderson's Model 272
7.4 Lloyd's Model 279
7.5 The Classical Diffusion Constant as the Quantum Localization Length 284
7.6 Absence of Localization for the Kicked Top 285
7.7 The Rotator as a Limiting Case of the Top 295
7.8 Problems 297
References 298
8 Dissipative Systems 300
8.1 Preliminaries 300
8.2 Hamiltonian Embeddings 300
8.3 Time-Scale Separation for Probabilities and Coherences 306
8.4 Dissipative Death of Quantum Recurrences 309
8.5 Complex Energies and Quasi-Energies 317
8.6 Different Degrees of Level Repulsion for Regular and Chaotic Motion 320
8.7 Poissonian Random Process in the Plane 323
8.8 Ginibre's Ensemble of Random Matrices 324
8.8.1 Normalizing the Joint Density 325
8.8.2 The Density of Eigenvalues 327
8.8.3 The Reduced Joint Densities 329
8.8.4 The Spacing Distribution 330
8.9 General Properties of Generators 335
8.10 Universality of Cubic Level Repulsion 340
8.10.1 Antiunitary Symmetries 340
8.10.2 Microreversibility 342
8.11 Dissipation of Quantum Localization 347
8.11.1 Zaslavsky's Map 347
8.11.2 Damped Rotator 350
8.11.3 Destruction of Localization 353
8.12 Problems 356
References 359
9 Classical Hamiltonian Chaos 361
9.1 Preliminaries 361
9.2 Phase Space, Hamilton's Equations and All That 361
9.3 Action as a Generating Function 363
9.4 Linearized Flow and Its Jacobian Matrix 364
9.5 Liouville Picture 366
9.6 Symplectic Structure 367
9.7 Lyapunov Exponents 368
9.8 Stretching Factors and Local Stretching Rates 370
9.9 Poincaré Map 372
9.10 Stroboscopic Maps of Periodically Driven Systems 374
9.11 Varieties of Chaos 375
9.12 The Sum Rule of Hannay and Ozorio de Almeida 375
9.12.1 Maps 376
9.12.2 Flows 377
9.13 Propagator and Zeta Function 379
9.14 Exponential Stability of the Boundary Value Problem 382
9.15 Sieber--Richter Self-Encounter and Partner Orbit 383
9.15.1 Non-technical Discussion 383
9.15.2 Quantitative Discussion of 2-Encounters 386
9.16 l-Encounters and Orbit Bunches 393
9.17 Densities of Arbitrary Encounter Sets 399
9.18 Problems 400
References 401
10 Semiclassical Roles for Classical Orbits 402
10.1 Preliminaries 402
10.2 Van Vleck Propagator 402
10.2.1 Maps 403
10.2.2 Flows 409
10.3 Gutzwiller's Trace Formula 412
10.3.1 Maps 413
10.3.2 Flows 417
10.3.3 Weyl's Law 423
10.3.4 Limits of Validity and Outlook 424
10.4 Lagrangian Manifolds and Maslov Theory 426
10.4.1 Lagrangian Manifolds 426
10.4.2 Elements of Maslov Theory 433
10.4.3 Maslov Indices as Winding Numbers 437
10.5 Riemann--Siegel Look-Alike 441
10.6 Spectral Two-Point Correlator 447
10.6.1 Real and Complex Correlator 447
10.6.2 Local Energy Average 448
10.6.3 Generating Function 451
10.6.4 Periodic-Orbit Representation 452
10.7 Diagonal Approximation 456
10.7.1 Unitary Class 456
10.7.2 Orthogonal Class 458
10.8 Off-Diagonal Contributions, Unitary Symmetry Class 458
10.8.1 Structures of Pseudo-Orbit Quadruplets 460
10.8.2 Diagrammatic Rules 463
10.8.3 Example of Structure Contributions: A Single 2-encounter 465
10.8.4 Cancellation of all Encounter Contributions for the Unitary Class 466
10.9 Semiclassical Construction of a Sigma Model, Unitary Symmetry Class 468
10.9.1 Matrix Elements for Ports and Contraction Lines for Links 469
10.9.2 Wick's Theorem and Link Summation 471
10.9.3 Signs 472
10.9.4 Proof of Contraction Rules, Unitary Case 475
10.9.5 Emergence of a Sigma Model 476
10.10 Semiclassical Construction of a Sigma Model, Orthogonal Symmetry Class 480
10.10.1 Structures 480
10.10.2 Leading-Order Contributions 481
10.10.3 Symbols for Ports and Contraction Lines for Links 483
10.10.4 Gauss and Wick 484
10.10.5 Signs 485
10.10.6 Proof of Contraction Rules, Orthogonal Case 487
10.10.7 Sigma Model 489
10.11 Outlook 490
10.12 Mixed Phase Space 491
10.13 Problems 494
References 496
11 Superanalysis for Random-Matrix Theory 500
11.1 Preliminaries 500
11.2 Semicircle Law for the Gaussian Unitary Ensemble 501
11.2.1 The Green Function and Its Average 501
11.2.2 The GUE Average 503
11.2.3 Doing the Superintegral 503
11.2.4 Two Remaining Saddle-Point Integrals 506
11.3 Superalgebra 509
11.3.1 Motivation and Generators of Grassmann Algebras 509
11.3.2 Supervectors, Supermatrices 510
11.3.3 Superdeterminants 512
11.3.4 Complex Scalar Product, Hermitianand Unitary Supermatrices 514
11.3.5 Diagonalizing Supermatrices 516
11.4 Superintegrals 517
11.4.1 Some Bookkeeping for Ordinary Gaussian Integrals 517
11.4.2 Recalling Grassmann Integrals 518
11.4.3 Gaussian Superintegrals 520
11.4.4 Some Properties of General Superintegrals 521
11.4.5 Integrals over Supermatrices,Parisi--Sourlas--Efetov--Wegner Theorem 523
11.5 The Semicircle Law Revisited 525
11.6 The Two-Point Function of the Gaussian Unitary Ensemble 528
11.6.1 The Generating Function 529
11.6.2 Unitary Versus Hyperbolic Symmetry 531
11.6.3 Efetov's Nonlinear Sigma Model 534
11.6.4 Implementing the Zero-Dimensional Sigma Model 541
11.6.5 Integration Measure of the Nonlinear Sigma Model 544
11.6.6 Back to the Generating Function 550
11.6.7 Rational Parametrization of the Sigma Model 552
11.6.8 High-Energy Asymptotics 558
11.7 Universality of Spectral Fluctuations:Non-Gaussian Ensembles 560
11.7.1 Delta Functions of Grassmann Variables 561
11.7.2 Generating Function 562
11.8 Universal Spectral Fluctuations of Sparse Matrices 565
11.9 Thick Wires, Banded Random Matrices,One-Dimensional Sigma Model 566
11.9.1 Banded Matrices Modelling Thick Wires 566
11.9.2 Inverse Participation Ratio and Localization Length 568
11.9.3 One-Dimensional Nonlinear Sigma Model 569
11.9.4 Implementing the One-Dimensional Sigma Model 574
11.10 Problems 583
References 585
Index 587

"Chapter 4 Random-Matrix Theory (p. 61 -62)

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 formany spectral characteristics. The 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)."