New Directions in Quantum Chaos

, by Fermi, Enrico; Guarneri, I.; Smilansky, U.; Casati, Giulio; Guarneri, I.; Smilansky, U.

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ISBN: 9781586030742 | 1586030744
Cover: Hardcover
Copyright: 8/1/2000

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The rapid progress of the research field of quantum chaos and its applications called for a book that keeps students abreast of the new developments and at the same time provides a solid bases in subjects which form the canon of the field. This book discusses the following topics: Spectral statistics and their semiclassical interpretation in terms of the Gutzwiller trace formula, Quantum chaos and its applications in mesoscopic physics, Spectral statistics and conductance fluctuations and Quantum chaos in systems with many degrees of freedom. The line of this book connects and continues past and present achievements and prepares the ground for a future full of intriguing and important developments.

Preface

G. Casati

I. Guarneri

U. Smilansky

Gruppo fotografico dei partecipanti al Corso

Classical and quantum zeta-functions and periodic orbits theory

(44)

R. Artuso

Introduction

(1)

Escape rates, zeta-functions and determinants

(10)

The strange repeller and its escape rate

(3)

Topological aspects, cycle expansions

(3)

Convergence properties of cycle expansion

(3)

Spectra and spaces

(1)

General structure of cycle expansion

(3)

Curvatures

(2)

Symbolic dynamics tricky points

(1)

Dynamical averages in terms of zeta-functions

(3)

Averages and generating functions

(1)

A model average calculation

(2)

Dynamical averages and deterministic diffusion

(3)

A cycle expansion for the diffusion constant

(2)

A piecewise linear example

(1)

The inclusion of marginal stability

(7)

One-dimensional intermittent maps

(2)

Anomalous diffusion

(2)

Zeta-functions without periodic orbits

(2)

Semiclassics: Gutzwiller trace formula

(17)

Density of states and Green's functions

(2)

The WKB ansatz

(2)

Van Vleck formula

(1)

The semiclassical Green's function

(2)

The short-time contribution to G

(1)

Gutzwiller trace formula

(2)

Quantum determinants

(1)

Appendix A

(1)

Manipulations on Van Vleck's determinant

(2)

Appendix B

(1)

From action to monodromy matrix

(4)

Spectral twinkling

(20)

M. V. Berry

Introduction

(3)

Examples of singularity-dominated strong fluctuations

(7)

Smells in random winds, and the sex life of moths

(2)

van Hove singularities and kin

(2)

Twinkling starlight

(3)

Spectral twinkling for integrable systems: superpoisson fluctuations

(3)

Spectral twinkling for mixed systems

(7)

Chaos, dissipation and quantal Brownian motion

(36)

D. Cohen

Introduction

(1)

Definition of the problem

(2)

Restricted versions of the problem

(1)

``History'' of the problem

(1)

Fluctuations: intensity and correlation time

(1)

Fluctuations: time-dependent Hamiltonian

(1)

Actual, parametric and reduced energy changes

(1)

The sudden and the adiabatic approximations

(1)

Ballistic and diffusive energy spreading

(1)

Energy spreading and dissipation

(1)

Application to the ``piston'' example

(2)

The route to stochastic behavior

(1)

The transition probability kernel

(2)

Limitations on quantal-classical correspondence (QCC)

(3)

The parametric evolution of P(n/m)

(1)

The time evolution of Pt(n/m)

(1)

Linear response theory

(1)

Actual and parametric dynamics

(1)

Perturbation theory

(2)

The over-simplified RMT picture

(1)

The perturbative core-tail spreading profile

(1)

An improved perturbation theory

(1)

Consequences of the improved perturbative treatment

(1)

The quantum-mechanical sudden approximation

(1)

The quantum-mechanical adiabatic approximation

(1)

Classical Brownian motion

(1)

The DLD Hamiltonian

(1)

The white-noise approximation (WNA)

(1)

Consequences of the WNA

(1)

The reduced propagator

(1)

Master equation

(1)

Brownian motion and dephasing

(1)

The open question

(3)

Quantum chaos in extended systems: Spreading wave packets and avoided band crossings

101

(20)

T. Geisel

R. Ketzmerick

K. Kruse

Introduction

101

(1)

Avoided band crossings

102

(10)

Modeling avoided band crossings

102

(5)

A perturbation calculation for the three-band model

107

(3)

A real example

110

(2)

Dynamics of quantum systems with fractal spectrum

112

(9)

The spectral function

112

(3)

Uniform scaling

115

(2)

Multiscaling

117

(4)

A brief introduction to random matrices

121

(24)

F. Haake

Introduction

121

(1)

Gaussian ensembles of Hermitian matrices

121

(1)

Integration measure

122

(1)

Eigenvalue distribution

123

(2)

Eigenvector distribution

125

(1)

Spacing distributions

125

(1)

Why universality of spectral fluctuations?

126

(2)

Wigner's semicircle law

128

(1)

Dyson's circular ensembles

129

(1)

Two-point correlator and form factor

130

(1)

CUE averages of symmetric functions

131

(1)

CUE form factor and pair correlator

132

(2)

How to process a single spectrum into a two-point correlator

134

(1)

Gaussian statistics and independence of finite-time traces

134

(2)

Ergodicity of the form factor

136

(1)

Poisson ensemble

137

(1)

Non-Gaussian ensembles

138

(1)

Universality of spectral fluctuations

139

(4)

Autocorrelations of secular determinants

143

(2)

The semiclassical tool in mesoscopic physics

145

(78)

Rodolfo A. Jalabert

Introduction

145

(4)

Quantum transport through classically chaotic cavities

149

(23)

Chaotic scattering

150

(3)

The scattering approach to the conductance

153

(3)

Quantum interference in ballistic cavities

156

(1)

Semiclassical transmission amplitudes

157

(3)

Transmission coefficients and average values

160

(2)

Conductance fluctuations

162

(7)

Weak localization in the ballistic regime

169

(3)

Scattering and integrability in quantum transport

172

(13)

Direct trajectories

173

(1)

Scattering through a rectangular cavity

174

(3)

Semiclassical transmission amplitudes for square cavities

177

(1)

Mean conductance in a square cavity

178

(3)

Conductance fluctuations in a square cavity

181

(2)

Circular billiards, diffraction and tunneling

183

(2)

Experiments on ballistic transport and other aspects of the theory

185

(6)

Conductance fluctuations and weak localization in ballistic microstructures

186

(2)

Semiclassics vs. random matrix theory

188

(2)

Mixed dynamics

190

(1)

Semiclassical approach to bulk conductivity

191

(1)

Orbital magnetism in clean systems

191

(8)

Thermodynamic formalism

192

(2)

Semiclassical treatment of susceptibilities

194

(2)

Square billiards

196

(2)

Integrable vs. chaotic behavior

198

(1)

Semiclassical approach to weak disorder

199

(9)

Disorder models

199

(1)

Single-particle Green function

200

(2)

Two-particle Green function

202

(1)

Fixed-size impurity average of the magnetic susceptibility

203

(2)

Combined impurity and energy average of the susceptibility

205

(3)

Relation to experiment and other theories

208

(1)

Electron-electron interactions in the ballistic regime

208

(7)

Screened Coulomb interaction in two dimensions

209

(1)

Thermodynamics and semiclassics of small interacting systems

210

(2)

First-order perturbation, diagonal and non-diagonal contributions

212

(1)

Higher-order terms

213

(2)

Conclusions

215

(8)

Statistics of energy levels and eigenfunctions in disordered and chaotic systems: Supersymmetry approach

223

(76)

A. D. Mirlin

Introduction

223

(1)

Introduction to the supersymmetry method and application to RMT

224

(19)

Green's function approach

224

(1)

Supermathematics

225

(3)

Average DOS from supersymmetry

228

(3)

Level correlations

231

(7)

Comments and generalizations

238

(1)

Structure of the saddle point manifold

238

(1)

Gaussian ensembles of different symmetry

239

(1)

Ensembles with non-Gaussian distributions of matrix elements

240

(2)

Random banded matrices

242

(1)

Parametric level statistics

243

(1)

Level statistics in a disordered sample: Diffusive σ-model

243

(15)

Derivation of the diffusive σ-model

243

(5)

Reduction to the 0D σ-model: Universal limit

248

(1)

Deviations from universality

249

(1)

Perturbation theory

249

(2)

Deviations from universality at ω «Ec

251

(3)

Stationary-point method

254

(2)

Spectral characteristics related to R2(s)

256

(1)

Spectral form factor

256

(1)

Level number variance

257

(1)

Eigenfunction statistics

258

(20)

Random matrix theory

258

(1)

Eigenfunction statistics in terms of the supersymmetric σ-model

259

(3)

Quasi-one-dimensional geometry

262

(1)

Exact solution of the σ-model

262

(1)

Short wire

263

(1)

Long wire

264

(1)

Metallic regime (arbitrary d)

265

(2)

2D: Weak multifractality of eigenfunctions

267

(2)

Spatial correlations of eigenfunction amplitudes

269

(1)

Zero-mode approximation

270

(1)

Quasi-1D geometry

271

(1)

Metallic regime (arbitrary d)

272

(1)

Anomalously localized states and long-time relaxation

273

(3)

Quasi-1D geometry

276

(1)

2D geometry

277

(1)

Random matrix model

278

(1)

Supersymmetry approach to the quantum chaos

278

(21)

Introduction: What we have learned from the diffusive problem

279

(1)

Ballistic σ-model

279

(1)

Heuristic arguments

279

(1)

Ballistic σ-model from disorder averaging

280

(2)

σ-model from energy averaging

282

(1)

Non-universal corrections and statistical noise

283

(1)

Problem of repetitions

284

(1)

Billiard with diffuse surface scattering

285

(1)

Level statistics

286

(2)

The level number variance

288

(1)

Eigenfunction statistics

288

(2)

σ-model for the kicked rotor

290

(2)

Concluding remarks

292

(7)

Experimental study of quantum chaos with cold atoms

299

(14)

M. G. Raizen

V. Milner

W. H. Oskay

D. A. Steck

Introduction

299

(1)

A two-level atom in a standing-wave potential

300

(2)

Experimental method

302

(2)

Kicked rotor

304

(6)

Current and future directions

310

(3)

Quantum stochasticity and the many-body problem

313

(20)

H. A. Weidenmuller

Introduction

313

(1)

Symmetries and the semiclassical approximation

314

(5)

Motivation

314

(1)

Pedestrain approach

315

(2)

Symmetry arguments

317

(2)

Invariant chaotic manifolds and collective motion

319

(4)

Invariant manifolds in interacting many-body systems

319

(3)

Quantum case

322

(1)

Classical model for the giant dipole resonance

322

(1)

Localization in Hilbert space

323

(10)

Anderson localization

323

(2)

Localization in polyatomic molecules

325

(2)

Fock-space localization in quantum dots

327

(3)

Fock-space localization in self-bound fermionic many-body systems

330

(3)

Spectral statistics and periodic orbits

333

(38)

E. B. Bogomolny

Introduction

333

(1)

Generalities

333

(7)

Trace formulas

333

(3)

Random matrix theory

336

(4)

Correlation functions

340

(5)

Diagonal approximation

341

(2)

Criterion of applicability of diagonal approximation

343

(2)

Beyond the diagonal approximation

345

(5)

The Hardy-Littlewood conjecture

347

(2)

Arithmetical systems

349

(1)

Construction of the density of states from a finite number of periodic orbits

350

(9)

Off-diagonal terms for the Riemann zeta-function

354

(2)

Off-diagonal contribution for dynamical systems

356

(3)

Random matrix universality

359

(4)

Riemann-Siegel form of density of states

363

(5)

Conclusion

368

(3)

Quantum chaos and thermalization for interacting particles

371

(60)

F. M. Izrailev

Introduction

371

(3)

Two-body random interaction model

374

(34)

Description of the model

374

(1)

Many-body Hamiltonian

374

(3)

Structure of the Hamiltonian matrix

377

(3)

Correlations in off-diagonal matrix elements

380

(1)

Density of states and spectrum statistics

381

(3)

Structure of exact eigenstates

384

(5)

Strength function

389

(6)

Analytical solution for the LDOS

395

(4)

Non-statistical properties of the TBRI-model

399

(9)

Thermalization and onset of chaos

408

(18)

Distribution of occupation numbers

408

(2)

Microcanonical vs. canonical distribution

410

(2)

Transition to the Fermi-Dirac distribution

412

(2)

Analytical approach to the ns-distribution

414

(3)

Effective Fermi-Dirac distribution for finite systems

417

(4)

Temperature vs. chaos

421

(3)

Transition to chaos and statistical equilibrium

424

(2)

Concluding remarks

426

(5)

Comparison between quantum and classical dynamics in the effective action formalism

431

(18)

F. Cametti

G. Jona-Lasinio

C. Presilla

F. Toninelli

Part I

432

(1)

Effective action in quantum mechanics

432

(2)

Loop expansion of the effective action

434

(2)

Derivative expansion of the effective action

436

(6)

Derivative expansion: Euclidean approach

437

(3)

Derivative expansion as a WKB-like approximation

440

(2)

Part II

442

(1)

2D anharmonic oscillator: classical

442

(1)

2D anharmonic oscillator: quantum

443

(6)

Torus maps, symmetries, and spectral statistics

449

(24)

J. P. Keating

F. Mezzadri

Introduction

449

(1)

Cat maps

450

(1)

Quantum cat maps

450

(3)

Kinematics

451

(1)

Dynamics

452

(1)

Trace formula

453

(1)

Spectral statistics and random matrix theory

453

(2)

Quantum symmetries

454

(1)

Classical symmetries

454

(1)

Examples

454

(1)

First puzzle

455

(1)

Genericity

455

(1)

Nonlinear perturbations

455

(7)

Second puzzle

462

(1)

Time-reversal and quantum phases

462

(2)

Third puzzle

464

(1)

Quantum symmetries with no classical limit

464

(7)

Outlook

471

(2)

Localization in billiards

473

(38)

T. Prosen

Introduction

473

(2)

Dynamical localization in conservative systems

475

(4)

Localization in 2D quantum billiards

479

(22)

Generic (KAM-type) billiard with perfect classical barriers (tori)

480

(6)

Rough billiards, and stadium billiard: partial classical barriers, slow classical diffusion, and cantori

486

(5)

Quantum dynamical localization

491

(6)

Local density of states

497

(4)

Localization in 3D quantum billiards

501

(4)

Scaling method for numerical solution of 2D Helmholz equation

505

(2)

Conclusions

507

(4)

Elenco dei partecipanti

511

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New Directions in Quantum Chaos

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Table of Contents

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New Directions in Quantum Chaos

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