Ebook: Supersymmetry and Trace Formulae: Chaos and Disorder

The motion of a particle in a random potential in two or more dimensions is chaotic, and the trajectories in deterministically chaotic systems are effectively random. It is therefore no surprise that there are links between the quantum properties of disordered systems and those of simple chaotic systems. The question is, how deep do the connec­ tions go? And to what extent do the mathematical techniques designed to understand one problem lead to new insights into the other? The canonical problem in the theory of disordered mesoscopic systems is that of a particle moving in a random array of scatterers. The aim is to calculate the statistical properties of, for example, the quantum energy levels, wavefunctions, and conductance fluctuations by averaging over different arrays; that is, by averaging over an ensemble of different realizations of the random potential. In some regimes, corresponding to energy scales that are large compared to the mean level spacing, this can be done using diagrammatic perturbation theory. In others, where the discreteness of the quantum spectrum becomes important, such an approach fails. A more powerful method, devel­ oped by Efetov, involves representing correlation functions in terms of a supersymmetric nonlinear sigma-model. This applies over a wider range of energy scales, covering both the perturbative and non-perturbative regimes. It was proved using this method that energy level correlations in disordered systems coincide with those of random matrix theory when the dimensionless conductance tends to infinity.

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This volume provides a representative overview of recent progress in the theories of quantum disordered and chaotic systems, and an introduction to the underlying concepts and techniques.

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This volume provides a representative overview of recent progress in the theories of quantum disordered and chaotic systems, and an introduction to the underlying concepts and techniques.
Content:
Front Matter....Pages i-ix
Periodic Orbits, Spectral Statistics, and the Riemann Zeros....Pages 1-15
Quantum Chaos: Lessons from Disordered Metals....Pages 17-57
Supersymmetric Generalization of Dyson’s Brownian Motion (Diffusion)....Pages 59-73
What Happens to the Integer Quantum Hall Effect in Three Dimensions?....Pages 75-83
Trace Formulas in Classical Dynamical Systems....Pages 85-102
Theory of Eigenfunction Scarring....Pages 103-132
Nonequilibrium Effects in the Tunneling Conductance Spectra of Small Metallic Particles....Pages 133-151
Pair Correlations of Quantum Chaotic Maps from Supersymmetry....Pages 153-172
Semiclassical Quantization of Maps and Spectral Correlations....Pages 173-192
Wave Functions, Wigner Functions and Green Functions of Chaotic Systems....Pages 193-225
Wave Functions in Chaotic Billiards: Supersymmetry Approach....Pages 227-243
Correlations of Wave Functions in Disordered Systems....Pages 245-260
Spatial Correlations in Chaotic Eigenfunctions....Pages 261-267
Level Curvature Distribution Beyond Random Matrix Theory....Pages 269-291
Almost-Hermitian Random Matrices: Applications to the Theory of Quantum Chaotic Scattering and Beyond....Pages 293-313
Topological Features of the Magnetic Response in Inhomogeneous Magnetic Fields....Pages 315-325
From Classical to Quantum Kinetics....Pages 327-341
Stochastic Scattering....Pages 343-353
H=xp and the Riemann Zeros....Pages 355-367
Parametric Random Matrices: Static and Dynamic Applications....Pages 369-399
Back Matter....Pages 401-404

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This volume provides a representative overview of recent progress in the theories of quantum disordered and chaotic systems, and an introduction to the underlying concepts and techniques.
Content:
Front Matter....Pages i-ix
Periodic Orbits, Spectral Statistics, and the Riemann Zeros....Pages 1-15
Quantum Chaos: Lessons from Disordered Metals....Pages 17-57
Supersymmetric Generalization of Dyson’s Brownian Motion (Diffusion)....Pages 59-73
What Happens to the Integer Quantum Hall Effect in Three Dimensions?....Pages 75-83
Trace Formulas in Classical Dynamical Systems....Pages 85-102
Theory of Eigenfunction Scarring....Pages 103-132
Nonequilibrium Effects in the Tunneling Conductance Spectra of Small Metallic Particles....Pages 133-151
Pair Correlations of Quantum Chaotic Maps from Supersymmetry....Pages 153-172
Semiclassical Quantization of Maps and Spectral Correlations....Pages 173-192
Wave Functions, Wigner Functions and Green Functions of Chaotic Systems....Pages 193-225
Wave Functions in Chaotic Billiards: Supersymmetry Approach....Pages 227-243
Correlations of Wave Functions in Disordered Systems....Pages 245-260
Spatial Correlations in Chaotic Eigenfunctions....Pages 261-267
Level Curvature Distribution Beyond Random Matrix Theory....Pages 269-291
Almost-Hermitian Random Matrices: Applications to the Theory of Quantum Chaotic Scattering and Beyond....Pages 293-313
Topological Features of the Magnetic Response in Inhomogeneous Magnetic Fields....Pages 315-325
From Classical to Quantum Kinetics....Pages 327-341
Stochastic Scattering....Pages 343-353
H=xp and the Riemann Zeros....Pages 355-367
Parametric Random Matrices: Static and Dynamic Applications....Pages 369-399
Back Matter....Pages 401-404
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