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Supersymmetry and Trace Formulae

Chaos and Disorder

  • Book
  • © 1999

Overview

Editors:
  1. Igor V. Lerner

    1. University of Birmingham, Birmingham, UK

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    You can also search for this editor in PubMed   Google Scholar

  2. Jonathan P. Keating

    1. University of Bristol and Hewlett-Packard Laboratories, Bristol, UK

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    You can also search for this editor in PubMed   Google Scholar

  3. David E. Khmelnitskii

    1. University of Cambridge, Cambridge, UK
      L. D. Landau Institute for Theoretical Physics, Moscow, Russia

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    You can also search for this editor in PubMed   Google Scholar

Part of the book series: NATO Science Series B: (NSSB, volume 370)

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Table of contents (20 chapters)

  1. Front Matter

    Pages i-ix
    Download chapter PDF

  2. Periodic Orbits, Spectral Statistics, and the Riemann Zeros

    • J. P. Keating
    Pages 1-15

  3. Quantum Chaos: Lessons from Disordered Metals

    • A. Altland, C. R. Offer, B. D. Simons
    Pages 17-57

  4. Supersymmetric Generalization of Dyson’s Brownian Motion (Diffusion)

    • Thomas Guhr
    Pages 59-73

  5. What Happens to the Integer Quantum Hall Effect in Three Dimensions?

    • J. T. Chalker
    Pages 75-83

  6. Trace Formulas in Classical Dynamical Systems

    • Predrag Cvitanović
    Pages 85-102

  7. Theory of Eigenfunction Scarring

    • Lev Kaplan, Eric J. Heller
    Pages 103-132

  8. Nonequilibrium Effects in the Tunneling Conductance Spectra of Small Metallic Particles

    • Oded Agam
    Pages 133-151

  9. Pair Correlations of Quantum Chaotic Maps from Supersymmetry

    • M. R. Zirnbauer
    Pages 153-172

  10. Semiclassical Quantization of Maps and Spectral Correlations

    • Uzy Smilansky
    Pages 173-192

  11. Wave Functions, Wigner Functions and Green Functions of Chaotic Systems

    • Shmuel Fishman
    Pages 193-225

  12. Wave Functions in Chaotic Billiards: Supersymmetry Approach

    • K. B. Efetov
    Pages 227-243

  13. Correlations of Wave Functions in Disordered Systems

    • Alexander D. Mirlin
    Pages 245-260

  14. Spatial Correlations in Chaotic Eigenfunctions

    • Mark Srednicki
    Pages 261-267

  15. Level Curvature Distribution Beyond Random Matrix Theory

    • V. E. Kravtsov, I. V. Yurkevich, C. M. Canali
    Pages 269-291

  16. Almost-Hermitian Random Matrices: Applications to the Theory of Quantum Chaotic Scattering and Beyond

    • Yan V. Fyodorov
    Pages 293-313

  17. Topological Features of the Magnetic Response in Inhomogeneous Magnetic Fields

    • E. Akkermans, R. Narevich
    Pages 315-325

  18. From Classical to Quantum Kinetics

    • D. E. Khmelnitskii, B. A. Muzykantskii
    Pages 327-341

  19. Stochastic Scattering

    • H. A. Weidenmüller
    Pages 343-353

  20. H=xp and the Riemann Zeros

    • M. V. Berry, J. P. Keating
    Pages 355-367
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Keywords

About this book

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.

Editors and Affiliations

  • University of Birmingham, Birmingham, UK

    Igor V. Lerner

  • University of Bristol and Hewlett-Packard Laboratories, Bristol, UK

    Jonathan P. Keating

  • University of Cambridge, Cambridge, UK

    David E. Khmelnitskii

  • L. D. Landau Institute for Theoretical Physics, Moscow, Russia

    David E. Khmelnitskii

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