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Supersymmetry and Trace Formulae

Chaos and Disorder

  • Igor V. Lerner
  • Jonathan P. Keating
  • David E. Khmelnitskii

Part of the NATO ASI Series book series (NSSB, volume 370)

Table of contents

  1. Front Matter
    Pages i-ix
  2. A. Altland, C. R. Offer, B. D. Simons
    Pages 17-57
  3. Predrag Cvitanović
    Pages 85-102
  4. Lev Kaplan, Eric J. Heller
    Pages 103-132
  5. Alexander D. Mirlin
    Pages 245-260
  6. Mark Srednicki
    Pages 261-267
  7. V. E. Kravtsov, I. V. Yurkevich, C. M. Canali
    Pages 269-291
  8. D. E. Khmelnitskii, B. A. Muzykantskii
    Pages 327-341
  9. H. A. Weidenmüller
    Pages 343-353
  10. M. V. Berry, J. P. Keating
    Pages 355-367
  11. Back Matter
    Pages 401-404

About this book

Introduction

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.

Keywords

Quantum Hall effect chaos diffusion disordered system dynamical systems kinetics metals quantization quantum chaos scattering supersymmetry symmetry wave

Editors and affiliations

  • Igor V. Lerner
    • 1
  • Jonathan P. Keating
    • 2
  • David E. Khmelnitskii
    • 3
    • 4
  1. 1.University of BirminghamBirminghamUK
  2. 2.University of Bristol and Hewlett-Packard LaboratoriesBristolUK
  3. 3.University of CambridgeCambridgeUK
  4. 4.L. D. Landau Institute for Theoretical PhysicsMoscowRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4615-4875-1
  • Copyright Information Kluwer Academic/ Plenum Publishers, New York 1999
  • Publisher Name Springer, Boston, MA
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4613-7212-7
  • Online ISBN 978-1-4615-4875-1
  • Series Print ISSN 0258-1221
  • Buy this book on publisher's site