Supersymmetry in quantum chaos and mesoscopic physics

TL;DR

This paper reviews the supersymmetry method and its application to quantum chaos and mesoscopic physics, highlighting how the zero-dimensional sigma-model relates to random matrix theory and enables analysis of conductance fluctuations and nuclear magnetic resonance in small metals.

Contribution

It demonstrates the equivalence of the zero-dimensional sigma-model to random matrix theory and its use in solving problems in quantum chaos and mesoscopic physics.

Findings

Zero-dimensional sigma-model is equivalent to random matrix theory.

The method allows calculation of distribution functions for physical quantities.

Applications include conductance fluctuations and nuclear magnetic resonance in small metal particles.

Abstract

A brief review of the supersymmetry method and its application to mesoscopic physics and quantum chaos is given. Alghough a non-linear supermatrix $% \sigma $-model in this approach was derived from models with random potential, it is emphasized that the zero-dimensional version of the $\sigma $-model is equivalent to the random matrix theory and can even be derived from the latter, too. This gives a possibility to use the zero -dimensional model for description of problems of quantum chaos and mesoscopic physics. A number of problems considered recently is presented. This includes nuclear magnetic resonance in small metal particles and statistics of conductance fluctuations in quantum dots. The solution of these problems became possible due to a new possibility to calculate distribution functions. (Expanded version of a lecture given at the 14 Annual CNLS Conference "Quantum Complexity in Mesoscopic Systems" (Los Alamos, May 1994))