Quantum chaotic systems: a random-matrix approach

TL;DR

This paper reviews how random matrix theory is applied to quantum chaotic systems, focusing on spectral analysis, symmetry classifications, and advanced statistical techniques.

Contribution

It clarifies the proper application of random matrix theory to quantum physics, including spectrum preparation, symmetry classification, and spectral statistics methods.

Findings

Explains the symmetry classification leading to Dyson's threefold and Altland-Zirnbauer's tenfold way.

Details the calculation of eigenvalue joint probability density functions from matrix probability densities.

Describes advanced techniques like orthogonal polynomials, determinantal processes, and supersymmetry in spectral analysis.

Abstract

We review the ideas of how random matrix theory has to be properly applied to quantum physics; particularly we focus on how the spectrum has to be properly prepared and the random matrix correctly identified before the random matrix and the physical eigenvalue spectrum can be compared. We explain the ideas of the symmetry classification of symmetric matrix spaces and how that yields Dyson's threefold and Altland-Zirnbauer's tenfold way. We also outline how the joint probability density function of the eigenvalues can be calculated from a given probability density function on the matrix space. Furthermore, we dive into the subtleties of the unfolding procedure. For this purpose, we explain the ideas of the local mean level spacing, the local level spacing distribution and the $k$-point correlation functions. We outline the techniques of orthogonal polynomials, determinantal and Pfaffian point processes and their related Fredholm determinants and Pfaffians as well as the supersymmetry method. Moreover, we relate the local spectral statistics to effective Lagrangians that give the relation to non-linear $\sigma$-models. In all these discussions, we also make brief excursions to non-Hermitian random matrix theory which are useful when studying open quantum systems, for instance.