Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature

TL;DR

This paper explores the spectral properties of a generalized random matrix model linked to the Calogero-Sutherland system, demonstrating critical statistics behavior relevant for complex quantum systems with fractal classical structures.

Contribution

It introduces a new random matrix model based on the Calogero-Sutherland Hamiltonian that captures critical spectral statistics at finite temperature.

Findings

Number variance is asymptotically linear with a model-dependent slope.

The model accurately describes spectral correlations in quantum systems with fractal classical structures.

Analytical results match numerical data for specific quantum examples.

Abstract

We investigate the spectral properties of a generalized GOE (Gaussian Orthogonal Ensemble) capable of describing critical statistics. The joint distribution of eigenvalues of this model is expressed as the diagonal element of the density matrix of a gas of particles governed by the Calogero-Sutherland Hamiltonian (C-S). Taking advantage of the correspondence between C-S particles and eigenvalues, we show that the number variance of our random matrix model is asymptotically linear with a slope depending on the parameters of the model. Such linear behavior is a signature of critical statistics. This random matrix model may be relevant for the description of spectral correlations of complex quantum systems with a self-similar/fractal Poincar\'e section of its classical counterpart. This is shown in detail for two examples: the anisotropic Kepler problem and a kicked particle in a well potential. In both cases the number variance and the $\Delta_3$-statistic is accurately described by our analytical results.