Finite-Range Coulomb Gas Models of Banded Random Matrices and Quantum Kicked Rotors

TL;DR

This paper introduces finite-range Coulomb gas models to analyze banded random matrices and quantum kicked rotors, revealing new universality classes and a transition from Poisson to random matrix statistics as the effective range increases.

Contribution

It presents a novel finite-range Coulomb gas framework for banded matrices and quantum rotors, expanding the understanding of eigenvalue statistics and universality classes.

Findings

Finite-range Coulomb gas models yield new universality classes.

Transition from Poisson to classical random matrix statistics with increasing range.

Effective range relates to bandwidth and chaos parameter in models.

Abstract

Dyson demonstrated an equivalence between infinite-range Coulomb gas models and classical random matrix ensembles for study of eigenvalue statistics. We introduce finite-range Coulomb gas (FRCG) models via a Brownian matrix process, and study them analytically and by Monte-Carlo simulations. These models yield new universality classes, and provide a theoretical framework for study of banded random matrices (BRM) and quantum kicked rotors (QKR). We demonstrate that, for a BRM of bandwidth b and a QKR of chaos parameter {\alpha}, the appropriate FRCG model has the effective range d = (b^2)/N = ({\alpha}^2)/N, for large N matrix dimensionality. As d increases, there is a transition from Poisson to classical random matrix statistics.