Random Matrix Ensemble for the Level Statistics of Many-Body Localization

TL;DR

This paper demonstrates that the Gaussian beta ensemble accurately models the level statistics of many-body localized systems, bridging the gap between Poisson and Wigner-Dyson statistics across different phases.

Contribution

It introduces the Gaussian beta ensemble as a universal model for level statistics in many-body localization, covering the entire crossover from thermal to localized phases.

Findings

Gaussian beta ensemble matches level statistics in MBL systems

Provides a continuous interpolation between Poisson and Wigner-Dyson statistics

Applicable to models with and without time-reversal symmetry

Abstract

We numerically study the level statistics of the Gaussian $\beta$ ensemble. These statistics generalize Wigner-Dyson level statistics from the discrete set of Dyson indices $\beta = 1,2,4$ to the continuous range $0 < \beta < \infty$. The Gaussian $\beta$ ensemble covers Poissonian level statistics for $\beta \to 0$, and provides a smooth interpolation between Poissonian and Wigner-Dyson level statistics. We establish the physical relevance of the level statistics of the Gaussian $\beta$ ensemble by showing near-perfect agreement with the level statistics of a paradigmatic model in studies on many-body localization over the entire crossover range from the thermal to the many-body localized phase. In addition, we show similar agreement for a related Hamiltonian with broken time-reversal symmetry.