Random Matrix Model for Eigenvalue Statistics in Random Spin Systems

TL;DR

This paper introduces a combined random matrix model approach to accurately describe eigenvalue statistics across the entire phase diagram of random spin systems, capturing the transition from thermal to many-body localized phases.

Contribution

The study develops a novel method using mixed and Gaussian β ensembles to model eigenvalue correlations, with a simple average providing near-perfect accuracy across different disorder regimes.

Findings

Mixed ensemble underestimates long-range correlations.

Gaussian β ensemble overestimates long-range correlations.

Average of both models accurately describes eigenvalue statistics.

Abstract

We propose a working strategy to describe the eigenvalue statistics of random spin systems along the whole phase diagram with thermal to many-body localization (MBL) transition. Our strategy relies on two random matrix (RM) models with well-defined matrix construction, namely the mixed (Brownian) ensemble and Gaussian $\beta$ ensemble. We show both RM models are capable of capturing the lowest-order level correlations during the transition, while the deviations become non-negligible when fitting higher-order ones. Specifically, the mixed ensemble will underestimate the longer-range level correlations, while the opposite is true for $\beta$ ensemble. Strikingly, a simple average of these two models gives nearly perfect description of the eigenvalue statistics at all disorder strengths, even around the critical region, which indicates the interaction range and strength between eigenvalue levels are the two dominant features that are responsible for the phase transition.