Random Matrix Model for Eigenvalue Statistics in Random Spin Systems

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.