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

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.