Model of level statistics for disordered interacting quantum many-body systems

We numerically study level statistics of disordered interacting quantum many-body systems. A two-parameter plasma model which controls level repulsion exponent $\beta$ and range $h$ of interactions between eigenvalues is shown to reproduce accurately features of level statistics across the transition from ergodic to many-body localized phase. Analysis of higher order spacing ratios indicates that the considered $\beta$-$h$ model accounts even for long range spectral correlations and allows to obtain a clear picture of the flow of level statistics across the transition. Comparing spectral form factors of $\beta$-$h$ model and of a system in the ergodic-MBL crossover, we show that the range of effective interactions between eigenvalues $h$ is related to the Thouless time which marks the onset of quantum chaotic behavior of the system. Analysis of level statistics of random quantum circuit which hosts chaotic and localized phases supports the claim that $\beta$-$h$ model grasps universal features of level statistics in transition between ergodic and many-body localized phases also for systems breaking time-reversal invariance.