Universal distribution of many-body localized eigenstates and selection-rule-related gap ratios

Recent studies of many-body localization (MBL) have highlighted the important but rarely explored regime with strong disorders, which were believed to reside deeply inside the localized side. However, resonances among eigenstates may have already destabilized the localization as fueled by the so-called avalanche mechanism. Understanding how these resonances arise and quantifying their universal behaviors analytically is desirable to unveil the nature of MBL in such regimes. This talk will introduce an analytical theory to quantitatively calculate the universal eigenstate distributions in Fock space. Our approach distinguishes from the previous analytical framework by explicitly incorporating a probability distribution function into the Fock space perturbation series. As a result, our closed-form formula directly predicts the values of physical quantities, such as the distribution of inverse-participation-ratio for eigenstates, that would be obtained in numerics after averaging over large numbers of disorder realization. It is found that pairwise resonance occurs in such strongly disordered regimes for eigenstates, which leads to two different scaling behaviors for eigenstates at the same parameter point. Moreover, based on our new analytical framework, we introduce the unconventional level spacing statistics (LSS) for {\em non-consecutive} levels related by Fock space selection rules, which sensitively capture resonance-induced level repulsions that already occur deep inside the localized regime. The non-consecutive gap ratios therefore show Gaussian Orthogonal Ensemble behaviors. This is in sharp contrast to conventional LSS for consecutive levels showing Poissonian distributions with the same parameters. Our analytical framework opens the door to viably calculating universal features for eigenstates in a strongly disordered interacting system, and points out a practical way to check selection-rule-related non-consecutive level spacings that can be essential in proximity to integrable points.