Signatures of a critical point in the many-body localization transition

Disordered interacting spin chains that undergo a many-body localization transition are characterized by two limiting behaviors where the dynamics are chaotic and integrable. However, the transition region between them is not fully understood yet. We propose here a signature that unambiguously identifies a possible finite-size precursor of a critical point, and distinguishes between two different stages of the transition. The kurtosis excess of the diagonal fluctuations of the full one-dimensional momentum distribution from its microcanonical average is maximum at this singular point in the paradigmatic disordered $J_1$-$J_2$ model. Both the particular value of this maximum and the disorder strength at which it is reached increase with the system size, as expected for a typical finite-size scaling. We completely characterize the short and long-range spectral statistics of the model and find that their behavior perfectly correlates with the properties of the diagonal fluctuations. For lower values of the disorder, we find a chaotic region in which the Thouless energy diminishes up to the transition point, at which it becomes equal to the Heisenberg energy. For larger values of disorder, spectral statistics are very well described by a generalized semi-Poissonian model, eventually leading to the integrable Poissonian behavior.