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 chaotic and integrable. However, region between them is not fully understood yet. We propose here possible finite-size precursor of critical point shows typical scaling distinguishes different dynamical phases. The kurtosis excess diagonal fluctuations full one-dimensional momentum distribution from its microcanonical average maximum at this singular in paradigmatic disordered $J_1$-$J_2$ model. For system sizes accessible to exact diagonalization, both position size scale linearly with size. Furthermore, we show found same disorder strength which Thouless Heisenberg energies coincide. Below point, spectral statistics follow universal random matrix behavior up energy. Above it, no traces remain, well described generalized semi-Poissonian model, eventually leading integrable Poissonian behavior. provide, thus, an integrated scenario for transition, conjecturing thermodynamic limit, if it exists, should be given value strength.