Critical properties of the many-body localization transition

The transition from a many-body localized phase to a thermalizing one is a dynamical quantum phasetransition that lies outside the framework of equilibrium statistical mechanics. We provide a detailed studyof the critical properties of this transition at finite sizes in one dimension. We find that the entanglemententropy of small subsystems looks strongly subthermal in the quantum critical regime, which indicates thatit varies discontinuously across the transition as the system size is taken to infinity, even though many otheraspects of the transition look continuous. We also study the variance of the half-chain entanglemententropy, which shows a peak near the transition, and find substantial variation in the entropy acrosseigenstates of the same sample. Furthermore, the sample-to-sample variations in this quantity are stronglygrowing and are larger than the intrasample variations. We posit that these results are consistent with apicture in which the transition to the thermal phase is driven by an eigenstate-dependent sparse resonant"backbone" of long-range entanglement, which just barely gains enough strength to thermalize the systemon the thermal side of the transition as the system size is taken to infinity. This discontinuity in a globalquantity-the presence of a fully functional bath-in turn implies a discontinuity even for local properties.We discuss how this picture compares with existing renormalization group treatments of the transition.