  The transition from a many-body localized phase to a thermalizing one is a
dynamical quantum phase transition which lies outside the framework of
equilibrium statistical mechanics. We provide a detailed study of the critical
properties of this transition at finite sizes in one dimension. We find that
the entanglement entropy of small subsystems looks strongly subthermal in the
quantum critical regime, which indicates that it varies discontinuously across
the transition as the system-size is taken to infinity, even though many other
aspects of the transition look continuous. We also study the variance of the
half-chain entanglement entropy which shows a peak near the transition, and
find substantial variation in the entropy across eigenstates of the same
sample. Further, the sample-to-sample variations in this quantity are strongly
growing, and larger than the intra-sample variations. We posit that these
results are consistent with a picture 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 system on the thermal side of the transition as the system size is taken to
infinity. This discontinuity in a global quantity --- 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.
