Abstract
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.
Highlights
Understanding the nature of quantum phases and phase transitions is part of the bedrock of condensed matter physics
We studied the finite-size quantum critical and crossover regimes of the many-body localization (MBL)-to-eigenstate thermalization hypothesis (ETH) phase transition and found evidence supporting a view of this transition as a hybrid between continuous and discontinuous phase transitions
We showed that SA, the entanglement entropy of subregions A much smaller than the system size, looks strongly subthermal in the critical regime, contrary to an established constraint that requires SA to be thermal at the transition if it is continuous [26]
Summary
Understanding the nature of quantum phases and phase transitions is part of the bedrock of condensed matter physics. This detailed parsing helps us identify the likely source of the observed violations of the CLO inequality and helps us formulate a possible picture of the universal critical properties of the transition Inspired by these data, we present a picture for the finitesize behavior near the phase transition, which is consistent with both the discontinuity in SA and the observed trends in the variance of the half-chain entropy: Essentially, the transition to the thermal phase appears to be driven by a sparse resonant “backbone” of long-range entanglement [23], which just barely gains enough strength to become a functional “bath” and thermalize the entire system in the L → ∞ limit on the thermal side of the transition.
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