Characterizing the many-body localization transition through correlations

Closed, interacting, quantum systems have the potential to transition to a many-body localized (MBL) phase under the presence of sufficiently strong disorder, hence breaking ergodicity and failing to thermalize. In this work we study the distribution of correlations throughout the ergodic-MBL phase diagram. We find the typical correlations in the MBL phase decay as a stretched exponential with range $r$ eventually crossing over to an exponential decay deep in the MBL phase. At the transition, the stretched exponential goes as $e^{-A\sqrt{r}}$, a decay that is reminiscent of the random singlet phase. While the standard deviation of the $\log(QMI)$ has a range dependence, the $\log(QMI)$ converges to a range-invariant distribution on all other moments (i.e., the skewness and higher) at the transition. The universal nature of these distributions provides distinct phenomenology of the transition different from both the ergodic and MBL phenomenologies. In addition to the typical correlations, we study the extreme correlations in the system, finding that the probability of strong long-range correlations is maximal at the transition, suggesting the proliferation of resonances there. Finally, we analyze the probability that a single bit of information is shared across two halves of a system, finding that this probability is non-zero deep in the MBL phase but vanishes at moderate disorder well above the transition.