Absence of true localization in many-body localized phases

Maximilian Kiefer-Emmanouilidis, Razmik Unanyan, Michael Fleischhauer, Jesko Sirker

We have recently shown that the logarithmic growth of the entanglement
entropy following a quantum quench in a many-body localized (MBL) phase is
accompanied by a slow growth of the number entropy, $S_N\sim\ln\ln t$. This
violates the standard scenario of MBL and raises the question whether the
observed behavior is transient or continues to hold at strong disorder in the
thermodynamic limit. Here we provide an in-depth numerical study of $S_N(t)$
for the disordered Heisenberg chain and find strong evidence that the system is
not fully localized even at strong disorder. Calculating the R\'enyi number
entropy $S_N^{(\alpha)}(t)$ for $\alpha\ll 1$---which is sensitive to large
number fluctuations occurring with low probability---we demonstrate that the
particle number distribution $p(n)$ in one half of the system has a small but
continuously growing tail. This indicates a steady increase of the number of
particles crossing between the partitions in the interacting case, and is in
sharp contrast to Anderson localization, where $S_N^{(\alpha)}(t)$ saturates.