The internal clock of many-body delocalization

After a decade of many claims to the opposite, there now is a growing consensus that generic disordered quantum wires, e.g. the XXZ-Heisenberg chain, do not exhibit many-body localization (MBL) - at least not in a strict sense within a reasonable window of disorder values $W$. Specifically, computational studies of short wires exhibit an extremely slow but unmistakable flow of physical observables with increasing time and system size (creep") that is consistently directed away from (strict) localization. Our work sheds fresh light on delocalization physics: Strong sample-to-sample fluctuations indicate the absence of a generic time scale, i.e. of a naive clock rate"; however, the concept of an ``internal clock" survives, at least in an ensemble sense. Specifically, we investigate the relaxation of the imbalance $\mathcal{I}(t)$ and its temporal fluctuations $\mathcal{F}(t)$, the entanglement and Renyi entropies, ${\mathcal{S}}_{\mathrm{e}}(t)$ and ${\mathcal{S}}_2(t)$, in a 1D system of interacting disordered fermions. We observe that adopting ${\mathcal{S}}_{\mathrm{e}}(t),{\mathcal{S}}_2(t)$ as a measure for the internal time per sample reduces the sample-to-sample fluctuations but does not eliminate them. However, a (nearly) perfect collapse of the average $\bar{\mathcal{I}}(t)$ and $\bar{\mathcal{F}}(t)$ for different $W$ is obtained when plotted against ${\bar{\mathcal{S}}}_{\mathrm{e}}(t)$ or ${\bar{\mathcal{S}}}_2(t)$, indicating that the average entropy appropriately models the ensemble-averaged internal clock. We take the tendency for faster-than-logarithmic growth of ${\bar{\mathcal{S}}}_{\mathrm{e}}(t)$ together with smooth dependency on $W$ of all our observables within the entire simulation window as support for the cross-over scenario, discouraging an MBL transition within the traditional parametric window of computational studies.