Relaxation at different length-scales in models of many-body localization

We study dynamical correlation functions in the random-field Heisenberg chain, which probes the relaxation times at different length scales. Firstly, we argue that the relaxation time associated with the dynamical imbalance (examining the relaxation at the smallest length scale) decreases with disorder much faster than the one related to the dc conductivity, which probes the global response of the system. We argue that the observed dependence of relaxation on the length scale originates from local nonresonant regions (islands). The latter has particularly long relaxation times or remains frozen, allowing for nonzero dc transport via higher-order processes. Based on the numerical evidence, we introduce a toy model that suggests that the nonresonant islands' asymptotic dynamics are essential for the proper understanding of the disordered chains with many-body interactions.