Extended slow dynamical regime close to the many-body localization transition

Many-body localization is characterized by a slow logarithmic growth of the entanglement entropy after a global quantum quench while the local memory of an initial density imbalance remains at infinite time. We investigate how much the proximity of a many-body localized phase can influence the dynamics in the delocalized ergodic regime where thermalization is expected. Using an exact Krylov space technique, the out-of-equilibrium dynamics of the random-field Heisenberg chain is studied up to L = 28 sites, starting from an initially unentangled high-energy product state. Within most of the delocalized phase, we find a sub-ballistic entanglement growth S(t) proportional to t(1/z) with a disorder-dependent exponent z >= 1, in contrast with the pure ballistic growth z = 1 of clean systems. At the same time, anomalous relaxation is also observed for the spin imbalance I(t) proportional to t(-zeta) with a continuously varying disorder-dependent exponent zeta, vanishing at the transition. This provides a clear experimental signature for detecting this nonconventional regime.