Evidence for Unbounded Growth of the Number Entropy in Many-Body Localized Phases

We investigate the number entropy S-N-which characterizes particle-number fluctuations between subsystems-following a quench in one-dimensional interacting many-body systems with potential disorder. We find evidence that in the regime which is expected to show many-body localization and where the entanglement entropy grows as S similar to In t as function of time t, the number entropy grows as S-N similar to In In t, indicating continuing subdiffusive particle transport at a very slow rate. We demonstrate that this growth is consistent with a relation between entanglement and number entropy recently established for non-interacting systems.