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We investigate the number entropy $S_N$---which characterizes particle-numberfluctuations between subsystems---following a quench in one-dimensionalinteracting many-body systems with potential disorder. We find evidence that inthe regime which is expected to show many-body localization (MBL) and where theentanglement entropy grows as $S\sim \ln t$ as function of time $t$, the numberentropy grows as $S_N\sim\ln\ln t$, indicating continuing particle transport ata very slow rate. We demonstrate that this growth is consistent with a relationbetween entanglement and number entropy recently established fornon-interacting systems.

Evidence for unbounded growth of the number entropy in many-body localized phases