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

In lattice systems with particle-number conservation the von Neumann entanglement entropy $S_{\textrm{ent}}$ is the sum of number entropy $S_{\textrm{n}}$ and configurational entropy $S_\textrm{conf}$. As shown recently both quantities can be obtained in an experiment from the full counting statistics. We numerically investigate the particle-number entropy $S_n$ following a quench in one-dimensional interacting many-body systems with potential disorder. We find evidence that in the regime which is expected to be many-body localized and where the von-Neumann entanglement entropy is known to grow as $S_{\textrm{ent}}\sim \ln t$ as function of time $t$, also the number entropy increases as function of time as $S_n\sim\ln\ln t$. If this growth continues in the thermodynamic limit for infinite times, it would signal (ultra-slow) ergodic behavior rather than localization of particles. We show furthermore that for free systems $S_{\textrm{ent}}$ is completely fixed by $S_{\textrm{n}}$.