Theory of growth of number entropy in disordered systems

We investigate the source of the recently reported slow growth of number entropy S­_N, in one-dimensional disordered many-body systems in their localized phase, which seems to point out a presence of particle transport. Using the disordered XXZ model, at various disorder strengths, our study shows the growth can be explained within the framework of many-body localization, and occurs mainly due to `resonant disorder' cases. We discuss the behavior of S­_N during three different timescales- small, intermediate and steady-state, and are able to semi-analytically predict the behaviour during the small and steady-state time scales for large disorder. The growth of S­_N is also heavily dependent on the choice of initial states, and in individual choices of disorder realization, the growth of S(half-chain Von-Neumann entropy) and S­_N is not correlated at intermediate time scales.