Roopayan Ghosh: Theory of growth of number entropy in disordered systems

We study the growth of the number entropy S_N in one-dimensional number-conserving interacting systems with strong disorder, which are believed to display many-body localization. Recently a slow and small growth of S_N has been numerically reported, which, if holding at asymptotically long times in the thermodynamic limit, would imply ergodicity and therefore the absence of true localization. By numerically studying S_N in the disordered isotropic Heisenberg model we first reconfirm that, indeed, there is a small growth of S_N. However, we show that such growth is fully compatible with localization. To be specific, using a simple model that accounts for expected rare resonances we can analytically predict several main features of numerically obtained S_N: trivial initial growth at short times, a slow power-law growth at intermediate times, and the scaling of the saturation value of S_N with the disorder strength. Because resonances crucially depend on individual disorder realizations, the growth of S_N also heavily varies depending on the initial state, and therefore S_N and von Neumann entropy can behave rather differently.