Resonance-induced growth of number entropy in strongly 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.