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Saturation (graph theory)

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Physical review. B./Physical review. B

We study the growth of the number entropy ${S}_{\mathrm{N}}$ 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 ${S}_{\mathrm{N}}$ 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 ${S}_{\mathrm{N}}$ in the disordered isotropic Heisenberg model we first reconfirm that, indeed, there is a small growth of ${S}_{\mathrm{N}}$. 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 ${S}_{\mathrm{N}}$: trivial initial growth at short times, a slow power-law growth at intermediate times, and the scaling of the saturation value of ${S}_{\mathrm{N}}$ with the disorder strength. Because resonances crucially depend on individual disorder realizations, the growth of ${S}_{\mathrm{N}}$ also heavily varies depending on the initial state, and therefore ${S}_{\mathrm{N}}$ and von Neumann entropy can behave rather differently.

Resonance-induced growth of number entropy in strongly disordered systems