Evidence for Unbounded Growth of the Number Entropy in Many-Body Localized Phases

We investigate the number entropy $S_N$—which characterizes particle-number fluctuations between subsystems—following a quench in one-dimensional interacting many-body systems with potential disorder. We find evidence that in the regime which is expected to show many-body localization and where the entanglement entropy grows as $S\sim \ln t$ as function of time $t$, the number entropy grows as $S_N\sim \ln \ln t$, indicating continuing subdiffusive particle transport at a very slow rate. We demonstrate that this growth is consistent with a relation between entanglement and number entropy recently established for noninteracting systems.

Figure 1

$S_N$ for $D<D_c\approx 14$: (a) $S_N(t)$ for $L=24$, with 500 disorder realizations and logarithmic fits, $S_N=(\nu /2)\ln t+b$. (b) $S_N(t\to \infty )$ for different system sizes $L$. (c) Prefactors $\nu$ and constants $b$ of the logarithmic fits as a function of $D$ for $L=24$.

Figure 2

$S_N$ for $D>D_c$: (a) $S_N(t)$ for $L=24$ and double logarithmic fits, $S_N=(\nu /2)\ln \ln t+b$. (b) $S_N(t\to \infty )$ for different system sizes $L$. (c) Prefactor $\nu$ and constant $b$ of the double logarithmic fits as a function of $D$ for $L=24$, see also [33].

Figure 3

$S_N^{(2)}$ and bound (7) for (a) $D<D_c$, and (b) $D>D_c$. The renormalization parameter $\gamma$ appears to decrease monotonically with increasing $D$.

Figure 4

$S_N^{(2)}$ for model (4) with binary disorder $D=20$ and segments with equal $D$ limited to four sites (1000 disorder realizations). For $t<t_{\mathrm{Th}}$ an unrenormalized bound ($\gamma =1$) holds while $\gamma$ is renormalized for $t>t_{\mathrm{Th}}$, see text.