Many-body localization in the Heisenberg $XXZ$ magnet in a random field

We numerically investigate Heisenberg $XXZ$ spin-$1∕2$ chain in a spatially random static magnetic field. We find that time-dependent density-matrix renormalization group simulations of time evolution can be performed efficiently, namely, the dimension of matrices needed to efficiently represent the time evolution increases linearly with time and entanglement entropies for typical chain bipartitions increase logarithmically. As a result, we show that for large enough random fields, infinite temperature spin-spin correlation function displays exponential localization in space indicating insulating behavior of the model.

Figure 1

Entanglement entropies $S_{0.5}\left(t\right)$ and $S_1\left(t\right)$ for (a) pure state evolution and (b) operator space evolution (see text for details). We compare cases of $\Delta =0.5$ (growing curves) and $\Delta =0$ (saturating curves), for $n=50$, average over (a) 100, (b) 21 (for $\Delta =0.5$), and (b) 1000 (for $\Delta =0$) disorder realizations with magnetic field magnitude $h=5$. In the insets, we compare logarithmic growth of $S_1\left(t\right)$ in disordered case (full curves, same data as in main frames) to linear growth in the case of staggered magnetic field $h_j={(-1)}^{j}5∕\sqrt{3}$ and $\Delta =0.5$ (dashed curves).

Figure 2

Dimension $D_{\epsilon }\left(t\right)$ for pure state (top) and operator evolution (bottom). Data for various $\epsilon$ clearly indicate linear growth with time. $\Delta =0.5$, $n=50$, and a single disorder realization with $h=5$ (the same realization for all data points).

Figure 3

Renyi entanglement entropy growth $S_{0.5}\left(t\right)$ for pure state evolution and different system sizes $n$. We show numerical results, averaged over 100 realizations of disorder, for $n=6$, 10, 30, and 50, with short-dashed, dotted, long-dashed, and full curves, respectively. Note that $n=30$ and $n=50$ cases almost overlap. In all cases, $\Delta =0.5$ and $h=5$, and we show the average entropy over all bipartite cuts in order to have smoother data.

Figure 4

(Color online) Spin-spin correlation function $C(r,t)$ (${\log }_{10}$ color code) for noninteracting case, $\Delta =0$, $n=500$, average over 1000 disorder realizations with $h=1$. Solid curves are isolines at ${10}^{-8}$, ${10}^{-6}$, ${10}^{-4}$, and ${10}^{-2}$ from right to left.

Figure 5

(Color online) Correlation function $C(r,t)$ for $n=50$, $\Delta =0.5$, averaged over 21 disorder realizations with $h=5$.