Many-body localization in infinite chains

We investigate the phase transition between an ergodic and a many-body localized phase in infinite anisotropic spin-$1/2$ Heisenberg chains with binary disorder. Starting from the Néel state, we analyze the decay of antiferromagnetic order $m_s\left(t\right)$ and the growth of entanglement entropy $S_{\mathrm{ent}}\left(t\right)$ during unitary time evolution. Near the phase transition we find that $m_s\left(t\right)$ decays exponentially to its asymptotic value $m_s\left(\infty \right)\ne 0$ in the localized phase while the data are consistent with a power-law decay at long times in the ergodic phase. In the localized phase, $m_s\left(\infty \right)$ shows an exponential sensitivity on disorder with a critical exponent $\nu \sim 0.9$. The entanglement entropy in the ergodic phase grows subballistically, $S_{\mathrm{ent}}\left(t\right)\sim t^{\alpha }$, $\alpha \le 1$, with $\alpha$ varying continuously as a function of disorder. Exact diagonalizations for small systems, on the other hand, do not show a clear scaling with system size and attempts to determine the phase boundary from these data seem to overestimate the extent of the ergodic phase.

Figure 1

Spin chain (1) with binary disorder. (a) Phase boundary for the infinite chain obtained from the order parameter $m_s\left(\infty \right)$. The relaxation time $\tau$ changes substantially near the transition line, see Sec. 2. (Inset) Thermalizing clusters (equal field $D$) are separated by barriers (staggered field $\pm D$). (b) ED phase boundary from finite-size extrapolation ${lim}_{L\to \infty }\overline{m_s}\left(L\right)$ and the line where the average gap in the energy spectrum for $L=16$ crosses the intermediate value $\langle r\rangle =0.4575$ between GOE and Poisson, see Sec. 4.

Figure 2

Decay of the order parameter $m_s\left(t\right)$ from LCRG for (a) $\mathrm{\Delta }=0.25$, (b) $\mathrm{\Delta }=1$, and (c) $\mathrm{\Delta }=4$. Left column $D=0.02$ and right column $D=0.3$. Error bars are smaller than the symbol size. The lines are fits with absolute statistical errors for $m_s\left(\infty \right)$ of the order of ${10}^{-3}–{10}^{-4}$. The relaxation time $\tau$ decreases with increasing disorder for $\mathrm{\Delta }=0.25$ and 4 while it increases for $\mathrm{\Delta }=1$.

Figure 3

(a) Data collapse of the magnetization $m_s\left(\infty \right)$. (b) Power-law exponent $\zeta$ [same symbols as in (a)] and (c) relaxation times $\tau$ for small disorder $D$. Symbols: results from fits of the LCRG data; lines in (b) and (c) are guides to the eye.

Figure 4

Entanglement entropy. (a) Exponent $S_{\mathrm{ent}}\sim t^{\alpha }$ for different anisotropies $\mathrm{\Delta }$ at the longest times accessible by LCRG. (b) $S_{\mathrm{ent}}\left(t\right)$ for $\mathrm{\Delta }=1$ and different disorder strengths $D$ on a log-log scale. The dashed lines are power-law fits.

Figure 5

(a) $\mathrm{\Delta }=1:\langle r\rangle$ values for open chains of length $L$. (b) Time averaged magnetizations $\overline{m_s}$ for disorder $D=0.08,0.24,0.48,0.72,...,2.16$ (from bottom to top). The dashed lines are linear extrapolations in $1/L$ for $L\ge 10$.

Figure 6

Time evolution of the staggered magnetization $m_s\left(t\right)$ at the isotropic point $\mathrm{\Delta }=1$ for small disorder $D=0.3$ from ED for $L\le 14$ with periodic boundary conditions (PBC) and LCRG ($L=\infty$). Already for short times $Jt\sim 4$, the time evolution differs visibly due to finite-size effects in ED.

Figure 7

Fit parameters for $\mathrm{\Delta }=0.5$ using the fit function Eq. (4). Fit 1: $t\ge 1$ with power law $B{t}^{-\zeta }$, fit 2: $t\ge 3$ with $m_s\left(\infty \right)$, and fit 3: $t\ge 5$ with $m_s\left(\infty \right)$. The values obtained for $m_s\left(\infty \right)$ are shown in Fig. 8.

Figure 8

(a) $m_s\left(t\right)$ for $\mathrm{\Delta }=0.5$ and $D=0.6,0.8$. The dashed lines denote the time average for $t\in [5,t_{\max }]$. (b) $m_s\left(\infty \right)$ extracted from the fits (see Fig. 7 for the other fitting parameters) for $D\le 0.5$ and from a time average for $D>0.5$.

Figure 9

$\langle r\rangle$ values for open XXZ chains for different anisotropies $\mathrm{\Delta }$ with $L=16$. The critical disorder strength where $\langle r\rangle$ crosses over from GOE to Poisson is used as criterion for the ergodic-MBL phase transition.

Figure 10

Time evolution of the staggered magnetization $m_s\left(t\right)$ at the isotropic point $\mathrm{\Delta }=1$ for disorder $D=0.9$ from ED for $L\le 16$ with open (OBC) and periodic boundary conditions (PBC) and LCRG ($L=\infty$). While OBC results converge slowly to the $L=\infty$ limit, the PBC results are quite accurate up to $Jt\sim 20$. Disorder averages are exact except for $L=16$ with OBC where 950 samples have been used.

Figure 11

Time averaged magnetizations at the isotropic point $\mathrm{\Delta }=1$ as a function of disorder $D$ for different lengths $L$ (ED with OBC).

Figure 12

Time averaged magnetizations at the isotropic point $\mathrm{\Delta }=1$ as a function of system size $1/L$ for different disorder $D$ (ED with PBC). The finite-size scaling is compatible with $\overline{m_s}\left(L\right)\sim L^{-3/2}$.

Figure 13

$\overline{m_s}$ as a function of disorder configuration for a chain with length $L=14$, $\mathrm{\Delta }=1$ (ED with PBC) where (a) $D=0.2$, and (b) $D=2.0$.