Relaxation at different length scales in models of many-body localization

We study dynamical correlation functions in the random-field Heisenberg chain, which probes the relaxation times at different length scales. First, we show that the relaxation time associated with the dynamical imbalance (examining the relaxation at the smallest length scale) decreases with disorder much faster than the one determined by the dc conductivity (probing the global response of the system). We argue that the observed dependence of relaxation on the length scale originates from local nonresonant regions. The latter have particularly long relaxation times or remain frozen, allowing for nonzero dc transport via higher-order processes. Based on the numerical evidence, we introduce a toy model that suggests that the nonresonant regions asymptotic dynamics are essential for the proper understanding of the disordered chains with many-body interactions.

Figure 1

(a) High-$T$ dc spin conductivities ${\tilde{\sigma }}_0$ within the random-field Heisenberg model vs disorder strength $W$ for different disorder realizations (30 samples for each $W$), as evaluated with MCLM on the $L=26$ system. The thick lines represent exponential fits to the median and average ${\tilde{\sigma }}_0$. (b) Cumulative distribution function (CDF) of ${\tilde{\sigma }}_0$ values for different $W$ (50 samples for each $W$). Curves represent log-normal distributions as a guide to the eye.

Figure 2

(a) Dynamical imbalance $I\left(\omega \right)$ within a single disorder sample at different strength $W$, calculated for $L=26$. Dashed line depicts the $1/\omega$ dependence. (b) Extracted dc spin conductivity ${\tilde{\sigma }}_{\pi }$ in comparison with the uniform ${\tilde{\sigma }}_0$ vs $W$ for three potential configurations.

Figure 3

(a) Local spin correlations $C_i\left(\omega \right)$ for all sites $i=1,L$ with potentials $h_i$ corresponding to $W=3$ (with the potential landscape also shown). (b), (c) Local correlations $C_i\left(\omega \right)$ for different $W=1-3$ for two characteristic sites $i=10,15$, representing resonant and localized islands, respectively. Dashed line in (b) represent $\propto 1/\omega$ dependence, while in (c) $\propto 1/\omega$ and $\propto 1/{\omega }^2$. (d) $\tilde{\sigma }\left(\omega \right)$ for flattened potentials with various thresholds $R=0-2.1$ for one configuration with $W=3$.

Figure 4

(a) Toy-model dc conductivity ${\tilde{\sigma }}_0$ for various $W$ and $L=25$. The inset show $L=20–16000$ dependence for $W=1.5$. (b),(c) Density ${\rho }_M$ and typical value of $J^{\mathrm{eff}}$ for nonresonant islands of length $M$ obtained for $L={10}^7$. (d) Distribution of $J^{\mathrm{eff}}$ in the toy model.