Anomalous Diffusion and Griffiths Effects Near the Many-Body Localization Transition

We explore the high-temperature dynamics of the disordered, one-dimensional $XXZ$ model near the many-body localization (MBL) transition, focusing on the delocalized (i.e., “metallic”) phase. In the vicinity of the transition, we find that this phase has the following properties: (i) local magnetization fluctuations relax subdiffusively; (ii) the ac conductivity vanishes near zero frequency as a power law; and (iii) the distribution of resistivities becomes increasingly broad at low frequencies, approaching a power law in the zero-frequency limit. We argue that these effects can be understood in a unified way if the metallic phase near the MBL transition is a quantum Griffiths phase. We establish scaling relations between the associated exponents, assuming a scaling form of the spin-diffusion propagator. A phenomenological classical resistor-capacitor model captures all the essential features.

Figure 1

The phase diagram of the random field, XXZ model for $J_z=1$. $h_{\max }$ characterizes disorder strength. As the disorder is increased, the system transitions smoothly into a subdiffusive Griffiths-like phase with an anomalous diffusion exponent $\beta$ and exponent $\alpha$ characterizing low-frequency optical conductivity, which satisfy the scaling relation $\alpha +2\beta =1$. The MBL transition is predicted to occur where $\sigma (\omega )\sim \omega$. Within precision, it coincides with the transition point determined from the level statistics parameter $r$ (see main text).

Figure 2

Behavior of (a) optical conductivity $\sigma (\omega )$, (b) return probability $C_{zz}(t)$, and (c) width $\mathrm{\Delta }\rho (\omega )$ of the distribution of resistivities as a function of frequency (magenta). All plots are for $J_z=0.8$, and disorder strengths as indicated in the legend. The fits (green) are power laws of the form $C_{zz}\sim 1/t^{\beta }$, $\sigma (\omega )\sim {\omega }^{\alpha }$, and $\mathrm{\Delta }\rho (\omega )\sim 1/{\omega }^{{\alpha }^{\prime }}$. The inset in (c) shows the relative power $\gamma ={\alpha }^{\prime }-\alpha$ governing the scaling of the ratio $\mathrm{\Delta }\rho (\omega )/⟨\rho (\omega )⟩$ of the width and the mean of the resistivity distribution.

Figure 3

(a) The RC model. (b) Width of the finite frequency resistivity distribution $\mathrm{\Delta }\rho (\omega )$ (magenta) and its asymptotic form $1/{\omega }^{{\alpha }^{\prime }}$ (green) plotted as a function of frequency. The inset shows $\gamma ={\alpha }^{\prime }-\alpha \gtrsim 0$, where $\alpha$ is the exponent of the average resistivity $\overline{\rho }(\omega )\sim 1/{\omega }^{\alpha }$. (c) $(\alpha +1)\tau /2$ as a function of $\tau$.