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 a.c. conductivity vanishes near zero frequency as a power law; (iii) the distribution of resistances 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 Griffiths phase. We establish scaling relations between the associated exponents, using exact linear-response arguments as well as a phenomenological resistor-capacitor model.