Anomalous diffusion and griffiths effects near the many-body localization transition.

Kartiek Agarwal,Sarang Gopalakrishnan,Michael Knap,Markus Müller,Eugene Demler

doi:10.1103/physrevlett.114.160401

Abstract

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.

Highlights

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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 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 many-body localization (MBL) transition is a quantum Griffiths phase

Summary

Conclusions

In this work we have numerically established the following facts about the delocalized phase near the MBL transition in the disordered XXZ chain. As the localized phase is approached, β → 0, and α → 1 while as the diffusive phase is approached β → 1/2 and α → 0. The distribution of resistivities becomes scale-free and presumably powerlaw in the d.c. limit. These general observations allow us to identify the phase as a Griffiths phase. We derived the central scaling relation α + 2β = 1 postulating a scaling form of the spin-diffusion propagator. Two intriguing aspects of the Griffiths phase that remain to be addressed in future work are: (i) whether it is ergodic; and (ii) whether any such phase exists in more than one dimension, where single local bottlenecks cannot block global transport

Finite Size scaling

Numerical Evaluation of Conductivity

Infinite-time Decay

Width of resistivity distribution in the RC Model

RC model simulations