On the stability of many-body localization in $d>1$

TL;DR

This paper investigates the stability of many-body localization in higher dimensions by modeling ergodic bubbles coupled to localized fermions, finding that large enough ergodic regions can prevent thermalization avalanches.

Contribution

The study provides a detailed analysis of the thermalization avalanche mechanism in higher dimensions, combining random matrix models and Hubbard models to assess stability of MBL.

Findings

Large ergodic bubbles can suppress avalanches in higher dimensions.

Back-action effects renormalize the critical size of ergodic regions.

Spectral functions show significant changes when coupling to localized states.

Abstract

Recent work by De Roeck et al. [Phys. Rev. B 95, 155129 (2017)] has argued that many-body localization (MBL) is unstable in two and higher dimensions due to a thermalization avalanche triggered by rare regions of weak disorder. To examine these arguments, we construct several models of a finite ergodic bubble coupled to an Anderson insulator of non-interacting fermions. We first describe the ergodic region using a GOE random matrix and perform an exact diagonalization study of small systems. The results are in excellent agreement with a refined theory of the thermalization avalanche that includes transient finite-size effects, lending strong support to the avalanche scenario. We then explore the limit of large system sizes by modeling the ergodic region via a Hubbard model with all-to-all random hopping: the combined system, consisting of the bubble and the insulator, can be reduced to an effective Anderson impurity problem. We find that the spectral function of a local operator in the ergodic region changes dramatically when coupling to a large number of localized fermionic states---this occurs even when the localized sites are weakly coupled to the bubble. In principle, for a given size of the ergodic region, this may arrest the avalanche. However, this back-action effect is suppressed and the avalanche can be recovered if the ergodic bubble is large enough. Thus, the main effect of the back-action is to renormalize the critical bubble size.