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

Ionut-Dragos Potirniche, Sumilan Banerjee, Ehud Altman

Published 2018-05-03Version 1

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 coupled to a large number of Anderson fermions, suggesting a possible way in which the avalanche can fail. While this fact per se does not preclude the thermalization avalanche, it violates a central assumption in the arguments of De Roeck et al. whereby the spectral function does not exhibit qualitative changes. We verify this conclusion in a solvable toy model in which describing the ergodic region using a Sachdev-Ye-Kitaev model allows us to obtain the exact solution of the fully coupled system in a suitable large-$N$ limit.