# Many-Body Localization: Stability and Instability

## TL;DR

This paper investigates the stability of many-body localization in disordered spin chains, constructing local integrals of motion and discussing the effects of rare regions and higher dimensions on ergodicity and slow dynamics.

## Contribution

It provides a non-perturbative construction of local integrals of motion for weakly interacting one-dimensional spin chains and discusses the implications for higher-dimensional systems.

## Key findings

- Local integrals of motion can be constructed in 1D weakly interacting systems.
- Rare Griffiths regions can destabilize localization.
- In higher dimensions, ergodicity is likely restored but with very slow dynamics.

## Abstract

Rare regions with weak disorder (Griffiths regions) have the potential to spoil localization. We describe a non-perturbative construction of local integrals of motion (LIOMs) for a weakly interacting spin chain in one dimension, under a physically reasonable assumption on the statistics of eigenvalues. We discuss ideas about the situation in higher dimensions, where one can no longer ensure that interactions involving the Griffiths regions are much smaller than the typical energy-level spacing for such regions. We argue that ergodicity is restored in dimension d > 1, although equilibration should be extremely slow, similar to the dynamics of glasses.