Criterion for the occurrence of many body localization in the presence of a single particle mobility edge

TL;DR

This paper investigates the conditions under which many-body localization occurs in one-dimensional models with single particle mobility edges, proposing a new criterion based on properties of the non-interacting models.

Contribution

It introduces a criterion to predict MBL in models with mobility edges using a measure of localization of states, based on non-interacting model properties.

Findings

Not all models with mobility edges exhibit MBL with weak interactions.

The proposed criterion effectively predicts MBL occurrence.

Features like localization length divergence do not reliably predict MBL.

Abstract

Non-interacting fermions in one dimension can undergo a localization-delocalization transition in the presence of a quasi-periodic potential as a function of that potential. In the presence of interactions, this transition transforms into a Many-Body Localization (MBL) transition. Recent studies have suggested that this type of transition can also occur in models with quasi-periodic potentials that possess single particle mobility edges. Two such models were studied in PRL 115,230401(2015) but only one was found to exhibit an MBL transition in the presence of interactions while the other one did not. In this work we investigate the occurrence of MBL in the presence of weak interactions in five different models with single particle mobility edges in one dimension with a view to obtaining a criterion for the same. We find that not all such models undergo a thermal-MBL phase transition in presence of weak interactions. We propose a criterion to determine whether MBL is likely to occur in presence of interaction based only on the properties of the non-interacting models. The relevant quantity $\epsilon$ is a measure of how localized the localized states are relative to how delocalized the delocalized states are in the non-interacting model. We also study various other features of the non-interacting models such as the divergence of the localization length at the mobility edge and the presence or absence of `ergodicity' and localization in their many-body eigenstates. However, we find that these features cannot be used to predict the occurrence of MBL upon the introduction of weak interactions.