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Non-interacting fermions in one dimension can undergo alocalization-delocalization transition in the presence of a quasi-periodicpotential 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 inmodels with quasi-periodic potentials that possess single particle mobilityedges. Two such models were studied in PRL 115,230401(2015) but only one wasfound to exhibit an MBL transition in the presence of interactions while theother one did not. In this work we investigate the occurrence of MBL in thepresence of weak interactions in five different models with single particlemobility edges in one dimension with a view to obtaining a criterion for thesame. We find that not all such models undergo a thermal-MBL phase transitionin presence of weak interactions. We propose a criterion to determine whetherMBL is likely to occur in presence of interaction based only on the propertiesof the non-interacting models. The relevant quantity $\epsilon$ is a measure ofhow localized the localized states are relative to how delocalized thedelocalized states are in the non-interacting model. We also study variousother features of the non-interacting models such as the divergence of thelocalization length at the mobility edge and the presence or absence of`ergodicity' and localization in their many-body eigenstates. However, we findthat these features cannot be used to predict the occurrence of MBL upon theintroduction of weak interactions.

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