Many-body localization proximity effect

We examine what happens when a strongly many-body localized system is coupled to a weak heat bath, with both system and bath containing similar numbers of degrees of freedom. Previous investigations of localized systems coupled to baths operated in regimes where the back action of the system on the bath is negligible and concluded that the bath generically thermalizes the system. In this paper we show that when the system is strongly localized and the bath is only weakly ergodic, the system can instead localize the bath. We demonstrate this both in the limit of weak coupling between system and bath and in the limit of strong coupling and also for two different types of ``weak'' bath---baths which are close to an atomic limit and baths which are close to a noninteracting limit. The existence of this ``many-body localization proximity effect'' indicates that many-body localization is more robust than previously appreciated, and it can not only survive coupling to a (weak) heat bath but can even destroy the bath.