  In the presence of strong disorder and weak interactions, closed quantum
systems can enter a many-body localized phase where the system does not
conduct, does not equilibrate even for arbitrarily long times, and robustly
violates quantum statistical mechanics. The starting point for such a many-body
localized phase is usually taken to be an Anderson insulator where, in the
limit of vanishing interactions, all degrees of freedom of the system are
localized. Here, we instead consider a model where in the non-interacting
limit, some degrees of freedom are localized while others remain delocalized.
Such a system can be viewed as a model for a many-body localized system brought
into contact with a small bath of a comparable number of degrees of freedom. We
numerically and analytically study the effect of interactions on this system
and find that generically, the entire system delocalizes. However, we find
certain parameter regimes where results are consistent with localization of the
entire system, an effect recently termed many-body proximity effect.
