Many body localization in the presence of a central qudit

We consider a many-body localized system coupled globally to a central d-level system. Under an appropriate scaling of d and system size, we find evidence that the localized phase survives. We argue for two possible thermalizing phases, depending on whether the qudit becomes fully ergodic. This system provides one of the first examples of many-body localization in the presence of long-range (non-confining) interactions. We discuss possible anomalous behavior in the system, such as an inverted mobility edge - localized at high temperature, but ergodic at low temperature.