Inverted many-body mobility edge in a central qudit problem

We study the disordered Ising model with transverse and longitudinal fields coupled globally to a d-level system (qudit). In the center of the many-body spectrum, earlier work [PRL 122, 240402 (2019)] found a regime where many-body localization (MBL) survives global coupling to the cavity. In this work, we study the dependence of MBL on energy. Most strikingly, we discover an inverted mobility edge, where high energy states are localized while low energy states are delocalized. Our results are supported by shift-and-invert eigenstate targeting and Krylov time evolution up to L=13 and 18 respectively, with large central spin d=12 that is effectively infinite. We argue for critical energy of the MBL to thermal phase transition which scales as E_c∝L^½, consistent with finite size numerics. We also show evidence for a reentrant MBL phase at even lower energies despite the presence of strong effects of the central qudit in this regime.