Inverted many-body mobility edge in a central qudit problem

Abstract

Many interesting experimental systems, such as cavity QED or central spin models, involve global coupling to a single harmonic mode. Out of equilibrium, it remains unclear under what conditions localized phases survive such global coupling. We study energy-dependent localization in the disordered Ising model with transverse and longitudinal fields coupled globally to a $d$-level system (qudit). Strikingly, we discover evidence for an inverted mobility edge where high-energy states are localized whereas low-energy states are delocalized. This prediction is supported by shift-and-invert eigenstate targeting and Krylov time evolution up to $L=13$ and 18, respectively. We argue for a critical energy of the localization phase transition which scales as ${E}_{c}\ensuremath{\propto}{L}^{1/2}$, consistent with finite-size numerics. We also show evidence for a reentrant many-body localization phase at even lower energies despite the presence of strong effects of the central mode in this regime. Similar results should occur in the central spin-$S$ problem at large $S$ and in certain models of cavity QED.