Anomalous localization in spin chains coupled to a non-local degree of freedom

It has recently been predicted that many-body localization survives the presence of coupling to a non-local degree of freedom, such as a cavity mode [PRL 122, 240402 (2019)]. Such a cavity-QED system can host anomalous phases unique to non-equilibrium systems. Here we present recent results on anomalous localization in such setups. First, we show that for the right choice of non-local degree of freedom, an inverted mobility edge occurs, meaning that infinite temperature states are localized while low energy states are delocalized. Second, we show a similar model can be used for realizing time crystals in cavity-QED systems and in the absence of drive, i.e., a time-crystalline phase in a static Hamiltonian. Finally, we study the stability of localization in the presence of non-zero but small photon loss. All the models presented are realizable on experimental AMO platforms.