Quantum origin of anomalous Floquet phases in cavity-QED materials

Anomalous Floquet topological phases are a hallmark of periodically-driven systems lacking a static analog. Inspired by the long-standing tradition of Floquet Engineering using classical electromagnetic radiation, Quantum Floquet Engineering has emerged as a promising tool to tailor the properties of quantum materials through the interaction with quantum light. While the later recovers the physics of Floquet materials in its semi-classical limit, the mapping between these two widely different scenarios remains mysterious in many aspects. In this work, we discuss the emergence of quantum anomalous topological phases in cavity-QED materials, and link the topological phase transitions in the electron-photon spectrum with those in the 0- and π-gaps of Floquet quasienergies in the semi-classical limit. Our results establish the microscopic origin of an emergent discrete time-translation symmetry in the matter sector, and link the physics of the isolated c-QED materials with that of periodically driven ones. Finally, we discuss the bulk-edge correspondence in terms of hybrid light-matter topological invariants.