Quantum origin of anomalous Floquet phases in cavity-QED materials

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