Abstract
Anomalous Floquet topological phases are a hallmark, without a static analog, of periodically driven systems. Recently, Quantum Floquet Engineering has emerged as an interesting approach to cavity-QED materials, which recovers the physics of Floquet engineering in its semi-classical limit. However, the mapping between these two widely different scenarios remains mysterious in many aspects. We discuss the emergence of anomalous topological phases in cavity-QED materials, and link topological phase transitions in the many-body spectrum with those in the 0- and π-gaps of Floquet quasienergies. Our results allow to establish the microscopic origin of an emergent discrete time-translation symmetry in the matter sector, and link the physics of isolated many-body systems with that of periodically driven ones. Finally, the relation between many-body and Floquet topological invariants is discussed, as well as the bulk-edge correspondence.