Topological phases of matter exhibit remarkable electronic properties that offer unique possibilities for applications. A prominent example is the robust quantization of the Hall conductivity in quantum Hall insulators. A widespread technique for generating topological band structures in synthetic systems, such as ultracold atoms in optical lattices, is Floquet engineering . This method relies on the periodic modulation of the system’s parameters to emulate the properties of a non-trivial static system and facilitated the realization of two paradigmatic topological lattice models: the Hofstadter and the Haldane model. Moreover, it inspired ideas for implementing complete lattice gauge theories .
The rich properties of Floquet systems, however, transcend those of their static counterparts . The associated quasienergy spectrum can exhibit a non-trivial winding number, which leads to the appearance of anomalous chiral edge modes even in situations where the bulk bands have zero Chern numbers, hence, altering the well-known bulk-edge correspondence. A full classification of Floquet phases requires a new set of topological invariants. We have studied the rich Floquet phase diagram of a periodically-modulated honeycomb lattice using bosonic atoms. The novel properties of anomalous Floquet phases mentioned above open the door to exciting new non-equilibrium phases without any static analogue .
 A. Eckardt, Phys. Mod. Phys. 89, 311 (2017)
 C. Schweizer et al., Nat. Phys. 15, 1168-1173 (2019)
 T. Kitagawa et al., Phys. Rev. B 82, 235114 (2010)
 K. Wintersperger et al., Nature Physics (2020)