Topological invariants of Floquet topological phases under periodical driving

Owing to the replication of Floquet bands and the presence of additional gaps in the quasienergy dimension, the topological phases in a periodically driven system cannot be fully characterized by the conventional topological invariants used in static systems. In particular, an anomalous strong topological phase can be a host in driven systems, featured by nontrivial counterpropagating edge modes which cannot be characterized by the bulk band structure. In this paper, we propose a scheme to obtain a complete characterization of Floquet topological phases using only information about bulk dispersions under the condition that both the location and chirality of Floquet band-touching points are not changed by the periodical driving. A set of topological invariants associated with the band-touching points are formulated to establish a one-to-one correspondence to the number of edge modes. Finally, we discuss the experimental realization and detection scheme using cold atomic gases.