Periodic table for Floquet topological insulators

Dynamical phaseswith novel topological properties are known to arise in driven systems of free fermions. In this paper, we obtain a 'periodic table' to describe the phases of such time-dependent systems, generalizing the periodic table for static topological insulators. Using K theory, we systematically classify Floquet topological insulators from the ten Altland-Zirnbauer symmetry classes across all dimensions. We find that the static classification scheme described by a group g becomes g(xn) in the time-dependent case, where n is the number of physically important gaps in the quasienergy spectrum (including any gaps at quasienergy pi). The factors of G may be interpreted as arising from the bipartite decomposition of the unitary time-evolution operator. Topologically protected edge modes may arise at the boundary between two Floquet systems, and we provide a mapping between the number of such edge modes and the topological invariant of the bulk.