Classification of topological phases in periodically driven interacting systems

We consider topological phases in periodically driven (Floquet) systems exhibiting many-body localization, protected by a symmetry G. We argue for a general correspondence between such phases and topological phases of undriven systems protected by symmetry Z where the additional Z accounts for the discrete time-translation symmetry. Thus, for example, the bosonic phases in d spatial dimensions without intrinsic topological order symmetry-protected topological (SPT) phases are classified by the cohomology group H^d+1Z, U (1). For unitary symmetries, we interpret the additional resulting Floquet phases in terms of the lower-dimensional SPT phases that are pumped to the boundary during one time step. These results also imply the existence of novel symmetry-enriched topological (SET) orders protected solely by the periodicity of the drive.