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 $\mathbb{Z}⋊G$ where the additional $\mathbb{Z}$ 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 $H^{d+1}[\mathbb{Z}⋊G,\mathrm{U}\left(1\right)]$. 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.