Abelian Floquet symmetry-protected topological phases in one dimension

Time-dependent systems have recently been shown to support novel types of topological order that cannot be realised in static systems. In this paper, we consider a range of time-dependent, interacting systems in one dimension that are protected by an Abelian symmetry group. We classify the distinct topological phases that can exist in this setting and find that they may be described by a bulk invariant associated with the unitary evolution of the closed system. In the open system, nontrivial phases correspond to the appearance of edge modes in the many-body quasienergy spectrum, which relate to the bulk invariant through a form of bulk-edge correspondence. We introduce simple models which realise nontrivial dynamical phases in a number of cases, and outline a loop construction that can be used to generate such phases more generally.