Phase Structure of 1d Interacting Floquet Systems I: Abelian SPTs

Recent work suggests that a sharp definition of `phase of matter' can be given for some quantum systems out of equilibrium---first for many-body localized systems with time independent Hamiltonians and more recently for periodically driven or Floquet localized systems. In this work we propose a classification of the finite abelian symmetry protected phases of interacting Floquet localized systems in one dimension. We find that the different Floquet phases correspond to elements of $\text{Cl}\times\mathcal{A}_G$, where $\text{Cl}$ is the undriven interacting classification, and $\mathcal{A}_G$ is a set of (twisted) 1d representations of $G$. We will address symmetry broken phases in a subsequent paper.