Phase structure of one-dimensional interacting Floquet systems. I. Abelian symmetry-protected topological phases

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 Cl-G x A(G), where Cl-G is the undriven interacting classification, and AG is a set of (twisted) one-dimensional representations corresponding to symmetry group G. We will address symmetry-broken phases in a subsequent paper C.W. von Keyserlingk and S.L. Sondhi, following paper, Phys. Rev. B 93, 245146 (2016).