Classification of Interacting Topological Floquet Phases in One Dimension

Periodic driving of a quantum system can enable new topological phases with no analog in static systems. In this paper we systematically classify one-dimensional topological and symmetry-protected topological (SPT) phases in interacting fermionic and bosonic quantum systems subject to periodic driving, which we dub Floquet SPTs (FSPTs). For physical realizations of interacting FSPTs, many-body localization by disorder is a crucial ingredient, required to obtain a stable phase that does not catastrophically heat to infinite temperature. We demonstrate that bosonic and fermionic FSPTs phases are classified by the same criteria as equilibrium phases, but with an enlarged symmetry group $\tilde G$, that now includes discrete time translation symmetry associated with the Floquet evolution. In particular, 1D bosonic FSPTs are classified by projective representations of the enlarged symmetry group $H^2({\tilde G},U(1))$. We construct explicit lattice models for a variety of systems, and then formalize the classification to demonstrate the completeness of this construction. We also derive general constraints on localization and symmetry based on the representation theory of the symmetry group, and show that symmetry-preserving localized phases are possible only for Abelian symmetry groups. In particular, this rules out the possibility of many-body localized SPTs with continuous spin symmetry.