  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.

Strongly Correlated Electrons;Disordered Systems and Neural Networks;Mesoscale and Nanoscale Physics