  Periodically driven quantum systems provide a novel and versatile platform
for realizing topological phenomena. Among these are analogs of topological
insulators and superconductors, attainable in static systems; however, some of
these phenomena are unique to the periodically driven case. Here, we show that
disordered, periodically driven systems admit an "anomalous" two dimensional
phase, whose quasi-energy spectrum consists of chiral edge modes that coexist
with a fully localized bulk - an impossibility for static Hamiltonians. This
unique situation serves as the basis for a new topologically-protected
non-equilibrium transport phenomenon: quantized non-adiabatic charge pumping.
We identify the bulk topological invariant that characterizes the new phase
(which we call the "anomalous Floquet Anderson Insulator", or AFAI). We provide
explicit models which constitute a proof of principle for the existence of the
new phase. Finally, we present evidence that the disorder-driven transition
from the AFAI to a trivial, fully localized phase is in the same universality
class as the quantum Hall plateau transition.
