  Time periodic driving serves not only as a convenient way to engineer
effective Hamiltonians, but also as a means to produce intrinsically dynamical
phases that do not exist in the static limit. A recent example of the latter
are 2D chiral Floquet (CF) phases exhibiting anomalous edge dynamics that pump
discrete packets of quantum information along one direction. In
non-fractionalized systems with only bosonic excitations, this pumping is
quantified by a dynamical topological index that is a rational number --
highlighting its difference from the integer valued invariant underlying
equilibrium chiral phases (e.g. quantum Hall systems). Here, we explore CF
phases in systems with emergent anyon excitations that have fractional
statistics (Abelian topological order). Despite the absence of mobile
non-Abelian particles in these systems, external driving can supply the energy
to pump otherwise immobile non-Abelian defects (sometimes called twist defects
or genons) around the boundary, thereby transporting an irrational fractional
number of quantum bits along the edge during each drive period. This enables
new CF phases with chiral indices that are square roots of rational numbers,
inspiring the label: "radical CF phases". We demonstrate an unexpected
bulk-boundary correspondence, in which the radical CF edge is tied to bulk
dynamics that exchange electric and magnetic anyon excitations during each
period. We construct solvable, stroboscopically driven versions of Kitaev's
honeycomb spin model that realize these radical CF phases, and discuss their
stability against heating in strongly disordered many-body localized settings
or in the limit of rapid driving as an exponentially long-lived pre-thermal
phenomena.
