  Nonequilibrium Floquet topological phases due to periodic driving are known
to exhibit rich and interesting features with no static analogs. Various known
topological invariants usually proposed to characterize static topological
systems often fail to fully characterize Floquet topological phases. This fact
has motivated extensive studies of Floquet topological phases to better
understand nonequilibrium topological phases and to explore their possible
applications. Here we present a theoretically simple Floquet topological
insulating system that may possess an arbitrary number of counter-propagating
chiral edge states. Further investigation into our system reveals another
related feature by tuning the same set of system parameters, namely, the
emergence of almost flat (dispersionless) edge modes. In particular, we employ
two-terminal conductance and dynamical winding numbers to characterize
counter-propagating chiral edge states. We further demonstrate the robustness
of such edge states against symmetry preserving disorder. Finally, we identify
an emergent chiral symmetry at certain sub-regimes of the Brillouin zone that
can explain the presence of almost flat edge modes. Our results have exposed
more interesting possibilities in Floquet topological matter.
