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Nonequilibrium Floquet topological phases due to periodic driving are knownto exhibit rich and interesting features with no static analogs. Various knowntopological invariants usually proposed to characterize static topologicalsystems often fail to fully characterize Floquet topological phases. This facthas motivated extensive studies of Floquet topological phases to betterunderstand nonequilibrium topological phases and to explore their possibleapplications. Here we present a theoretically simple Floquet topologicalinsulating system that may possess an arbitrary number of counter-propagatingchiral edge states. Further investigation into our system reveals anotherrelated feature by tuning the same set of system parameters, namely, theemergence of almost flat (dispersionless) edge modes. In particular, we employtwo-terminal conductance and dynamical winding numbers to characterizecounter-propagating chiral edge states. We further demonstrate the robustnessof such edge states against symmetry preserving disorder. Finally, we identifyan emergent chiral symmetry at certain sub-regimes of the Brillouin zone thatcan explain the presence of almost flat edge modes. Our results have exposedmore interesting possibilities in Floquet topological matter.

Counter-propagating edge states in Floquet topological insulating phases