  Electromagnetic driving in a honeycomb lattice can induce gaps and
topological edge states with a structure of increasing complexity as the
frequency of the driving lowers. While the high frequency case is the most
simple to analyze we focus on the multiple photon processes allowed in the low
frequency regime to unveil the hierarchy of Floquet edge-states. In the case of
low intensities an analytical approach allows us to derive effective
Hamiltonians and address the topological character of each gap in a
constructive manner. At high intensities we obtain the net number of edge
states, given by the winding number, with a numerical calculation of the Chern
numbers of each Floquet band. Using these methods, we find a hierarchy that
resembles that of a Russian nesting doll. This hierarchy classifies the gaps
and the associated edge states in different orders according to the
electron-photon coupling strength. For large driving intensities, we rely on
the numerical calculation of the winding number, illustrated in a map of
topological phase transitions. The hierarchy unveiled with the low energy
effective Hamiltonians, alongside with the map of topological phase transitions
discloses the complexity of the Floquet band structure in the low frequency
regime. The proposed method for obtaining the effective Hamiltonian can be
easily adapted to other Dirac Hamiltonians of two dimensional materials and
even the surface of a 3D topological insulator.
