Bulk-edge correspondence and new topological phases in periodically driven spin-orbit coupled materials in the low frequency limit

We study the topological phase transitions induced in spin-orbit coupled materials with buckling like silicene, germanene, stanene, etc, by circularly polarised light, beyond the high frequency regime, and unearth many new topological phases. These phases are characterised by the spin-resolved topological invariants, $C_0^{\uparrow }$, $C_0^{\downarrow }$, $C_{\pi }^{\uparrow }$ and $C_{\pi }^{\downarrow }$, which specify the spin-resolved edge states traversing the gaps at zero quasi-energy and the Floquet zone boundaries respectively. We show that for each phase boundary, and independently for each spin sector, the gap closure in the Brillouin zone occurs at a high symmetry point.