Floquet topological insulators with spin-orbit coupling

In a milestone paper F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988), Haldane elaborated a model of graphene within the time-reversal symmetry breaking is achieved by next-nearest-neighbors imaginary counter-rotating hopping, hence conferring topological properties. In recent years, the time-reversal symmetry turned out to be broken also by light irradiation in so-called Floquet topological insulators (FTIs). On the other hand, Kane and Mele introduced a spin-orbit coupling (SOC) model C. L. Kane et al., Phys. Rev. Lett. 95, 226801 (2005) inspired by the Haldane's mechanism. In this paper, we present the topological properties of a FTI possessing SOC, using graphene as the playground. It was found that the interplay between sublattice subspace and the spin one triggers interesting topological phase transitions. Basically, in a FTI with SOC, two topological phases may be excited: charge quantum Hall effect (CQHE) and, respectively, spin quantum Hall effect (SQHE) phases. Also, it was demonstrated that the CQHE and SQHE coexistence is forbidden by the topology of the system. As well, it was identified a special driving regime of spin filter (SF), in which only one spin state is topological and, consequently, will be filtered in quantum transport.