Second-order Floquet topological phases and corner states based on spatial symmetries in honeycomb lattices in the presence of spin-orbit coupling

We investigate the second-order Floquet topological (SOFT) phase transitions, from the perspective of spatial symmetries. In this respect, we consider a generic honeycomb lattice Floquet topological insulator (FTI), realized by circularly polarized light irradiation, in the presence of spin-orbit coupling (Kane-Mele model). We find that our studied FTI presents chiral symmetry on a preferential direction in Fourier space, the same property that protects the topological phases in the Su-Schrieffer-Heeger (SSH) model. Thus we were allowed to characterize the SOFT phases in terms of mirror-graded winding numbers (Zak phase). Moreover, our model exhibits ${C}_{2}$ and ${C}_{3}$ symmetry in Fourier space, a property which lead us to investigate two finite structures having the aforementioned symmetries, namely, rhombic and triangular shapes. Indeed, we find that both of them undergo SOFT phase transitions, characterized by the appearance of 0D corner states symmetrically localized over the whole sheet. Finally, we investigate the ${C}_{2}$ and ${C}_{3}$ symmetry breaking. Interestingly, we reveal that the corner states are not destroyed, but localize at preferential corners instead, giving rise to a corner polarization.