Pseudo-spin-valley coupled topological states protected by different symmetries in photonic crystals

The quantum spin Hall effect protected by C₆ symmetry [realized in the domain wall (DW) formed by a trivial-photonic crystal (TPC) and a nontrivial-PC (NPC)] and the quantum valley Hall effect protected by C₃ symmetry [realized in the DW formed by two valley PCs (VPCs)] have been widely researched due to their excellent topological properties. The topological edge states (TESs) and topological corner states (TCSs) at DWs between different symmetric structures remain to be explored, which is essential for connecting waveguides with different symmetries to construct optical communication devices. In this Letter, there is (are) one TES (two TESs) for the DW1 and DW3 (DW2 and DW4) between the TPC (NPC) and two VPCs. Through simulation calculations of the Wilson-loop of the TPC and NPC and the Berry curvature distribution of VPCs, the corresponding relationship between the topological invariant and the number of TESs is obtained. Based on the TPC, NPC, and two VPCs, the waveguides are constructed to verify the realization of TESs. The parity of the gapped TESs is analyzed, and its relationship with the TCSs is obtained. Moreover, box-shaped structures are constructed to verify the appearance of TCSs. These results have a guiding significance for the research of the interaction between topological states protected by different symmetries.