Type-\textbf{III} corner state in second-order topological insulator by distinctly hybridized photonic Wannier functions

In the presence of crystalline symmetries, second-order topological insulators can be featured by the polarization which is believed identical to the Wannier center. In this Letter, we show that second-order topological insulators are present in the full process of topological phase transition between a pair of degenerate photonic bands resulting from the non-symmorphic glide symmetry in all-dielectric photonic crystals. Due to the coupling (hybridization) of the Wannier functions, the center of the maximally localized Wannier function always locates at the origin, not equal to the addition of constituent polarization. In this case, the corner states are ascribed to the hybridized Wannier functions obstructed by the boundary. In combination with the local density of states, the second-order topology is clearly confirmed and characterized. That ensures corner states which are fundamentally different to either the type-I corner state localized in particular sublattice, or the type-II corner state originated from interacting topological edge states. This type of corner states is regarded as type-III here and provides an alternative method to explore higher-order topological physics, which may lead to interesting applications in integrated and quantum photonics.