Type-III corner states in second-order topological insulators enabled by distinct hybridization of photonic Wannier functions

In periodic crystalline systems, topological insulators are uniquely characterized by modern polarization concepts, analogous to Wannier centers. Here, we show that second-order topological insulators (SOTIs) arise throughout the full spectrum of a phase transition between degenerate photonic bands, due to the combination of nonsymmorphic glide symmetry and time-reversal symmetry in all-dielectric photonic crystals. We find that the hybridization of Wannier functions causes the center of the maximally localized Wannier function to lie at the unit cell origin rather than a site with nonzero net polarization. This mechanism leads to corner states originating from obstructed Wannier functions. By examining the local density of states, we confirm and characterize the second-order topology, revealing an additional type of corner state - type-III - which differs from the conventional type-I and type-II. These findings broaden the exploration of higher-order topological physics and hold promise for integrated and quantum photonics.