Absence of Topological Protection of the Interface States in Z_{2} Photonic Crystals.

Inspired from electronic systems, topological photonics aims to engineer new optical devices with robust properties. In many cases, the ideas from topological phases protected by internal symmetries in fermionic systems are extended to those protected by crystalline symmetries. One such popular photonic crystal model was proposed by Wu and Hu in 2015 for realizing a bosonic Z_{2} topological crystalline insulator with robust topological edge states, which led to intense theoretical and experimental studies. However, a rigorous relationship between the bulk topology and edge properties for this model, which is central to evaluating its advantage over traditional photonic designs, has never been established. In this Letter, we revisit the expanded and shrunken honeycomb lattice structures proposed by Wu and Hu and show that they are topologically trivial in the sense that symmetric, localized Wannier functions can be constructed. We show that the Z and Z_{2} type classifications of the Wu-Hu model are equivalent to the C_{2}T protected Euler class and the second Stiefel-Whitney class, respectively, with the latter characterizing the full valence bands of the Wu-Hu model, indicating only a higher order topological insulator. Additionally, we show that the Wu-Hu interface states can be gapped by a uniform topology preserving C_{6} and T symmetric perturbation, which demonstrates the trivial nature of the interface. Our result reveals that topology is not a necessary condition for the reported helical edge states in many photonics systems and opens new possibilities for interface engineering that may not be constrained by topological considerations.