Evolution of the edge states and corner states in a multilayer honeycomb valley-Hall topological metamaterial

The valley-Hall effect provides topological protection to a broad class of defects in valley-Hall photonic topological metamaterials. Unveiling precisely how such protection is achieved and its implications in practical implementations is paramount to move from fundamental science to applications. To this end, we investigate a honeycomb valley-Hall topological metamaterial and monitor the evolution of the topological valley-Hall edge states and higher-order corner states under different perturbation δR. The evolutions of the edge states of the armchair and zigzag interfaces are demonstrated, respectively. By adjusting the geometric parameters and introducing disturbances to break the inversion symmetry, we achieve the edge states with different modes including the conventional crossed edge state and the specific gapped edge state. It is found that the edge states of topological valley kinking will gradually separate with the increase of δR, and finally a complete gap between the edge states appears. The gap has rarely been reported previously in topological materials fabricated by printed circuit board technology. In addition, the higher-order topological corner states can also be observed in the proposed topological metamaterial. The higher-order topological phase is theoretically characterized by nontrivial bulk polarization and the Wannier centers. Our results show that the corner state localization becomes stronger with the increase of δR. It is expected that our results will provide a platform for the realization of optical topological insulators.