Disparities of topological states at multiple interconvertible domain walls

With the emergence of rich topological phases in artificial photonic crystals (PCs), numerous peculiar physical phenomena occur within domain walls (DWs) constructed by PCs with different topological phases. These phenomena include robust topological edge states (TESs), strongly localized topological corner states (TCSs), the quantum spin Hall effect, the quantum valley Hall effect, filling anomalies, and fractional charges at corners and edges. In this paper, we propose triangular lattice PCs composed of six triangle-shaped silicon rods to form two pairs of interconvertible DWs under two types of perturbations. We show that topological phase transitions occur with perturbations affecting 3 out of 6 rods. The first perturbation opens a band gap at the \ensuremath{\Gamma} point, supporting a pair of pseudospin helical edge states. The second perturbation opens another band gap at the K point, supporting valley edge states while preserving the initial topological band gap. We find that all designed topological PCs exhibit Wannier centers partially deviating from the center, causing different filling anomalies, thereby marking PCs with different topological indices corresponding to distinct higher-order topological phases. The results from mode charge distribution, calculated by integrating the local density of states (LDOS), are consistent with the Wannier center position analysis. By analyzing the LDOS under different boundary conditions at the corners and edges, we uncover significant disparities in the distribution of TES and TCS. Intriguingly, under certain conditions, the corner states can hybridize with edge states or even with bulk states. The rich topological physics presented not only provides insights into topological phases but also opens avenues for engineering topological states with potential applications.