Topological edge and corner states in honeycomb-kagome photonic crystals.

We systematically study the first- and second-order band topologies, which are tied to the pseudospin and valley degree of freedoms (DOFs), in honeycomb-kagome photonic crystals (HKPCs). We first demonstrate the quantum spin Hall phase as the first-order pseudospin-induced topology in HKPCs by observing the partial pseudospin-momentum locked edge states. By employing the topological crystalline index, we also discover the multiple corner states emerging in the hexagon-shaped supercell as the manifestation of the second-order pseudospin-induced topology in HKPCs. Next, by gapping the Dirac points, a lower band gap associated with the valley DOF emerges, in which the valley-momentum locked edge states are observed as the first-order valley-induced topology. Such HKPCs without inversion symmetry are proved to be Wannier-type second-order topological insulators, which manifested with valley-selective corner states. Additionally, we also discuss the symmetry breaking effect on pseudospin-momentum locked edge states. Our work realizes both pseudospin-induced and valley-induced topologies in a higher-order manner and thus provides more flexibility in manipulating electromagnetic waves, which may find potential applications in topological routings.