Quadrupole photonic topological corner states in generalized non-square lattices with translation symmetry

Quadrupole topological insulators have recently attracted great attention in the field of topological physics, while they are limited to square and hexagonal lattices. In this work, we theoretically show that nontrivial quadrupole topology can be obtained in generalized non-square lattice photonic crystals with translation symmetry, which are composed of parallelogram-shaped unit cells. The translation symmetry is described by the fractional linear combination of primary lattice vectors, leading to the quantization of fractional quadrupole moment in conjunction with an additional symmorphic symmetry. For parallelogramatic lattice with inversion symmetry, in particular, the quantization of the quadrupole moment is independent of the choice of primary lattice vectors, enabling cavity structures with arbitrary angles. For the change of structural parameters, quadrupole bandgaps undergo second-order topological phase transitions, accompanying with double band inversions. Nontrivial quadrupole phases are manifested by the appearance of disorder-immune in-gap corner states localized at the topological interfaces. Furthermore, the proposed parallelogramatic lattice photonic crystal has multiple quadrupole bandgaps for proper structural parameters, exhibiting multiband second-order topological corner states. The presented results will further extend the class of quadrupole topological photonic crystals and pave a broad way towards their practical applications due to improved design flexibility.