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.

中文翻译：

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具有平移对称性的广义非方晶格中的四极光子拓扑角态

四极拓扑绝缘体最近在拓扑物理领域引起了极大的关注，但它们仅限于正方形和六角形晶格。在这项工作中，我们从理论上证明，可以在由平行四边形晶胞组成的具有平移对称性的广义非方晶格光子晶体中获得非平凡的四极拓扑。平移对称性由主晶格向量的分数线性组合来描述，导致分数四极矩的量化以及附加的同态对称性。特别是对于具有反演对称性的平行四边形晶格，四极矩的量子化与主晶格矢量的选择无关，从而实现具有任意角度的空腔结构。随着结构参数的变化，四极带隙发生二阶拓扑相变，并伴随双能带反转。非平凡的四极相表现为拓扑界面处局部无序免疫带隙角态的出现。此外，所提出的平行四边形晶格光子晶体具有多个四极带隙，具有适当的结构参数，表现出多带二阶拓扑角态。所提出的结果将进一步扩展四极拓扑光子晶体的类别，并由于提高的设计灵活性为其实际应用铺平了广阔的道路。