He Li, Addison Zachariah, Mele Eugene J, Zhen Bo
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, 19104, USA.
Nat Commun. 2020 Jun 19;11(1):3119. doi: 10.1038/s41467-020-16916-z.
Quadrupole topological phases, exhibiting protected boundary states that are themselves topological insulators of lower dimensions, have recently been of great interest. Extensions of these ideas from current tight binding models to continuum theories for realistic materials require the identification of quantized invariants describing the bulk quadrupole order. Here we identify the analog of quadrupole order in Maxwell's equations for a gyromagnetic photonic crystal (PhC) through a double-band-inversion process. The quadrupole moment is quantized by the simultaneous presence of crystalline symmetry and broken time-reversal symmetry, which is confirmed using three independent methods: analysis of symmetry eigenvalues, numerical calculations of the nested Wannier bands and the expectation value of the quadrupole operator. Furthermore, we reveal the boundary manifestations of quadrupole phases as quantized edge polarizations and fractional corner charges. The latter are the consequence of a filling anomaly of energy bands as first predicted in electronic systems.
四极拓扑相因其具有受保护的边界态（这些边界态本身是低维拓扑绝缘体）而近来备受关注。将这些概念从当前的紧束缚模型扩展到实际材料的连续体理论，需要确定描述体四极序的量子化不变量。在此，我们通过双能带反转过程，在旋磁光子晶体（PhC）的麦克斯韦方程组中确定了四极序的类似物。四极矩通过晶体对称性和时间反演对称性破缺的同时存在而被量子化，这一点通过三种独立方法得以证实：对称本征值分析、嵌套万尼尔能带的数值计算以及四极算符的期望值。此外，我们揭示了四极相的边界表现为量子化的边缘极化和分数角电荷。后者是能带填充反常的结果，这一现象最早在电子系统中被预测。
