A second-order topological insulator in $d$ dimensions is an insulator which has no $d-1$ dimensional topological boundary states but has $d-2$ dimensional topological boundary states. It is an extended notion of the conventional topological insulator. Higher-order topological insulators have been investigated in square and cubic lattices. In this Letter, we generalize them to breathing kagome and pyrochlore lattices. First, we construct a second-order topological insulator on the breathing Kagome lattice. Three topological boundary states emerge at the corner of the triangle, realizing a $1/3$ fractional charge at each corner. Second, we construct a third-order topological insulator on the breathing pyrochlore lattice. Four topological boundary states emerge at the corners of the tetrahedron with a $1/4$ fractional charge at each corner. These higher-order topological insulators are characterized by the quantized polarization, which constitutes the bulk topological index. Finally, we study a second-order topological semimetal by stacking the breathing kagome lattice.

中文翻译：

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呼吸Kagome和烧绿石格子上的高阶拓扑绝缘子和半金属

中的二阶拓扑绝缘子 $d$ 尺寸是没有绝缘体 $d-\mathrm{1个}$ 维拓扑边界状态，但具有 $d-\mathrm{2个}$维拓扑边界状态。它是常规拓扑绝缘体的扩展概念。已经研究了正方形和立方晶格中的高阶拓扑绝缘子。在这封信中，我们将它们概括为呼吸的kagome和烧绿石晶格。首先，我们在呼吸的Kagome格上构造了二阶拓扑绝缘体。在三角形的拐角处出现了三个拓扑边界状态，从而实现了$\mathrm{1个}/3$每个角落的分数电荷。第二，我们在呼吸的烧绿石晶格上构造了一个三阶拓扑绝缘体。在四面体的角出现四个拓扑边界状态，其中$\mathrm{1个}/4$每个角落的分数电荷。这些高阶拓扑绝缘子的特征是量化极化，该极化构成整体拓扑指数。最后，我们通过叠加呼吸的孔形格来研究二阶拓扑半金属。