Higher-Order Topological Insulators and Semimetals on the Breathing Kagome and Pyrochlore Lattices

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.

Figure 1

Illustration of triangles made of the breathing kagome lattice with (a) $t_a=0$, (b) $0<t_a<t_b$, and (c) $t_b=0$. A triangle contains many small triangles. There are three isolated atoms at the corner of the triangle for $t_a=0$, while there are no isolated atoms for $t_b=0$. The size of the triangle is $L=5$. Illustration of tetrahedrons made of the breathing pyrochlore lattice with (d) $t_a=0$, (e) $t_a=t_b$, and (f) $t_b=0$. A tetrahedron contains many small tetrahedrons. There are four isolated atoms at the corner of the tetrahedron for $t_a=0$, while there are none for $t_b=0$. The size of the tetrahedron is $L=5$. See Fig. 5 for the unit cell of the pyrochlore lattice ($t_a=t_b$).

Figure 2

Band structure of breathing kagome lattices with (a1) $t_a/t_b=-1.5$, (b1) $t_a/t_b=-1$, (c1) $t_a/t_b=-0.25$, (d1) $t_a/t_b=0.25$, (e1) $t_a/t_b=1$, and (f1) $t_a/t_b=1.5$. The horizontal axes are $k_x$ and $k_y$. (a2)–(f2) The corresponding band structure of nanoribbons, where the horizontal axis is $k$. In each figure, two red curves represent boundary modes while a cyan line represents a perfect flat band belonging to the bulk. Although the nanoribbon spectrum indicates that the bulk must be a trivial insulator for (c1) and (d1), it is actually a HOTI.

Figure 3

(a) Energy spectrum of the triangle made of the breathing kagome lattice with $L=20$. The horizontal axis is $t_a/t_b$. There emerge zero-energy states (marked in red) for $-1<t_a/t_b<1/2$. They are topological boundary states. (b) The square root of the local density of states $\sqrt{{\rho }_i}$ for the triangle with $L=5$ and $t_a/t_b=1/2$. The amplitude is represented by the radius of the spheres. The local density of states becomes arbitrarily small except for the three corners for $L\gg 1$.

Figure 4

The Wannier centers in the topological and trivial phases.

Figure 5

(a) Brillouin zone of the pyrochlore lattice. Letters $\mathrm{\Gamma }$, $X$, $W$, $U$, $L$, and $K$ represent high symmetry points. (b) Unit cell of the pyrochlore lattice ($t_a=t_b$).

Figure 6

Band structure of breathing pyrochlore lattices along the $\mathrm{\Gamma }\text{−}X\text{−}W\text{−}U\text{−}L\text{−}K\text{−}\mathrm{\Gamma }$ line. (a1) $t_a/t_b=-1.5$, (b1) $t_a/t_b=-1$, (c1) $t_a/t_b=-0.25$, (d1) $t_a/t_b=0.25$, (e1) $t_a/t_b=1$, and (f1) $t_a/t_b=1.5$. Cyan lines represent perfect flat bands belonging to the bulk. (a2)–(f2) The corresponding band structure of a thin film, where the horizontal axis is momentum along the $X\text{−}\mathrm{\Gamma }\text{−}Y$ line. (a3)–(f3) The corresponding band structure of a triangular prism, whose size of the triangle is $L=9$. There are no topological boundary states for thin films and triangular prisms. Nevertheless, it is not a trivial insulator but a HOTI for (c1) and (d1).

Figure 7

(a) Energy spectrum of the tetrahedron made of the breathing pyrochlore lattice with $L=9$. The horizontal axis is $t_a/t_b$. There emerge zero-energy states (marked in red) for $-1<t_a/t_b<1/2$. They are topological boundary states. (b) The square root of the local density of states $\sqrt{{\rho }_i}$ for the tetrahedron with $L=5$ and $t_a/t_b=1/2$. The amplitude is represented by the radius of the spheres. The local density of states becomes arbitrarily small except for the four corners for $L\gg 1$. The localized states emerge only at the four corners of the tetrahedron.