Tetrahedron

Combinatorics

Physics

Topological insulator

Topological index

Topology (electrical circuits)

Acoustics

Geometry

Condensed matter physics

Pyrochlore

Lattice (music)

Mathematics

Phase (matter)

Topological order

Quantum mechanics

Topological quantum number

Quantum

Symmetry protected topological order

Physical Review Letters

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 paper, 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.

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