Fractional hinge and corner charges in various crystal shapes with cubic symmetry

Recent studies suggested that even topologically trivial insulators may exhibit fractionally quantized charges localized at hinges or corners. We study hinge and corner charges for five crystal shapes of vertex-transitive polyhedra with the cubic symmetry such as a cube, an octahedron and a cuboctahedron [1]. We derive real-space formulas for the hinge and corner charges in terms of the electric charges associated with bulk Wyckoff positions. We find that both the hinge and corner charges can be predicted from the bulk perspective only modulo certain fractions depending on the crystal shape, because the relaxation near boundaries of the crystal may affect the fractional parts. We also derive momentum-space formulas for the hinge and corner charges by using the method of elementary band representation (EBR) matrix. It turns out that the irreducible representations of filled bands at high-symmetry momenta are not sufficient to determine the corner charge. In order to resolve this issue, we introduce an additional Wilson-loop invariant. We show that the momentum-space formulas for the corner charges can be obtained by combining the EBR matrix with the new Wilson-loop invariant.

[1] K. Naito, R. Takahashi, H. Watanabe, and S. Murakami, arXiv:2109.14914 (2021).