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

TL;DR

This paper systematically analyzes fractional hinge and corner charges in three-dimensional cubic-symmetric insulators of various polyhedral shapes, deriving real-space and momentum-space formulas, and revealing shape-dependent fractional quantization effects.

Contribution

It introduces real-space formulas for hinge and corner charges in 3D insulators with cubic symmetry and explores shape-dependent fractional quantization, including a new Wilson-loop invariant.

Findings

Fractional charges depend on crystal shape and boundary relaxation.

A fractional charge of 1/24 mod 1/12 can appear in certain shapes.

Momentum-space invariants require an additional Wilson-loop to determine corner charges.

Abstract

Higher-order topological insulators host gapless states on hinges or corners of three-dimensional crystals. Recent studies suggested that even topologically trivial insulators may exhibit fractionally quantized charges localized at hinges or corners. Although most of the previous studies focused on two-dimensional systems, in this work, we take the initial step toward the systematic understanding of hinge and corner charges in three-dimensional insulators. We consider five crystal shapes of vertex-transitive polyhedra with the cubic symmetry such as a cube, an octahedron and a cuboctahedron. 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. In particular, we show that a fractionally quantized charge $1/24$ mod $1/12$ in the unit of elementary charge can appear in a crystal with a shape of a truncated cube or a truncated octahedron. We also investigate momentum-space formulas for the hinge and corner charges. It turns out that the irreducible representations of filled bands at high-symmetry momenta are not sufficient to determine the corner charge. We introduce an additional Wilson-loop invariant to resolve this issue.