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

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