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Higher-order topological insulators host gapless states on hinges or cornersof three-dimensional crystals. Recent studies suggested that even topologicallytrivial insulators may exhibit fractionally quantized charges localized athinges or corners. Although most of the previous studies focused ontwo-dimensional systems, in this work, we take the initial step toward thesystematic understanding of hinge and corner charges in three-dimensionalinsulators. We consider five crystal shapes of vertex-transitive polyhedra withthe cubic symmetry such as a cube, an octahedron and a cuboctahedron. We derivereal-space formulas for the hinge and corner charges in terms of the electriccharges associated with bulk Wyckoff positions. We find that both the hinge andcorner charges can be predicted from the bulk perspective only modulo certainfractions depending on the crystal shape, because the relaxation nearboundaries of the crystal may affect the fractional parts. In particular, weshow that a fractionally quantized charge $1/24$ mod $1/12$ in the unit ofelementary charge can appear in a crystal with a shape of a truncated cube or atruncated octahedron. We also investigate momentum-space formulas for the hingeand corner charges. It turns out that the irreducible representations of filledbands at high-symmetry momenta are not sufficient to determine the cornercharge. We introduce an additional Wilson-loop invariant to resolve this issue.

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