Theory of Fractional Corner Charges and Its Application to Vertex-Transitive Polyhedra with Higher Genus

In obstructed atomic insulators, fractional charges appear at the corners of the crystals in the shape of vertex-transitive polyhedra, and are given by the filling anomaly divided by the number of corners. Recent studies reveal that the filling anomaly for the cases with genus 0 is universally given by the total charge at the Wyckoff position 1a. In this study, we rewrite the formula in terms of the degree of sharpness of the corner, and show that the corner charge formula also holds for cases with higher genus. We also extend our formula to vertex-transitive shell polyhedra, which are three-dimensional closed surfaces made only of polygons. Then, we show that the corner charges of such shell polyhedra are equal to the two-dimensional disclination charges of the corresponding disclinations. By identifying it with the disclination charge under the Wen-Zee action, we show that the coupling constant of the Wen-Zee action for a given crystalline insulator is given by the total charge at the Wyckoff position 1a.