Theory of fractional corner charges in cylindrical crystal shapes

Recent studies showed that topologically trivial insulators may have fractionally quantized corner charges due to the topological invariant called a filling anomaly. Such crystal shapes in three dimensions are restricted to vertex-transitive polyhedra, which are classified into spherical and cylindrical families. The previous works derived formulas of the fractional corner charge for the spherical family, which corresponds to the tetrahedral and cubic space groups (SGs). In this study, we derive all the corner charge formulas for the cylindrical family, which corresponds to the orthorhombic, tetragonal, hexagonal, and trigonal crystal shapes. We show that all the real-space formulas of the filling anomaly for the cylindrical SGs are universally determined by the total charges at the Wyckoff position $1a$. Moreover, we derive the $k$-space formulas of the corner charge for the cylindrical cases with time-reversal symmetry. From our results, we also show that ${\mathrm{CsLi}}₂{\mathrm{Cl}}₃, {\mathrm{KN}}₃$, and ${\mathrm{Li}}₃\text{N}$ are candidate materials with a quantized corner charge by using the ab initio calculations. Together with our previous work, we exhaust corner charge formulas for all the SGs and crystal shapes having quantized corner charges.