Filling anomaly for general two- And three-dimensional C4 symmetric lattices

In this paper, we derive symmetry indicator formulas for the filling anomaly on two-dimensional (2D) C4 symmetric square lattices with and without time reversal, inversion symmetry, or their product, in the presence of spin-orbit coupling. We go beyond previous work by considering lattices with atoms occupying multiple Wyckoff positions. We also provide an algorithm using the Smith normal form that systematizes the derivation. The formulas determine the corner charge in 2D atomic or fragile topological insulators, as well as in three-dimensional (3D) insulators and semimetals by studying their 2D slices in momentum space. We apply our results to a 3D tight-binding model on a body-centered tetragonal lattice, whose projection into the 2D plane has two atoms in the unit cell. Our symmetry indicators correctly describe the higher-order hinge states and Fermi arcs in cases where the existing indicators do not apply.