Filling anomaly in 2D and 3D C4 symmetric lattices

We derive the symmetry indicator formulas for the filling anomaly on the 2D square lattice with spinful atoms occupying multiple Wyckoff positions. We consider two sets of symmetries: C4, T, I or C4, TI. Our symmetry indicators generalize previous work where only one atom in the unit cell is considered. The formulas are valid for atomic insulators and fragile topological insulators and can be used to determine the corner charge. They can also be used to analyze the second order topology in C4 symmetric 3D insulators and semimetals with time-reversal symmetry and spin-orbit coupling, by studying their two dimensional kz slices. As an example, we build an explicit tight-binding model on a body-centered tetragonal lattice and numerically calculate the energy spectrum on a rod that is finite in two dimensions and infinite in the third. Our symmetry indicators correctly describe the higher-order hinge states and Fermi arcs in cases where the existing indicators do not apply.