Mirror Chern number in the hybrid Wannier representation

We formulate the mirror Chern number (MCN) of a two-dimensional insulator with reflection symmetry M_z in terms of hybrid Wannier functions (the eigenfunctions of PzP, the position operator projected onto the valence bands). Because PzP and M_z anticommute, the spectrum of "Wannier bands" is symmetric about the mirror plane, and an excess of one mirror eigenvalue over the other in the occupied manifold leads to the appearance of flat bands on the mirror plane. (This structure is reminiscent of the energy bands of a bipartite lattice, where the Hamiltonian anticommutes with the sublattice symmetry operator.) In the absence of flat bands, pairs of dispersive bands may touch at isolated points on the mirror plane. These Dirac cones are protected by symmetry, and the MCN is given by the sum of their winding numbers. When flat bands are present the Dirac cones are no longer protected, and the MCN is related instead to the Chern number of the flat bands. In three dimensions, the present formalism reveals a simple relation between the MCNs and the quantized axion angle θ, whose expression in the hybrid Wannier representation was previously obtained.