Mirror Chern numbers in the hybrid Wannier representation

The topology of electronic states in band insulators with mirror symmetry can be classified two different ways. One is terms the Chern number, an integer that counts number protected Dirac cones Brillouin zone high-symmetry surfaces. other via a $\mathbb{Z}_2$ index distinguishes between systems have nonzero quantized orbital magnetoelectric coupling ("axion-odd"), and those do not ("axion-even"); this classification also induced by symmetries magnetic point group, including time reversal inversion. A systematic characterization axion has previously been obtained representing valence hybrid Wannier functions localized along one chosen crystallographic direction, inspecting associated structure. Here we focus on symmetry, extend to number. We choose direction orthogonal plane as Wannierization show determined from winding numbers touching points bands mirror-invariant planes, flat pinned planes. In representation, relation readily established. formalism illustrated means $\textit{ab initio}$ calculations for SnTe monolayer bulk forms, complemented tight-binding toy model.