Surface theorem for the Chern-Simons axion coupling

The Chern-Simons axion coupling of a bulk insulator is only defined modulo a quantum of e²/h. The quantized part of the coupling is uniquely defined for a bounded insulating sample, but it depends on the specific surface termination.Working in a slab geometry and representing the valence bands in terms of hybridWannier functions, we show how to determine that quantized part from the excess Chern number of the hybridWannier sheets located
near the surface of the slab. The procedure is illustrated for a tight-binding model consisting of coupled quantum anomalous Hall layers. By slowly modulating the model parameters it is possible to transfer one unit of Chern number from the bottom to the top surface over the course of a cyclic evolution of the bulk Hamiltonian, changing the surface anomalous Hall conductivity by a quantum of conductance e²/h. When the evolution of the surface Hamiltonian is also cyclic, the Chern pumping is obstructed by chiral touchings between valence and conduction surface bands.