Surface theorem for the Chern-Simons magnetoelectric coupling

The magnetoelectric response $\alpha_{ij}=\partial M_j/\partial {\cal E}_i$ of insulators has an isotropic geometric contribution, $\alpha_{ij}^{\rm CS}=(\theta e^2/2\pi h)\delta_{ij}$. For crystals that respect neither inversion nor time-reversal symmetry the Chern-Simons (CS) axion coupling $\theta$ can take arbitrary values, which however can only be determined modulo $2\pi$ from bulk calculations. Once an insulating surface termination is specified it becomes possible to resolve the quantum of indeterminacy, as with the spontaneous electric polarization. We prove this "surface theorem" by considering the $\theta$ coupling of a finite slab from the viewpoint of the hybrid Wannier representation. Each Wannier sheet carries a Chern number, and tiling up the periodic sheet structure close to the surface and counting the leftover Chern amount gives the excess quantized surface anomalous Hall conductivity (AHC). We illustrate these ideas for a tight-binding model consisting of Haldane-model layers with alternating Chern numbers. For appropriate choices of the interlayer couplings, this model realizes an adiabatic pump of CS axion coupling. Over a pumping cycle, one quantum of surface AHC gets transferred from the bottom to the top surface, changing $\theta$ by $2\pi$.