So far we have discussed the mathematical properties of the axion coupling. Physically, a non-trivial axion coupling ( θ θ = π π ) will result in a half-quantized surface anomalous Hall conductivity ( σ σ AHC surf = e 2 / 2 h ) if the surface states are gapped. To see this, note that in general σ σ AHC surf has two contribution. One comes from the axion coupling θ θ , a quantity that is determined from bulk considerations as we have seen, while the other is the Berry phase ϕ ϕ of the surface states at the Fermi level and therefore depends on the surface. In summary for a given surface termination the perpendicular component of the surface anomalous Hall conductivity to the surface will be