On the other hand, if time-reversal symmetry is absent, other symmetries can quantize θ θ = π π and but not force σ σ AHC surf to vanish. The most extreme case is the case of inversion symmetry (I). Inversion is never a surface symmetry and therefore a non-zero σ σ AHC surf is valid. In the case that a surface is gapped, we have ϕ ϕ = 0 which results in a half-quantized surface AHC σ σ AHC surf = − − e 2 2 h .