Chern Number & Hall Conductance

The Berry curvature integrated over the Brillouin zone torus gives an integer Chern number C. The Hall conductance is exactly σxy = C·e²/h — topology quantizes transport.

C = 0
σ_xy = 0 · e²/h
1.00
0.30
Haldane-like 2-band: H(k) = d(k)·σ where d_z = M − 2t₂(cos kx + cos ky).
Berry curvature: F = ε_{ijk} d̂·(∂_x d̂ × ∂_y d̂)/2. Chern number: C = (1/2π)∫F dk.
Phase transitions occur when gap closes: M/t₂ = ±4. Map shows F(kx,ky) — warm=positive, cool=negative.