Two-Band Model
Chern number C: —
σ_xy: — (e²/h)
Berry flux: —
Ω(k) = −2 Im[⟨∂_kx u|∂_ky u⟩]
C = (1/2π) ∫∫_BZ Ω(k) d²k
σ_xy = (e²/h) · C
C = (1/2π) ∫∫_BZ Ω(k) d²k
σ_xy = (e²/h) · C
The Berry curvature Ω(k) is a "magnetic field" in momentum space that deflects electrons, giving rise to the anomalous Hall effect. The Hall conductivity σ_xy = (e²/h)C where C is the Chern number — an integer topological invariant equal to the total Berry flux through the Brillouin zone.