Berry Curvature & Band Topology

Quantum geometry in k-space — Chern number and anomalous Hall conductivity

2-Band Model (Haldane)

Topology

C = 0
σ_xy (e²/h)0.00
Band gap (t)
PhaseTrivial

Berry curvature:

Ω_n(k) = −2 Im Σ_{m≠n} <n|∂_kx H|m><m|∂_ky H|n>/(E_m−E_n)²

Chern number:

C = (1/2π)∫∫_BZ Ω(k) d²k

Hall conductivity:

σ_xy = C·e²/h

Haldane model: graphene + NNN hopping + flux.
Color: +Ω (red/hot) → −Ω (blue/cool)