Topological Band Theory — Berry Curvature

Two-band model H(k) = d(k)·σ where d = (sin kx, sin ky, m+cos kx+cos ky). Chern number C = (1/2π)∫ Ω(k)d²k counts how many times d̂ wraps around S². Tune m through topological transitions at m=0,±2.

Berry curvature Ω(kx,ky) over Brillouin zone
C = +1
Chern number (topological invariant)
m<−2: C=0 | −2<m<0: C=+1
0<m<2: C=−1 | m>2: C=0
Ω(k) = −2Im⟨∂kx u|∂ky u⟩
= (d̂ · ∂kx d̂ × ∂ky d̂) / 2

|C| = # chiral edge modes (bulk-boundary)

Transitions: m = 0, ±2 (gap closes)
Color scale:
Ω<00Ω>0
d̂ vector winding on S² (kx slice)