Berry Curvature & Band Topology
Quantum geometry, anomalous Hall effect & Chern number in 2D bands
Model Parameters
Mass term m:
0.50
Hopping t:
1.00
SOC λ:
0.50
Animate k-path
Key Equations
Ω
n
(k) = -2 Im Σ
m≠n
⟨n|∂
kx
H|m⟩⟨m|∂
ky
H|n⟩ / (E
m
-E
n
)²
C = (1/2π) ∫
BZ
Ω(k) d²k
σ
xy
= C·e²/h (Anomalous Hall)
H(k) = d(k)·σ, d = (sin kx, sin ky, m+cos kx+cos ky)
Chern Number
C = +1
Topological invariant — robust to smooth deformations. Changes at gap-closing transitions.