Berry Curvature & Chern Number

Qi-Wu-Zhang model — topological band theory in the Brillouin zone

QWZ Model

Topology

C = ?
Band gap: —
Phase: —
∮ F_xy dk: —

Theory

QWZ Hamiltonian:
H(k) = dᵢ(k)σᵢ
d₁=sin kₓ, d₂=sin kᵧ
d₃=m+cos kₓ+cos kᵧ

Berry curvature:
F_xy = ε_ijk d̂·(∂_kx d̂ × ∂_ky d̂)/2
Chern number:
C = 1/2π ∫∫ F_xy d²k
C=1 for -2 < m < 0
C=-1 for 0 < m < 2
C=0 otherwise