Orbital Magnetism — Modern Theory & Berry Curvature
M_orb = (e/2ℏ)∫Ω(k)ε(k) dk/(2π)² — Berry curvature as source of orbital magnetization
The modern theory of orbital magnetization (Thonhauser, Ceresoli, Xiao, Resta 2005) expresses M_orb via Berry curvature Ω(k) and band energy: M = (e/2ℏ)(1/2π)² ∫[Ω(k)·(ε(k)−μ)] d²k. The anomalous Hall conductivity σ_xy = (e²/h)·C where C is the Chern number, a topological invariant equal to (1/2π)∫Ω d²k.