Orbital Magnetization — Modern Theory & Berry Phase

Parameters

B-field / band gap: 0.5 Fermi energy E_F: 0.3 Haldane parameter φ: 0.3

Key Formulas

M = -(e/2ℏc) Σ_nk f_nk × Im⟨∂_k u_nk|×(H-E_nk)|∂_k u_nk⟩

= Berry-phase term + circulation term

Ω_nk = -2Im⟨∂_kx u|∂_ky u⟩

C = (1/2π)∫_BZ Ω dk²

Modern theory: Orbital magnetization requires Berry curvature. The naive Bloch-band formula diverges — the correct formula has two terms: Berry-phase part + energy-weighted Chern number.

Haldane model: Broken T-symmetry without B-field gives anomalous Hall conductance σ_xy=Ce²/h and orbital magnetization from Berry curvature.