Dirac Mass Terms Govern Chern Insulator Phase Transitions — E8 Intelligence Research

FINDING: Chern insulator phase transitions are governed by Dirac mass terms whose sign change closes and reopens the bulk gap, with the Chern number changing by ±1 per Dirac cone — the Haldane mass (intrinsic, time-reversal-breaking) and Semenoff mass (sublattice/staggered, inversion-breaking) are the two canonical mass terms controlling this transition. | MATH: Low-energy Dirac Hamiltonian H(k) = v_F(σ_x k_x + σ_y k_y) + m σ_z. Gap = 2|m|. Chern number C = (1/2)Σ_n sgn(m_n) over Dirac cones. Haldane model: m_H = 3√3 t₂ sin(φ) (breaks T, preserves inversion); Semenoff mass: m_S = M (breaks inversion, preserves T). Transition at m_H = ±m_S: gap closes at Dirac point, C changes by ±1. Haldane phase: C = ±1 for |m_H| > |m_S|, trivial C = 0 for |m_H| < |m_S|. | CONNECTION: The Haldane model lives on a honeycomb lattice — the 2D projection of the hexagonal close-packed / graphene structure with 6-fold (C₆) crystallographic symmetry. Dirac points sit at K, K′ = (4π/3a)(±1, 0) — vertices of t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com