Topological Phase Boundary in Haldane Model Set by Mass-to-Hopping Ratio — E8 Intelligence Research

Abstract

FINDING: The Haldane model's topological phase transition is governed by the ratio of the sublattice mass (M) to the next-nearest-neighbor hopping amplitude (t₂), with the critical boundary defined by |M/t₂| = 3√3, a value directly tied to the hexagonal Brillouin zone's geometry. | MATH: Critical condition: |M/t₂| = 3√3 ≈ 5.196. The hexagonal reciprocal lattice has primitive vectors b₁ = (2π/a)(1, 1/√3), b₂ = (2π/a)(1, −1/√3) in Cartesian coordinates, giving area |b₁×b₂| = 4π²/(a²√3). The high-symmetry points K and K′ sit at (2π/3a)(1, √3) and (2π/3a)(1, −√3), with the distance from Γ to K being 4π/(3a). The ratio 3√3 emerges from the product of the coordination number (3) and the √3 factor inherent to the triangular/hexagonal lattice's reciprocal space. | CONNECTION: The 3√3 critical ratio is a crystallographic fingerprint of the hexagonal lattice (root system A₂). Note that 3√3 = 3 × √3, and √3 ≈ 1.732, which relates to the golden ratio via √3 = 2cos(π/6) — a hexagonal symmetry angle Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com