Goldstone Bosons & Zero Modes — 2D XY Model

Spin waves, Goldstone modes, vortex-antivortex pairs, and BKT transition

T (T_BKT≈0.89): — ⟨cos θ⟩ (order): — Vortex pairs: — Energy: —
About: The 2D XY model has continuous U(1) symmetry: each spin has an angle θ ∈ [0,2π). Goldstone's theorem guarantees massless spin-wave modes (Goldstone bosons) when symmetry is spontaneously broken. At T below T_BKT ≈ 0.893 J/k_B (Berezinskii-Kosterlitz-Thouless), vortex-antivortex pairs are bound; above T_BKT they unbind and destroy quasi-long-range order. The structure factor S(k) ∝ 1/k² is a hallmark of Goldstone modes.