H = −J Σ cos(θᵢ−θⱼ) — topological vortex-antivortex pairs unbind at T_KT ≈ 0.89 J/k_B
Vortex count—
Antivortex count—
Helicity ⟨cos Δθ⟩—
T_KT ≈0.893
Phase—
KT transition (Nobel Prize 2016): 2D XY model has no conventional order but shows a topological transition. Below T_KT, vortex-antivortex pairs are bound → quasi-long-range order (power-law correlations). Above T_KT, free vortices proliferate → exponential decay. Arrows show spin direction, colored by angle. +/− mark vortex cores.