Topological Defects — Vortex Annihilation in 2D XY Model
Kosterlitz-Thouless transition: vortex-antivortex pairs, winding numbers, and defect dynamics
0.50
1.00
3
Vortices (+1): 0
Antivortices (−1): 0
Energy: —
Steps: 0
In the 2D XY model, each site carries a planar spin θ ∈ [0,2π). Vortices are topological defects where the angle winds ±2π around a plaquette (winding number ±1). Below the Kosterlitz-Thouless (KT) temperature T_KT ≈ 0.89J, vortex-antivortex pairs are tightly bound; above T_KT they unbind, destroying quasi-long-range order. The XY Hamiltonian: H = −J∑cos(θᵢ−θⱼ). Vortex pairs feel a logarithmic attractive force V(r) = −(J/π)ln(r).