XY Model — Topological Defect Annihilation

The XY model assigns a planar spin angle θ ∈ [0, 2π) to each lattice site. Unlike the Ising model, it supports topological defects: vortices (winding number +1) and antivortices (-1), where ∮∇θ·dl = ±2π around a closed loop. Below the BKT critical temperature T_BKT ≈ 0.89 J/k_B, vortex-antivortex pairs are bound; above it they unbind and proliferate, destroying quasi-long-range order. Bound pairs attract and annihilate; free vortices drive the superfluid-to-normal transition seen in thin helium films and 2D superconductors. Colors show spin angle; markers indicate vortex cores.