2D XY spins (arrows show direction)
Vortex density (+red, −blue)
The Berezinskii-Kosterlitz-Thouless (BKT) transition (Nobel Prize 2016, Kosterlitz & Thouless) is a topological phase transition in 2D systems with continuous symmetry. The 2D XY model has spins θ_i ∈ [0,2π) on a lattice with H = −J Σ cos(θ_i − θ_j).
Mermin-Wagner: no spontaneous symmetry breaking in 2D
BKT: vortex unbinding at T_KT = πJ/2 ≈ 0.898J
Vortices are topological defects where the spin winds by ±2π around a plaquette (charge ±1). Below T_KT: vortex-antivortex pairs are tightly bound (logarithmic confinement), producing quasi-long-range order with algebraically decaying correlations G(r) ~ r^{-η(T)}. Above T_KT: pairs unbind, free vortices proliferate, correlations decay exponentially. The transition has an essential (BKT) singularity — no diverging order parameter, only the correlation length ξ ~ exp(b/√(T-T_KT)).