BKT Transition: Vortex–Antivortex Unbinding

Berezinskii-Kosterlitz-Thouless (BKT) transition: The 2D XY model cannot have long-range order (Mermin-Wagner theorem), but has a topological phase transition driven by vortex unbinding.

Below T_BKT: vortices and antivortices are bound in pairs. Their logarithmic interaction energy V(r) = 2πJ ln(r/a) confines them — free energy F = (2πJ − 2T) ln(r/a), and since 2πJ > 2T below T_BKT, pairs stay bound. The correlation function decays as power law: ⟨S(0)·S(r)⟩ ~ r^(−η) with η = T/(2πJ).

Above T_BKT: free vortices proliferate. The stiffness jumps discontinuously: K(T_BKT⁺) = 0, K(T_BKT⁻) = 2/π. This is the universal jump. The correlation length diverges as ξ ~ exp(c/√|T−T_BKT|) — essential singularity, not power law.