BKT Transition (Berezinskii 1971, Kosterlitz–Thouless 1973):
The 2D XY model has spins that are unit vectors — no long-range order exists (Mermin–Wagner theorem),
yet a topological phase transition occurs at T_BKT ≈ 0.89 J/k_B.
Below T_BKT: vortex–antivortex pairs (±1 winding number) are bound — algebraic order, quasi-LRO.
Above T_BKT: pairs unbind — free vortices proliferate, exponential decay of correlations.
Vortices (red +1) and antivortices (blue −1) are detected from the winding number of each plaquette.
The BKT transition was a breakthrough in topological physics, earning the 2016 Nobel Prize.