2D lattice spins with continuous O(2) symmetry, each a unit vector (angle 0 to 2π). Mermin-Wagner theorem: no spontaneous order in 2D with continuous symmetry. But below T_BKT, vortex pairs bind, giving quasi-long-range order. Above: vortices unbind → disordered.
T_BKT ≈ 0.89 J/k_B for the XY model. Above T_BKT vortex (+1) and antivortex (−1) topological defects unbind and proliferate — destroying quasi-LRO. This is a topological phase transition without a local order parameter. Nobel Prize 2016 (Kosterlitz, Thouless).