2-D XY model · Kosterlitz–Thouless

Can a 2-D system have a phase transition with NO long-range order to switch on — and what drives it if not a broken symmetry?

How the lab tests it

Planar spins on a lattice, neighbours aligning (E=−Σcos Δθ), evolved by Metropolis Monte-Carlo. Across a temperature sweep, measure the helicity modulus ϒ(T) (the spin stiffness) and the vortex density ρ_v(T). The Nelson–Kosterlitz criterion says ϒ jumps to zero across the universal line ϒ=(2/π)T — the crossing locates T_KT. Live, a lattice oscillates its temperature through T_KT with vortices flagged ± (red/cyan).

What it checks

a TOPOLOGICAL transition driven by vortex unbinding, not symmetry breaking. In 2-D the continuous symmetry forbids true long-range order at any T>0 (Mermin–Wagner), so the magnetisation never sharply switches on; instead the order is quasi-long-range (power-law) and is destroyed when vortices, bound in ± pairs below T_KT, UNBIND into a free plasma above it. ϒ(T) crosses 2T/π at T_KT (≈0.95 from the naive crossing on this finite lattice; the ∞-limit 0.893 is recovered only by finite-size scaling with logarithmic corrections, not by a larger lattice alone) and ρ_v rises sharply there. A phase transition with no local order parameter