XY Model & the Kosterlitz–Thouless TransitionEach point on this grid is a tiny compass needle free to point in any direction, not just up or down. Neighbours want to line up while heat jostles them at random. Watch for rainbow whirlpools — vortices — where every hue swirls together at one point.Continuous spins in 2D can never fully line up at any nonzero temperature (Mermin–Wagner theorem). Yet Kosterlitz and Thouless showed in 1973 that a real transition still happens: at low heat, vortex–antivortex pairs stay tightly bound and a weak, quasi-long-range order survives; above a critical temperature, thermal energy tears the pairs apart into a gas of free vortices, scrambling the field. It is a transition driven by unbinding topological defects, not by symmetry breaking — work that won the 2016 Nobel Prize in Physics.