2D XY model: BKT vortex unbinding

What you are seeing: a 2D grid where each site holds an angle (a "compass needle") that can point anywhere from 0 to 360 degrees. Neighbors prefer to point in the same direction ($E = -J\cos(\theta_i - \theta_j)$ per bond). At low $T$ all needles align approximately; at high $T$ they go random. The interesting feature: the transition is not a sharp ordering but a Kosterlitz- Thouless transition where pairs of vortices (rotational defects) unbind.

Each cell is colored by the angle (rainbow modulo $2\pi$). Red and blue dots mark $+2\pi$ and $-2\pi$ vortices, found by computing the winding number around each plaquette. The transition temperature is $T_\text{BKT} \approx 0.893 J$; below it vortices pair up; above it they roam free. Watch the vortex count climb past $T_\text{BKT}$.