2D XY Model and the BKT Vortex Transition

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 $T_\text{BKT}$ .

Figure 1. 2D XY model angles colored by direction; vortices marked by colored dots. Method: single-spin Metropolis.

T0.70

L64

speed3

WHAT TO TRY

Cool below the BKT temperature: the compass needles lock into smooth swirls and the bound vortex-antivortex pairs stay tightly paired. Above it the pairs unbind and free vortices proliferate.
Raise the temperature T through the transition: watch isolated vortices (the pinwheel defects) appear and wander, the topological mechanism behind the Berezinskii-Kosterlitz-Thouless transition.
Enlarge the lattice L: the quasi-long-range order below the transition shows as slowly twisting domains, correlations that decay as a power law rather than locking rigidly or vanishing.