Topological Charge Density — XY Vortices

2D XY model: winding number / topological charge computed around each plaquette — vortex-antivortex pairs.

Parameters

Vortices (+1): --
Antivortices (-1): --
Total charge Q: --
Energy/spin: --
In the 2D XY model, spins are unit vectors with angle theta(i,j).

Topological charge on plaquette (i,j):
q = (1/2pi) * sum of angle differences around the square (mod pi to [-pi,pi]).

Vortices (q=+1) and antivortices (q=-1) come in pairs. Below the Kosterlitz-Thouless temperature T_KT, pairs are bound; above T_KT they unbind and destroy quasi-long-range order.

Red = vortex (+1), Blue = antivortex (-1).