Vortex–Antivortex Pairs
XY Model — Kosterlitz-Thouless Topological Phase Transition
Controls
Temperature T/T_KT:
0.80
Spin coupling J:
1.0
Grid size N:
40
+ Vortex pair
Reset
Statistics
Free vortices
0
Bound pairs
0
Total vortices
0
MCS steps
0
Phase
—
Theory
H = −J Σ cos(θᵢ − θⱼ)
Vortex charge: q = ±1
q = (1/2π)∮▽θ·dl
Below T_KT: vortices bound in ±1 pairs — quasi-LRO, power-law correlations
Above T_KT: unbinding transition, free vortices → exponential decay, no order
T_KT ≈ πJ/2
Visual Guide
🔴 Vortex (q=+1)
CCW swirl
🔵 Antivortex (q=−1)
CW swirl
Arrow
spin direction
Click canvas
add pair