Vortex-antivortex binding

In the phase with quasi-long-range order, vortex defects of opposite winding remain bound in pairs. Their unbinding produces the disordered phase with a finite correlation length.

For two unit vortices of opposite charge separated by $R$, the vortex Coulomb gas gives $E_{+-}(R)\simeq 2E_c+2\pi {\rho }_s\log (R/a)$ before screening. At distances much greater than $R$, their phase gradients cancel at leading order: the pair field decays as $R/r^2$, whereas a lone phase vortex has a $1/r$ gradient. A neutral pair therefore has finite energy as $L\to \infty$ at fixed $R$. Bound does not mean absent: small pairs remain thermally excited and reduce the phase stiffness through their response to a twist.