= 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 $E_{+-}(R)\simeq2E_c+2\pi\rho_s\log(R/a)$ 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.
\Image[/bkt-vortex-pair-phase-geometry.png]
{title=Opposite phase windings and the geometry of a neutral vortex pair}
{description=Original phase field $\theta=\arg(x+1+iy)-\arg(x-1+iy)$. Each arrow is the unit vector $(\cos\theta,\sin\theta)$, not the superfluid velocity. Counterclockwise loops around the left and right cores have winding $+1$ and $-1$; a loop enclosing both has zero winding. The circles on the right show the corresponding motion in phase space.}
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