Vortex-pair renormalization flow ($d{K}^{-1}/d\ell =4{\pi }^3{y}^2,dy/d\ell =(2-\pi K)y$)

In the dilute-vortex description of the two-dimensional XY model, $K={\rho }_s/(k_{B}T)$ is dimensionless stiffness and $y$ is phase vortex fugacity. A phase vortex has logarithmic energy $\pi {\rho }_s\log (L/a)$, while its placement entropy is $2k_B\log (L/a)$. Small phase vortex pairs screen the logarithmic interaction, lowering the stiffness. Near $K=2/\pi$, the rescaled variables $X=2-\pi K$, $Y=4\pi y$ obey $X^{\prime }=Y^2$, $Y^{\prime }=XY$ to leading order. The invariant $X^2-Y^2$ leads to an essential correlation-length singularity $\xi \sim a\exp (C/\sqrt{T-T_c})$, not a finite power-law exponent.