= Vortex-pair renormalization flow
{title2=$dK^{-1}/d\ell=4\pi^3y^2,\quad dy/d\ell=(2-\pi K)y$}
In the dilute-vortex description of the two-dimensional XY model, $K=\rho_s/(k_BT)$ 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 $X'=Y^2$, $Y'=XY$ 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.
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