= 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.
Here $\ell=\log(r/a)$ increases toward larger observation distances, and the <vortex fugacity> $y$ is a dimensionless weight per unit core, conventionally initialized as $y_0\simeq e^{-E_c/(k_BT)}$ with the microscopic measure fixed. Numerical coefficients depend on this normalization. In the displayed convention, differentiating $X=2-\pi K$ using the original flow gives
$$
X'=4\pi^4K^2y^2=\frac{(2-X)^2}{4}Y^2,
\qquad Y'=XY.
$$
Only replacing the prefactor $(2-X)^2/4$ by one near $X=0$ gives the scaled leading model. Its <first integral> follows directly:
$$
\frac{d}{d\ell}(X^2-Y^2)=2XY^2-2Y(XY)=0.
$$
Thus this <first integral> belongs to the stated scaled model, not exactly to the original $K,y$ equations. For $Y\geq0$, the critical <separatrix> is $X=-Y$. The bound side has $X<-Y$ and flows to $Y=0$, $X<0$; the unbinding side ultimately runs to large <vortex fugacity>, where the dilute approximation ceases to be quantitative.
To derive the unusual <correlation length> scaling without fitting an exponent, write the invariant on a nearby unbinding trajectory as $X^2-Y^2=-c^2$. Then $X'=X^2+c^2$ and the time to cross fixed small limits $-b$ to $b$ is $2\arctan(b/c)/c\sim\pi/c$. If a physical one-parameter path crosses the <separatrix> transversely, its $c^2$ is proportional to the positive <reduced temperature>. Since distance grows as $e^\ell$, this gives $\xi\sim a\exp(b_0/\sqrt{t})$, with a nonuniversal $b_0>0$. This is the dimensionless-temperature form of the singularity above.
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