 Vortex-pair renormalization flow

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.

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 $y_0\simeq e^{-E_c/(k_{B}T)}$ with the microscopic measure fixed. Numerical coefficients depend on this normalization. In the displayed convention, differentiating $X=2-\pi K$ using the original flow givesOnly replacing the prefactor $(2-X{)}^2/4$ by one near $X=0$ gives the scaled leading model. Its first integral follows directly:Thus this first integral belongs to the stated scaled model, not exactly to the original $K,y$ equations. For $Y\ge 0$, 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 $X^{\prime }=X^2+c^2$ 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.