= Solution
The <Berezinskii–Kosterlitz–Thouless transition> is a defect-unbinding transition of a two-dimensional system with a compact continuous phase, such as the short-range <XY model>. At long wavelengths, smooth <spin waves> have energy
$$
F_{\rm sw}=\frac{\rho_s}{2}\int d^2x\,|\nabla\theta|^2,
$$
where $\rho_s$ is the <phase stiffness>. Gaussian averaging gives
$$
\langle[\theta(\mathbf r)-\theta(0)]^2\rangle=\frac{k_BT}{\pi\rho_s}\log(r/a)+O(1),\qquad
\langle e^{i[\theta(\mathbf r)-\theta(0)]}\rangle\sim(a/r)^\eta,\quad
\eta=\frac{k_BT}{2\pi\rho_s}.
$$
Thus smooth fluctuations already destroy true nonzero magnetization in an infinite two-dimensional system, in accordance with the <Mermin-Wagner theorem>. The low-temperature phase instead has <quasi-long-range order>, with algebraically decaying correlations. The stiffness in its asymptotic exponent is the renormalized long-distance stiffness, not merely the microscopic coupling.
The phase is periodic modulo $2\pi$, so it also permits <topological defects> with integer winding $n$. A <phase vortex> has $|\nabla\theta|=|n|/r$, and integration gives
$$
E_n=\pi\rho_s n^2\log(L/a)+E_{\rm core}.
$$
A singly charged <phase vortex> can be placed in roughly $(L/a)^2$ locations, giving entropy $2k_B\log(L/a)$. This energy-entropy balance suggests that isolated <vortices> become favorable when the effective stiffness falls to $2k_BT/\pi$. Below the transition, <vortex-antivortex binding> predominates; above it their unbinding screens the logarithmic interaction and produces a finite <correlation length>. A single-vortex estimate identifies the mechanism but does not include screening by smaller pairs.
The dilute-vortex <vortex-pair renormalization flow> incorporates this screening. With $K=\rho_s/(k_BT)$ and dimensionless core fugacity $y$, one conventional leading-order normalization gives
$$
\boxed{\frac{dK^{-1}}{d\ell}=4\pi^3y^2,\qquad
\frac{dy}{d\ell}=(2-\pi K)y.}
$$
The first equation says polarizable <phase vortex> pairs reduce stiffness. The second compares the <phase vortex>'s position scaling dimension two with its logarithmic energy cost $\pi K$. Numerical coefficients in $y$ depend on its definition, but the zero-fugacity threshold $K=2/\pi$ is invariant. For $K>2/\pi$, sufficiently small fugacity flows to zero, leaving a line of Gaussian spin-wave <renormalization-group fixed points> with varying algebraic exponent. On the other side of the separatrix, fugacity grows and the dilute approximation eventually ceases to apply.
Near the endpoint set $X=2-\pi K$ and $Y=4\pi y$. To leading order $X'=Y^2$ and $Y'=XY$, so $X^2-Y^2$ is invariant. The critical separatrix approaches $(0,0)$. Just on the disordered side this invariant is $-\varepsilon^2$, proportional to the reduced temperature deviation for a generic path through the transition. Hence $X'=X^2+\varepsilon^2$, whose integration gives a time in logarithmic scale of order $1/\varepsilon$. Stopping when the fugacity becomes order one yields
$$
\boxed{\xi\sim a\exp(B/\sqrt\tau),\qquad \tau=(T-T_c)/T_c\downarrow0^+,}
$$
with nonuniversal $B>0$. This derives the essential singularity and explains why assigning a finite ordinary correlation-length exponent is inappropriate.
At the transition the limiting stiffness from below has the <universal stiffness jump>
$$
\boxed{\rho_s(T_c^-)=\frac{2k_BT_c}{\pi},\qquad \eta(T_c)=1/4.}
$$
Above $T_c$, the thermodynamic long-distance stiffness vanishes and correlations decay exponentially. The singular free-energy contribution is of order $\xi^{-2}$ and is exponentially small in $1/\sqrt\tau$, so this infinite-order transition is unlike a Landau transition with a discontinuous <order parameter> or an ordinary power-law heat-capacity singularity. The renormalization mechanism is developed in https://doi.org/10.1088/0022-3719/7/6/005[Kosterlitz's original vortex analysis]. \b[The transition separates bound <vortices> and algebraic order from unbound <vortices> and short-range order, while neither finite-temperature phase has conventional spontaneous XY magnetization.]
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