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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 energywhere ${\rho }_s$ is the phase stiffness. Gaussian averaging givesThus 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 givesA 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 $2k_{B}T/\pi$. 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 $K={\rho }_s/(k_{B}T)$ and dimensionless core fugacity $y$, one conventional leading-order normalization givesThe 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 $X^{\prime }=Y^2$ and $Y^{\prime }=XY$, so $X^2-Y^2$ is invariant. The critical separatrix approaches $(0,0)$. Just on the disordered side this invariant is $-{\epsilon }^2$, proportional to the reduced temperature deviation for a generic path through the transition. Hence $X^{\prime }=X^2+{\epsilon }^2$, whose integration gives a time in logarithmic scale of order $1/\epsilon$. Stopping when the fugacity becomes order one yieldswith 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 jumpAbove $T_c$, 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 Kosterlitz's original vortex analysis. 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.