Critical and near-critical XY scaling and BKT universality

We prove the critical logarithmic correction for the nearest-neighbor cosine XY model on the square lattice. Let \(C_{b_c}(r)\) be the two-point correlation at critical inverse temperature, obtained by taking the free-box thermodynamic limit while the two sites remain r lattice steps apart along a coordinate axis. Then \(\displaystyle C_{b_c}(r)=B_{\mathrm{XY}}r^{-1/4}(\log r)^{1/8}(1+o(1)), \qquad B_{\mathrm{XY}}\in(0,\infty),\) as \(r\to\infty\).

Assuming the stated critical-height, local-renormalization, and spin-field inputs from the companion papers, we determine the center magnetization of the planar XY model in a square with aligned boundary spins at its mass-defined critical threshold. As \(n\to\infty\), the magnetization is \(A_{\mathrm{XY}}n^{-1/8}(\log n)^{1/16}(1+o(1))\), where \(A_{\mathrm{XY}}\) is a finite, strictly positive model-specific constant.

Using the companion critical Bessel-height and pin-limit theorems, we prove that at the mass-defined critical inverse temperature of the nearest-neighbor XY model on the square lattice, the spin field on a square with boundary angles fixed to zero converges along the full sequence to the full-variance imaginary exponential of a zero-Dirichlet Gaussian free field with stiffness \(2/\pi\). The normalization uses the exact center magnetization and the lattice Green-function factor. Convergence holds in law in \(H^{-3}_{\mathrm{loc}}((-1,1)^2)\); all mixed moments and joint laws of fields smeared against smooth compactly supported test functions in \((-1,1)^2\) converge as well.

We prove the Berezinskii–Kosterlitz–Thouless essential singularity for the correlation length of the nearest-neighbor cosine XY model on the square lattice. Let \(m(b)\) be the mass obtained by taking first the free-box thermodynamic limit and then the separation limit along a coordinate axis. As the inverse temperature b approaches b_c from the massive side, \(\displaystyle \sqrt{b_c-b}\log\frac{1}{m(b)} \longrightarrow A_{\mathrm{XY}},\) where \(A_{\mathrm{XY}}\) is a finite, strictly positive model-specific constant.

For the ordinary nearest-neighbor XY model on the square lattice, we prove the Berezinskii–Kosterlitz–Thouless prediction that the critical spin-correlation exponent is 1/4. More precisely, at the mass-defined critical inverse temperature, the infinite-volume correlation is \(n^{-1/4+o(1)}\). The infinite-volume limit through free square boxes is taken before the separation limit.

We prove Gaussian scaling limits for the two-dimensional discrete Gaussian height model throughout its rough phase, including the physical roughening threshold, for every finite square-symmetric interaction set containing the nearest neighbors. After the natural lattice normalization, the critical effective temperature has the universal value \(8\pi\). For the ordinary nearest-neighbor Villain and XY models at sufficiently low fixed temperatures, we also prove that their Green-function-normalized spin fields converge to the imaginary exponential of a Dirichlet Gaussian free field.