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_{XY}r^{-1/4}(log r)^{1/8}(1+o(1)), B_{XY}∈(0,∞), as r→∞.

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→∞, the magnetization is A_{XY}n^{-1/8}(log n)^{1/16}(1+o(1)), where A_{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/π. The normalization uses the exact center magnetization and the lattice Green-function factor. Convergence holds in law in H^{-3}_{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_{XY}, where A_{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 8π. 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.