Critical and near-critical XY scaling and BKT universality

What OpenAI claims

For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations C_βc(r) ∼ Ar^-1/4(log r)^1/8 and the Berezinskii–Kosterlitz–Thouless essential singularity √(β_c-β) log ξ(β) → B, with A,B > 0 after the free-box thermodynamic limit. For finite square-symmetric interactions containing nearest neighbors, discrete Gaussian heights converge to Gaussian fields throughout the rough phase, including its threshold, along geometric torus sizes. Critical center-magnetization and spin-field conclusions retain their stated height, renormalization, and field-input assumptions.

Papers (6)

1. The critical logarithmic correction for the planar XY model

   5 October 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove the critical logarithmic correction for the nearest-neighbor cosine XY model on the square lattice. Let C_bc(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

2. Critical Center Magnetization in the Planar XY Model

   5 October 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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_XYn^-1/8(log n)^1/16(1+o(1)), where A_XY is a finite, strictly positive model-specific constant.

3. The Critical Spin Field of the Planar XY Model

   5 October 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

4. Essential Singularity of the Correlation Length in the Planar XY Model

   5 October 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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 bc from the massive side,

5. The critical correlation exponent of the planar XY model

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

6. BKT universality for height and planar spin fields

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.