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The Critical Spin Field of the Planar XY Model
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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 $2/\pi$. 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.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Possibly relevant literature (found by the agent, not checked)
- Christophe Garban and Avelio Sepúlveda, Statistical reconstruction of the GFF and KT transition (J. Eur. Math. Soc. 26 (2024), 639–694). Studies the GFF observed modulo 2π/T, motivated by the XY and Villain models; the companion BKT-universality paper cites it for a joint limit of the discrete GFF and its Wick exponential, which relates to the imaginary-chaos limit in Theorem 1.1, and it is not in this paper's bibliography.
Digests and expositions
- Latent Seminar, The planar XY model: critical logarithms and the BKT transition (2026-10-06). Explains center normalization and the field-limit argument, without independent proof verification. [family-level]
Elsewhere
- Emergent Mind: family 216
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026,
author = {{OpenAI}},
title = {{The Critical Spin Field of the Planar XY Model}},
howpublished = {OpenAI Math Release preprint \href{https://github.com/openai/math/blob/main/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/paper.pdf}{OAI:The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026}},
year = {2026}
}Family 216: Critical and near-critical XY scaling and BKT universality
For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations $C_{\beta_c}(r)\sim Ar^{-1/4}(\log r)^{1/8}$ and the Berezinskii–Kosterlitz–Thouless essential singularity $\sqrt{\beta_c-\beta}\log\xi(\beta)\to B$, with $A,B\gt 0$ 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.
Other papers in this family:
- The critical logarithmic correction for the planar XY model · October 5, 2026
- Critical Center Magnetization in the Planar XY Model · October 5, 2026
- Essential Singularity of the Correlation Length in the Planar XY Model · October 5, 2026
- The critical correlation exponent of the planar XY model · September 24, 2026
- BKT universality for height and planar spin fields · September 24, 2026