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Essential Singularity of the Correlation Length in the Planar XY Model

OpenAI · October 5, 2026

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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, $\displaystyle \sqrt{b_c-b}\log\frac{1}{m(b)} \longrightarrow A_{\mathrm{XY}},$ where $A_{\mathrm{XY}}$ is a finite, strictly positive model-specific constant.

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Cite

@misc{OAI:Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026,
  author = {{OpenAI}},
  title = {{Essential Singularity of the Correlation Length in the Planar XY Model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/paper.pdf}{OAI:Essential-Singularity-of-the-Correlation-Length-in-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.

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