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Critical Center Magnetization in the Planar XY Model
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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 $n\to\infty$, 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.
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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)
- Martin Hasenbusch, The two dimensional XY model at the transition temperature: A high precision Monte Carlo study (arXiv:cond-mat/0502556, J. Phys. A 38 (2005) 5869–5884). Monte Carlo study of logarithmic corrections to the magnetic susceptibility at β_KT, a numerical counterpart to the Kenna–Irving prediction discussed in the paper's historical context and a related observable distinct from the aligned-boundary center magnetization; not in the paper's bibliography.
Digests and expositions
- Latent Seminar, The planar XY model: critical logarithms and the BKT transition (2026-10-06). Explains the center asymptotic and companion assumptions, without independent proof verification. [family-level]
Elsewhere
- Emergent Mind: family 216
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Critical-Center-Magnetization-in-the-Planar-XY-Model-October-5-2026,
author = {{OpenAI}},
title = {{Critical Center Magnetization in the Planar XY Model}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Critical-Center-Magnetization-in-the-Planar-XY-Model-October-5-2026/paper.pdf}{OAI:Critical-Center-Magnetization-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.
Other papers in this family:
- The critical logarithmic correction for the planar XY model · October 5, 2026
- The Critical Spin Field of 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