View source for The Berezinskii–Kosterlitz–Thouless transition in the planar XY model

The '''Berezinskii–Kosterlitz–Thouless (BKT) transition''' of the two-dimensional XY model, in which unit vectors in the plane interact through the cosine of their angle differences, separates a high-temperature phase with exponentially decaying correlations from a low-temperature phase with power-law decay and no long-range order. Physics predicts that at the transition the spin correlation decays with exponent 1/4 and a logarithmic correction, that the correlation length has an essential singularity, and that the renormalized stiffness takes the universal value 2/π. Six manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), listed as its result 216 and dated 24 September and 5 October 2026, claim these predictions for the nearest-neighbour cosine model on the square lattice, at the critical inverse temperature defined by vanishing mass. One of them also claims Gaussian scaling limits for discrete Gaussian height models up to and including the roughening threshold. None of the six claims has a formal statement in the collection. This page summarizes the published claims and lists publicly available follow-up work.

== Background ==
* As recounted in the manuscripts' introductions:
** ''Predictions.'' Berezinskii derived algebraic spin correlations with a temperature-dependent exponent in a low-temperature quadratic approximation. Kosterlitz and Thouless separated spin waves from vortices and described the transition as the unbinding of vortex pairs. Kosterlitz's recursion for the vortex activity and the stiffness predicted the essential singularity of the correlation length, and Nelson and Kosterlitz predicted the universal limit 2/π of the renormalized stiffness from the low-temperature side.
** ''The rigorous phase picture.'' McBryan and Spencer proved power-law upper bounds for planar spin correlations. Fröhlich and Spencer proved the existence of the low-temperature algebraic phase for the plane rotator and Villain models by a Coulomb-gas multiscale analysis; Kharash and Peled give a modern account for the discrete Gaussian and Villain cases. Van Engelenburg and Lis obtained an exponential-versus-polynomial dichotomy by currents and dual heights, with a polynomial lower bound at the threshold whose exponent is not sharp. Aizenman, Harel, Peled and Shapiro developed integer-restricted Gaussian and level-set methods for depinning and the dual spin phase. Lammers related the XY mass exactly to the mass of the dual height field and proved a localization dichotomy with a universal variance gap, and Durand and Lammers expressed the XY mass through a derivative of the dual-height surface tension.
** ''Renormalization and height fields.'' Dimock and Hurd controlled the infrared flow of the sine-Gordon model with a fixed ultraviolet cutoff, and Falco constructed a critical trajectory at small activity for the lattice Coulomb gas, with logarithmic corrections for fractional charges. Bauerschmidt, Park and Rodriguez proved the torus Gaussian limit of discrete Gaussian heights at sufficiently high temperature; their Conjecture 1.3 predicts the limit through a critical temperature, with the endpoint value and an infinite right slope, for sufficiently spread-out interactions.
** ''Spin fields.'' Imaginary Gaussian multiplicative chaos was constructed by Lacoin, Rhodes and Vargas and developed by Junnila, Saksman and Webb. Garban and Sepúlveda stated and sketched a joint limit of the zero-boundary discrete Gaussian free field and its Wick exponential and discussed the expected connection with the Villain model.

== The original results ==
* '''The critical exponent (24 September 2026).''' ''The critical correlation exponent of the planar XY model'' claims that, at the critical inverse temperature defined by vanishing mass, the infinite-volume spin correlation at separation ''n'' is ''n''<sup>−1/4+o(1)</sup>, with the infinite-volume limit taken through free square boxes before the separation limit. It uses the van Engelenburg–Lis lower bound at the threshold as an external input, and the BKT-universality companion for finite Gaussian comparisons, annular estimates and local analytic maps.
* '''Height fields and low-temperature spin fields (24 September 2026).''' ''BKT universality for height and planar spin fields'' claims Gaussian scaling limits for the two-dimensional discrete Gaussian height model throughout its rough phase, including the roughening threshold, for every finite square-symmetric interaction set containing the nearest neighbours, with the critical effective temperature equal to the universal value 8π after the natural lattice normalization. It states that its height theorem gives, for every admissible interaction including nearest neighbours, the fixed-interaction conclusions predicted by Conjecture 1.3 of Bauerschmidt, Park and Rodriguez and the discussion following it. It also claims that, for the nearest-neighbour Villain and XY models at sufficiently low fixed temperatures, the Green-function-normalized spin fields converge to the imaginary exponential of a Dirichlet Gaussian free field.
* '''The logarithmic correction (5 October 2026).''' ''The critical logarithmic correction for the planar XY model'' claims that the critical two-point correlation along a coordinate axis, taken by the free-box thermodynamic limit, satisfies ''C''(''r'') = ''B''<sub>XY</sub> ''r''<sup>−1/4</sup>(log ''r'')<sup>1/8</sup>(1 + ''o''(1)) with a constant 0 < ''B''<sub>XY</sub> < ∞.
* '''The essential singularity (5 October 2026).''' ''Essential singularity of the correlation length in the planar XY model'' claims that the mass ''m''(''b''), taken by the free-box thermodynamic limit followed by the separation limit along an axis, satisfies √(''b''<sub>c</sub> − ''b'') log(1/''m''(''b'')) → ''A''<sub>XY</sub> as the inverse temperature ''b'' increases to ''b''<sub>c</sub>, where ''A''<sub>XY</sub> is a finite, strictly positive model-specific constant that is not evaluated in closed form. It uses the critical-exponent and logarithmic-correction companions as inputs.
* '''The critical spin field (5 October 2026).''' ''The critical spin field of the planar XY model'' claims that, at the critical inverse temperature, the spin field on a square with boundary angles fixed to zero, normalized by the exact center magnetization and the lattice Green-function factor, converges in law in ''H''<sup>−3</sup><sub>loc</sub> to the full-variance imaginary exponential of a zero-Dirichlet Gaussian free field with stiffness 2/π, together with all mixed moments and the joint laws of smeared fields. It uses the companions' critical height and pin-limit theorems.
* '''The center magnetization (5 October 2026).''' ''Critical center magnetization in the planar XY model'' claims, assuming stated critical-height, local-renormalization and spin-field inputs from the companion papers, that the magnetization at the center of a square of side ''n'' with aligned boundary spins at the critical threshold is a finite, strictly positive model-specific constant times ''n''<sup>−1/8</sup>(log ''n'')<sup>1/16</sup>(1 + ''o''(1)).
* '''Sources.''' Commit <code>7ef5c0c</code> of the openai/math repository:
** [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/The-critical-correlation-exponent-of-the-planar-XY-model-September-24-2026 ''The critical correlation exponent of the planar XY model''];
** [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/BKT-universality-for-height-and-planar-spin-fields-September-24-2026 ''BKT universality for height and planar spin fields''];
** [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/The-critical-logarithmic-correction-for-the-planar-XY-model-October-5-2026 ''The critical logarithmic correction for the planar XY model''];
** [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026 ''Essential singularity of the correlation length in the planar XY model''];
** [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026 ''The critical spin field of the planar XY model''];
** [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/Critical-Center-Magnetization-in-the-Planar-XY-Model-October-5-2026 ''Critical center magnetization in the planar XY model''].
* '''Formalization.''' The repository has no Lean scope note or Comparator challenge for result 216.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 30 October 2026)
|-
| 2026-10-06 || [https://lowrank.github.io/latent-seminar/OpenAI-Math/216-xy-bkt ''The planar XY model: critical logarithms and the BKT transition''], an article of the Latent Seminar blog at lowrank.github.io || Reading (exposition) || Explains family 216: correlations, correlation length and dual heights; the claims ordered by strength, which connect the critical power law, the logarithmic correction, the spin field and the rapidly diverging correlation length; why a marginal scale flow produces a logarithm; the essential singularity and the continuum field; and key steps and proof outlines. It notes that some companions explicitly use field-limit and renormalization inputs supplied elsewhere in the family. || Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs.
|-
| 2026-10-16
| J. Jardine, Lattice24, referee reviews [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_306.json rank_306] (critical exponent), [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_150.json rank_150] (height and spin fields), [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_095.json rank_095] (logarithmic correction), [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_151.json rank_151] (essential singularity), [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_148.json rank_148] (critical spin field) and [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_104.json rank_104] (center magnetization) in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| All six reviews report no gap and classify their issues as minor. Each states which parts it read in full and which it only skimmed.
* The five reviews of the papers other than the height-field paper record that load-bearing inputs are imported from companion papers in the family and were not re-audited there, among them the critical coefficient 8π of the dual height field and the analytic local maps of the renormalization flow.
* The review of the critical-exponent paper states that the hard-pin transfer and the vanishing band interaction rest on a vanishing-reach argument and band-density bounds that are only sketched, and that openness relies on a uniformity which the paper derives from the companion's proof rather than its stated theorem.
* The review of the height-field paper names the entry theorem, on which every endpoint conclusion depends, as the main load-bearing input that it could not check line by line.
* The review of the center-magnetization paper notes that the result is explicitly conditional and that it only spot-checked the analytic entry construction.
| Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.
|-
| 2026-10-17 || ''AI Proved What?'', episode 216 [https://aiprovedwhat.com/episodes/216-xy-critical-scaling-bkt-universality/ Exactly one quarter?], remade version of 22 October 2026 || Reading (exposition) || Video explainer of six OpenAI preprints, 461 pages, on the square-lattice nearest-neighbour XY model: the critical exponent exactly 1/4, the logarithmic correction with power 1/8, the essential singularity of the correlation length, the critical spin field as a continuum object with stiffness 2/π, and the magnetization at the centre of a critical square, together with a 169-page foundation paper claiming that integer height models become Gaussian fields up to and including their roughening point for every finite symmetric range, which it says covers a conjecture of Bauerschmidt, Park and Rodriguez. The episode reports its own simulations at the critical point on grids up to 512 across, in which the decay looks like a power near −0.21 rather than −1/4, and states that the papers do not compute the critical temperature, that they call the constants in front model-specific, and that no other lattice is claimed for the spin results. || Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as preprints: OpenAI, six papers, 461 pages, dated September 24 and October 5, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: not formalized. The page states that none of the six preprints has been revised since the repository's initial commit of 6 October 2026, that none is among those OSVerify reviewed, that it found no expert reaction or earlier claim, and that it checked that all 302 pinpoint citations between the papers point to existing statements and checked the constants, but not the analytic estimates, which it puts at about 400 of the pages.
|}

== Notes ==
* The background summarizes the manuscripts' own accounts of earlier work.
* For the O(''n'') models with ''n'' ≥ 3, where mass generation is predicted at every temperature, see [[Mass generation in two-dimensional O(n) models]].
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]