The spin-one Haldane gap
The spin-one Haldane gap is the subject of two manuscripts dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math). Haldane predicted that antiferromagnetic Heisenberg chains of integer spin have a positive spectral gap above the ground energy, unlike chains of half-odd-integer spin. The manuscripts claim a uniform gap for the pure spin-one chain on even periodic rings, and a gap for odd open chains with a fixed field at both ends. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
The original results
- Claim (periodic rings). For the pure antiferromagnetic spin-one Heisenberg chain HL = Σj Sj · Sj+1 on an even periodic ring of length L, the ground state is unique for every even L ≥ 60, and the gap γL between the least and the next distinct energy satisfies γL > (4/105) log(80/79) for every even L ≥ 60 and γL > (log 20)/784 for every even L ≥ 2304 (Theorem 1.1). The manuscript states that this resolves the spin-one Haldane gap conjecture in its even-periodic finite-volume formulation. Source: the manuscript The periodic spin-one Haldane gap, dated 24 September 2026. Its proof combines partition functions written as spatial moments, two purity estimates and two finite thermal calculations (appendices A and B).
- Claim (boundary field). For odd open chains of 2L + 1 sites with the same field h = 3/5 at both endpoints, the gap exceeds (log 10)/392 for every L ≥ 960 (Theorem 1.1); the manuscript then applies Tasaki's index theorem to obtain index −1 for every subsequential local limit of these boundary-selected ground states. Source: the manuscript A boundary-field gap for the spin-one Heisenberg chain, dated 24 September 2026.
- Formalization. The collection's Lean documentation has no scope note for this family.
Earlier published results
- As recounted in the periodic manuscript: Lieb, Schultz and Mattis, and Affleck and Lieb, gave the rigorous obstruction for half-odd-integer chains; the Affleck–Kennedy–Lieb–Tasaki chain, with an added biquadratic interaction, is rigorously gapped; Yarotsky proved that the gap persists under small translation-invariant perturbations of that interaction, which does not reach the pure bilinear point. Numerical studies (for example White and Huse) estimate the bulk gap of the pure chain at about 0.4105.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 15 October 2026) |
|---|---|---|---|---|
| 2026-10-07 (repository), 2026-10-08 (version 1.7.0) | amachino/haldane-gap-refinement (GitHub) | Extension (conditional) | A technical note described in its README as "autonomously researched and written by GPT-6 Astra Max" and "unreviewed". Assuming the spatial-transfer identities of the OpenAI manuscripts and the certified finite inputs specified in the note, it states γL > 303/2000 = 0.1515 for every integer L ≥ 24, with a unique ground state, including odd lengths. The README states that these are lower bounds, not estimates of the physical gap, and that it claims neither priority nor an independent proof of the Haldane gap. | Unreviewed note (version 1.7.0). |
| 2026-10-10 | arXiv:2609.34119 (v2), H. Tasaki; titled The equivalence of topological indices for U(1)⋊Z2-symmetric quantum spin chains | Use; partial reading | Version 2 combines the boundary-field gap theorem of the OpenAI manuscript (field 3/5) with the author's earlier twist-index calculation, and a new variance estimate, to obtain a spin-rotation-invariant locally-unique gapped ground state of the infinite spin-one Heisenberg chain with both topological indices equal to −1. The paper states that only this application uses the OpenAI gap theorem and that the periodic result is context only. A footnote states: "At the time of this revision, I am studying these proofs in detail. I have so far found no logical gap in the portions I have examined, but my verification is not yet complete." Version 1 was posted on 28 September 2026; the paper discloses use of ChatGPT. | arXiv v2. |
Notes
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.