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A boundary-field gap for the spin-one Heisenberg chain

OpenAI · September 24, 2026

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Abstract

We prove a uniform spectral gap for odd open spin-one antiferromagnetic Heisenberg chains with the same fixed magnetic field at both endpoints. The field $h=3/5$ gives a gap greater than $\log(10)/392$ on every chain of $2L+1$ sites with L ≥ 960. Tasaki's index theorem then gives index −1 for every subsequential local limit of these boundary-selected ground states.

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-23) and not checked by a person. May be wrong or incomplete. Report an error.

Follow-up works

  • Hal Tasaki, The equivalence of topological indices for U(1)⋊Z₂-symmetric quantum spin chains (arXiv:2609.34119v2, 10 Oct 2026). Version 2 combines this paper's boundary-field gap theorem (which it says "establishes the boundary-field gap hypothesis used in our twist-index calculation") with an index-equivalence result to obtain a spin-rotation-invariant, locally-unique gapped ground state of the infinite S=1 Heisenberg chain with Ogata and twist indices −1, without assuming infinite-volume uniqueness; its revision note says of the OpenAI proofs: "I have so far found no logical gap in the portions I have examined, but my verification is not yet complete."
  • GPT-6 Astra Max / amachino, Exact-arithmetic refinements of periodic spin-one Haldane-gap bounds (GitHub technical note, version 1.7.0, 8 October 2026). This explicitly unreviewed note uses inputs from both manuscripts to claim a periodic gap greater than 0.1515 for every integer length at least 24, conditional on the cited OpenAI transfer identities and certified finite inputs. [AI-generated per source] [family-level]

Comments and discussion

  • Masaki Oshikawa (@MasakiOshikawa), post on X (6 Oct 2026). In Japanese, Oshikawa asks Hal Tasaki whether the results should be understood as proving the existence of the Haldane gap and that the S=1 Haldane phase is SPT. [family-level]

Possibly relevant literature (found by the agent, not checked)

  • T. Kennedy, Exact diagonalisations of open spin-1 chains (J. Phys.: Condens. Matter 2, 5737, 1990). This paper studies numerically the exponentially small splitting of the lowest levels of open spin-1 chains (including the Heisenberg point), the edge-state degeneracy that the endpoint fields of this paper are meant to lift.

Digests and expositions

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Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026,
  author = {{OpenAI}},
  title = {{A boundary-field gap for the spin-one Heisenberg chain}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026/paper.pdf}{OAI:A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-September-24-2026}},
  year = {2026}
}

From OpenAI's README for this paper:

Verification

The finite-computation programs require Python 3.11 or later with assertions enabled. The scalar and trial programs use the standard library; the thermal program also requires NumPy and SciPy. From this paper directory, use a fresh replay directory so the shipped certificates are not overwritten:

article_dir="$PWD"
verification_output="$article_dir/.verification-output/computation-replay"
mkdir -p "$verification_output"
env -u PYTHONOPTIMIZE python3 -B "$article_dir/verification/scalar_checks.py" "$verification_output"
env -u PYTHONOPTIMIZE python3 -B "$article_dir/verification/trial_verify.py" "$verification_output"
(cd "$verification_output" && env -u PYTHONOPTIMIZE python3 -B "$article_dir/verification/thermal_certificate_check.py")

Successful commands exit with status zero. Use the exact stored-certificate comparison in verification/README.md, which also lists prerequisites, expected files, and the finite checks' scope and limitations. These computations do not replace the analytic arguments.

Family 268: The spin-one Haldane gap

Proves the spin-one Haldane gap conjecture for the pure antiferromagnetic Heisenberg chain on even periodic rings: the spectral gap stays uniformly positive as the chain grows. A companion establishes a gap for odd open chains with endpoint field $h=3/5$ and gives boundary-selected infinite-volume states with topological index −1.

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