The spin-one Haldane gap

2 papers

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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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

The periodic spin-one Haldane gap

OpenAI · September 24, 2026

We prove a uniform positive spectral gap for the pure antiferromagnetic spin-one Heisenberg chain on even periodic rings, establishing the positive even-periodic formulation of the spin-one Haldane conjecture.

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@misc{OAI:The-periodic-spin-one-Haldane-gap-September-24-2026,
  author = {{OpenAI}},
  title = {{The periodic spin-one Haldane gap}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/paper.pdf}{OAI:The-periodic-spin-one-Haldane-gap-September-24-2026}},
  year = {2026}
}

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

OpenAI · September 24, 2026

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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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}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.