Result 268, Mathematical physics

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=3/5h=3/5 and gives boundary-selected infinite-volume states with topological index −1.

Proof

The bigger picture

Why it matters

A long chain of quantum spins can retain a nonzero energy cost for excitation even as it grows. These unreviewed manuscripts claim this behavior for a central spin-one model, with carefully specified boundary conditions.

What changes?

The periodic manuscript reports a uniform positive spectral gap, the energy separation above the ground state, for the pure antiferromagnetic spin-one Heisenberg chain on even periodic rings. Each site has three local spin states; antiferromagnetic interactions favor opposing neighboring spins. Uniform means a positive lower bound independent of length. The companion reports a gap greater than the natural logarithm of 10 divided by 392 for every odd open chain of at least 1,921 sites, with identical endpoint magnetic fields of strength 3/5.

What does that help mathematicians do?

For even rings, the claimed bound rules out excitation energies collapsing toward the ground-state energy as the system grows. The companion also uses Tasaki's index theorem to assign topological index -1 to every subsequential local limit of its boundary-selected ground states. These limits describe infinite chains through observations on finite regions. This connects the controlled open-chain construction to a specific topological classification, rather than merely establishing an energy bound.

Are there practical applications?

The immediate value is foundational for quantum magnetism: the claims would establish a persistent excitation threshold in the specified Heisenberg chains and identify a topological index for boundary-selected infinite-chain states. They provide mathematical information about this model, not a demonstrated technology or a guarantee for real magnetic materials with additional interactions.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The periodic spin-one Haldane gap

September 24, 2026 30 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 24, 2026 28 pages

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=3/5h=3/5 gives a gap greater than log⁡(10)/392\log(10)/392 on every chain of 2L+12L+1 sites with L ≥ 960. Tasaki's index theorem then gives index −1 for every subsequential local limit of these boundary-selected ground states.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.