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.

Manuscripts (2)

The periodic spin-one Haldane gap

September 24, 2026 · source · cite as \cite{OAI:The-periodic-spin-one-Haldane-gap-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.

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

September 24, 2026 · source · cite as \cite{OAI:A-boundary-field-gap-for-the-spin-one-Heisenberg-chain-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.

Result 268 · Mathematical physics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.