The spin-one Haldane gap

What OpenAI claims

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.

Papers (2)

1. The periodic spin-one Haldane gap

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

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

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.