The Haldane gap conjecture for the spin-one Heisenberg antiferromagnetic chain
For the spin-one antiferromagnetic Heisenberg chain with spin-one matrices normalized by , Haldane predicted in 1981 (published 1983) that integer-spin chains, unlike half-odd-integer ones, have a unique ground state separated from the rest of the spectrum by a gap that stays positive in the thermodynamic limit. AKLT proved such a gap for a modified model with a biquadratic term, Yarotsky for small perturbations of it, and numerics put the Heisenberg gap near , but no proof existed for the pure bilinear model. Writing for the gap above the ground energy on a periodic ring of even length : is ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Quantum spin chains; spectral gaps
- Posed by
- F. D. M. Haldane (1981 ILL preprint; Phys. Rev. Lett. and Phys. Lett. A, 1983)
- Year posed
- 1981
- Years open
- 45y
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 62 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims that the pure bilinear spin-one Heisenberg chain on even periodic rings has a unique ground state for and a gap bounded below by explicit constants, so , with a corresponding lower bound for local excitations in subsequential thermodynamic limits. The companion proves Tasaki's boundary-field gap for odd open chains at and the nontrivial index . The constants are far below the numerical value of about . It does NOT treat odd periodic rings, spin above one, or the bilinear-biquadratic family away from the Heisenberg point.
What the AI did
The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the Hodge conjecture for CM abelian varieties and the zeta zero-free region work) do not concern this family. Both manuscripts are credited to OpenAI with no human author named. Each ships finite certificates (thermal trace enclosures and matrix-product trial energies) with a Python verifier in its verification folder.
Verification
No independent mathematician has checked this yet. Theorem 1.1 was read against the posed problem: it claims a unique ground state and for every even , and for even , so the even-periodic gap is uniformly positive. The manuscript calls this the positive even-periodic formulation; odd periodic rings are not covered. The proof rests on finite rigorous computations (interval enclosures of matrix-exponential traces and explicit integer matrix-product trial states) fed into an analytic bootstrap; the verifier was not run here. No Lean formalization. The companion was read at the level of its main theorem: a gap above for odd open chains of sites, , with endpoint field , which is Tasaki's Assumption 2 and gives Tasaki index .