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The spin-one Haldane gap

Claims the spin-one Haldane gap: the Heisenberg chain on even rings has an energy gap that stays above log(20)/784.

2 papers September 24, 2026 Explained by AI

Haldane predicted in 1983 that chains of quantum spins behave differently for whole-number and half-number spin. For spin one, the lowest energy level should stay a fixed distance below the next, however long the chain. Numerical work strongly supports this, but no proof existed for the plain Heisenberg chain. Two papers claim to prove it with explicit numerical bounds, on rings and on open chains with end fields. There is no Lean check.

First step
1983
F. D. M. Haldane, prediction that integer-spin antiferromagnetic chains have a gap
Significance
Major
The Haldane gap conjecture for the spin-one Heisenberg chain is a famous open problem in mathematical physics.
Trust
Claimed, not checked
Field
Mathematical physics
2 papers

The explanation

Take two spin-one particles, tiny quantum magnets, joined by the Heisenberg interaction S₁ · S₂. Their total spin S can be 0, 1 or 2. The energy is (S(S + 1) − 4)/2, so the levels are −2, −1 and +1. The pair prefers opposite alignment, the singlet with S = 0. The gap, the step from the lowest energy to the next, is 1. Now put L such spins on a ring, each joined to its two neighbours. Does the gap stay above a fixed size as L grows, or shrink to zero?

Haldane predicted that the answer depends on the spin. For half-number spins, like spin one-half, the gap closes. Lieb, Schultz and Mattis gave the rigorous half-number result for spin one-half. For spin one, Haldane predicted a gap that stays positive. White and Huse estimated it numerically at 0.41050. The Affleck–Kennedy–Lieb–Tasaki chain, a modified model with an extra interaction term, has a proven gap. Small changes of that model keep the gap. But that does not give a gapped path to the plain Heisenberg chain.

The obstacle is that the plain chain is not frustration-free. That means its ground state does not minimize every neighbour interaction at once. The standard finite-size methods for gaps work for frustration-free chains. The plain Heisenberg ring needs a different tool.

The first paper claims a uniform positive gap on even rings. For every even L ≥ 60, the ground state is unique and the gap exceeds (4/105) log(80/79), about 0.000479. For every even L ≥ 2304, the gap exceeds log(20)/784, about 0.00382. These bounds are far below the numerical 0.41, but they do not shrink with L. The paper says this resolves the conjecture in the even-periodic formulation.

The second paper treats open chains with 2L + 1 sites. A magnetic field of strength 3/5 acts on both end spins. The claimed gap exceeds log(10)/392, about 0.00587, for every L ≥ 960. This is the setting Tasaki used to link the gap to a topological index. With the gap, Tasaki's theorem gives index −1 for the limiting states.

How sure is this? There is no Lean formalization for these papers. The proofs combine an analytic argument with stated finite certificates for the starting lengths. They rest on two papers, and no specialist review of them has been published.

What it doesn't show. This does not compute the true size of the gap, about 0.41, and it does not cover other spin values or other interactions.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds. There is no Lean check; the gap bounds rest on two papers with finite certificates and no published specialist review.

Words used

Spin one
A quantum magnet whose spin along any axis can take three values: −1, 0 or +1.
Heisenberg chain
A line or ring of spins where each neighbouring pair has energy S_j · S_(j+1), favouring opposite alignment.
Spectral gap
The step from the lowest energy to the next distinct energy.
Haldane gap
The predicted positive gap of integer-spin antiferromagnetic chains that does not shrink with chain length.
Frustration-free
A system whose ground state minimizes every local interaction term at the same time.
AKLT chain
A spin-one chain with an extra squared interaction term, built so its gap can be proved exactly.
Topological index
A whole number attached to a gapped phase that cannot change without closing the gap.

The original papers (2)

All quoted lines (12)
  1. Haldane predicted that antiferromagnetic Heisenberg chains have different low-energy behavior for integer and half-odd-integer spins.
    The periodic spin-one Haldane gap Back to text
  2. The half-odd-integer side has a rigorous antecedent in the low-energy excitation bound of Lieb, Schultz and Mattis for the cyclic spin-one-half chain.
    The periodic spin-one Haldane gap Back to text
  3. For the pure spin-one chain, White and Huse's density-matrix renormalization-group study estimated a bulk gap of 0.41050(2)0.41050(2) and exhibited effective spin-one-half degrees of freedom at open ends.
    The periodic spin-one Haldane gap Back to text
  4. The Affleck–Kennedy–Lieb–Tasaki construction established a rigorous gapped spin-one chain with a biquadratic interaction.
    The periodic spin-one Haldane gap Back to text
  5. This controls a neighborhood of that interaction; it does not establish a gapped path to the pure bilinear point.
    The periodic spin-one Haldane gap Back to text
  6. Finite-volume methods for uniform spectral gaps have a substantial history, including the criteria of Knabe and Nachtergaele for frustration-free chains, whose ground states minimize every local interaction.
    The periodic spin-one Haldane gap Back to text
  7. 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.
    The periodic spin-one Haldane gap Back to text
  8. For the Hamiltonians (…), the ground state is unique for every even L≥60L\ge60, and γL>4105log⁡8079for every even L≥60,γL>log⁡20784for every even L≥2304.\begin{aligned}\gamma_L&>\frac4{105}\log\frac{80}{79} &&\text{for every even }L\ge60,\\ \gamma_L&>\frac{\log20}{784} &&\text{for every even }L\ge2304.\end{aligned}
    The periodic spin-one Haldane gap Back to text
  9. Theorem … resolves the spin-one Haldane gap conjecture positively in the even-periodic finite-volume formulation Δ1>0\Delta_1>0.
    The periodic spin-one Haldane gap Back to text
  10. 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≥960L\ge960.
    A boundary-field gap for the spin-one Heisenberg chain Back to text
  11. This is the boundary formulation used by Tasaki to connect the Haldane gap problem to a nontrivial topological index.
    A boundary-field gap for the spin-one Heisenberg chain Back to text
  12. Tasaki's index theorem then gives index −1-1 for every subsequential local limit of these boundary-selected ground states.
    A boundary-field gap for the spin-one Heisenberg chain Back to text

Other sources

  1. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

Browse Mathematical physics Eigenvalues and spectral theoryOpen since the 1980sClaimed, not yet checked