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Spontaneous magnetization of the three-dimensional quantum Heisenberg ferromagnet at low temperature (Lieb's Problem A)

The nearest-neighbour quantum Heisenberg ferromagnet on Zd\mathbb Z^d has Hamiltonian H=−∑∣x−y∣=1Sx⋅SyH=-\sum_{|x-y|=1}\mathbf S_x\cdot\mathbf S_y with spin-SS matrices at each site. Mermin and Wagner excluded order at positive temperature for d≤2d\le2; Frohlich-Simon-Spencer proved ordering for the classical model in d≥3d\ge3 and Dyson-Lieb-Simon for quantum antiferromagnets, but the reflection-positivity method gives no infrared bound for the quantum ferromagnet. Lieb listed the problem in 1999 (Problem A), and it was still described as open in 2025. Does the quantum Heisenberg ferromagnet in dimension d≥3d\ge3, for every spin SS, order at sufficiently low positive temperature, i.e. have an equilibrium state with nonzero spontaneous magnetization at zero field?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Quantum spin systems; phase transitions
Posed by
Elliott H. Lieb (1999, Problem A of his list of statistical-mechanics problems); obstruction noted by Dyson, Lieb and Simon (1978)
Year posed
1999
Years open
27y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
58 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every d≥3d\ge3 and S∈{1/2,1,3/2,… }S\in\{1/2,1,3/2,\dots\} there is β0(d,S)\beta_0(d,S) such that for every β≥β0\beta\ge\beta_0 the zero-field nearest-neighbour dynamics has a translation-invariant β\beta-KMS state with ω(S0z)≥S/4\omega(S_0^z)\ge S/4. The proof uses an exchange-cycle (random interchange) representation with boundary pins, stable-polynomial negative dependence and a determinant identity for conditioned walks. Not shown: long-range order in the finite-volume two-point formulation, the critical temperature, or anything in d≤2d\le2. The companion manuscripts in the family prove Bloch's law and the spherical magnetization law in d=3d=3.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family has four manuscripts; this entry's principal one is dated September 24, 2026, and the three Bloch-law manuscripts (October 5, 2026) build on its finite-set algebra and lattice geometry.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of 'Spontaneous magnetization in the quantum Heisenberg ferromagnet' read against Lieb's problem as the manuscript cites it. Lean: the challenge lean/ComparatorChallenges/Heisenberg.json (solution_module OAI.MathematicalPhysics.Heisenberg.Main, present at the pinned commit) is not in the formalization catalogue lean/formalization.yaml. The statement OAI.Heisenberg.spontaneous_magnetization was read here: for d≥3d\ge3 and ℓ=2S≥1\ell=2S\ge1, finite-volume dynamics converge, and for all large β\beta there is a translation-invariant β\beta-KMS state with ω(S0z)\omega(S^z_0) real and at least ℓ/8=S/4\ell/8=S/4. That is the headline claim. Not rebuilt here. The paper itself says it resolves the problem in the KMS-state formulation and that the finite-volume two-point criterion in Seiringer's account is a different formulation it does not address.

Sources

Changelog1 change

Discussion