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Bloch's T3/2T^{3/2} law for the three-dimensional quantum Heisenberg ferromagnet

Bloch's spin-wave theory (1930) predicts that the spontaneous magnetization of a three-dimensional ferromagnet falls below saturation by an amount proportional to T3/2T^{3/2}, with a coefficient fixed by the quadratic one-magnon dispersion; for the nearest-neighbour spin-SS Heisenberg model the deficit is ζ(3/2)(8π3/2S3/2)−1T3/2\zeta(3/2)(8\pi^{3/2}S^{3/2})^{-1}T^{3/2} to leading order. Dyson (1956) analysed spin-wave interactions and the higher-order terms, but not as a rigorous fixed-spin proof, and Correggi-Giuliani-Seiringer (2015) proved the corresponding free-energy asymptotics, which do not control the zero-field derivative. Is it true, for the quantum Heisenberg ferromagnet on Z3\mathbb Z^3 at fixed spin, that S−m(β)∼c β−3/2S-m(\beta)\sim c\,\beta^{-3/2} as β→∞\beta\to\infty with Bloch's coefficient, where mm is the spontaneous magnetization?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Quantum spin systems; spin-wave theory
Posed by
Felix Bloch (1930), spin-wave prediction; developed by Holstein-Primakoff (1940) and Dyson (1956)
Year posed
1930
Years open
96y
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
32 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every fixed spin SS and every nonnegative symmetric finite-range coupling JJ on Z3\mathbb Z^3 whose support generates Z3\mathbb Z^3, lim⁡β→∞β3/2(S−mS,J(β))=ζ(3/2)/(8π3/2det⁡DS,J)\lim_{\beta\to\infty}\beta^{3/2}(S-m_{S,J}(\beta))=\zeta(3/2)/(8\pi^{3/2}\sqrt{\det D_{S,J}}), with DS,J=S2∑zJ(z)zzTD_{S,J}=\frac S2\sum_z J(z)zz^{\mathsf T}; nearest-neighbour gives ζ(3/2)/(8π3/2S3/2)\zeta(3/2)/(8\pi^{3/2}S^{3/2}). Companions: the first lattice correction 3ζ(5/2)(βS)−5/2/(128π3/2)3\zeta(5/2)(\beta S)^{-5/2}/(128\pi^{3/2}) for nearest-neighbour couplings, deficit equal to the ideal-magnon density up to o(β−5/2)o(\beta^{-5/2}); and a spherical magnetization law for the symmetric zero-field magnetization on periodic cubes. Not shown: Dyson's T4T^4 interaction term, dimensions other than three, or uniformity in SS.

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 principal manuscript and two companions (lattice correction, spherical law) are dated October 5, 2026; they reuse the finite-set algebra and lattice geometry of the September 24 ordering manuscript in the same family.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the Bloch's law manuscript read against Bloch's prediction as the manuscript states it. The family's only Lean challenge (Heisenberg) states spontaneous magnetization with ω(S0z)≥S/4\omega(S^z_0)\ge S/4, not the T3/2T^{3/2} asymptotic, so this entry is unreviewed. The magnetization is defined as the right derivative at zero field of the infinite-volume pressure; the order of limits (volume, then field, then temperature) is part of the claim. The proof imports results of the companion ordering manuscript, also unrefereed.

Sources

Changelog1 change

Discussion