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Spontaneous magnetization in the quantum Heisenberg ferromagnet
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Abstract
For every dimension d ≥ 3 and every spin $S\in\{\frac12,1,\frac32,\ldots\}$, we prove that the nearest-neighbor isotropic quantum Heisenberg ferromagnet has a translation-invariant, spontaneously magnetized equilibrium state at every sufficiently low positive temperature. The same state satisfies the KMS condition for the zero-field dynamics and has magnetization at least $S/4$. This resolves the low-temperature ordering problem in the spontaneous-magnetization formulation.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI, Family 271 Lean scope note (OpenAI math release, 2026-10-06). The release's Lean development covers this paper's Theorem 1.1 (nearest-neighbor, translation-invariant zero-field β-KMS state with magnetization at least $S/4$, every $d\ge3$ and every spin) and convergence of the finite-volume dynamics; it does not cover the companion Bloch-law, lattice-correction or spherical-law results. [family-level]
- OpenAI, OAI/MathematicalPhysics/Heisenberg/Main.lean (openai/math, 2026). Main Lean file of the release, stating
spontaneous_magnetizationfor d ≥ 3 and ℓ = 2S ≥ 1 slots, with the bound ℓ/8 on the one-site z-spin expectation.
Possibly relevant literature (found by the agent, not checked)
- B. Simon, Fifteen Problems in Mathematical Physics (Perspectives in Mathematics, Anniversary of Oberwolfach 1984, Birkhäuser). Problem 5D asks for long-range order of the quantum Heisenberg model for ν ≥ 3 and large β in a two-point-function formulation, a different formulation from the one-point KMS-state statement proved here.
- P. Federbush, For the Quantum Heisenberg Ferromagnet, Tao to the Proof of a Phase Transition (arXiv:math-ph/0202044, 2002). Outlines an unfinished programme for the 3D phase transition via a polymer expansion over permutation cycles, an earlier cycle-based approach to the problem this paper addresses.
- D. Ueltschi, Random loop representations for quantum spin systems (J. Math. Phys. 54, 083301, 2013). Proves occurrence of macroscopic loops on the cubic lattice in d ≥ 3 for certain SU(2)-invariant spin-1 models via a loop representation, a related use of probabilistic representations to obtain long-range order.
Digests and expositions
- Azmat, Quantum Heisenberg Ferromagnet: How OpenAI’s Proof Settled a Decades-Old 3D Ordering Problem (Binary Verse AI, 2026-10-15). This exposition follows the manuscript’s exchange-loop, pinning, return-estimate and KMS-state construction and states that the site has not independently compiled or audited the complete formalization.
News and blogs
- N. Baf, OpenAI Dumped 722 Math Papers on Mathematicians. Three Were Pulled in 24 Hours. (The NovTech, 2026-10-12). Cites the catalogue's claim for this paper and recounts the Dyson–Lieb–Simon history of the problem; it quotes no expert assessment of the paper.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 271
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026,
author = {{OpenAI}},
title = {{Spontaneous magnetization in the quantum Heisenberg ferromagnet}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026/paper.pdf}{OAI:Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/271.md. The Lean statements have not been checked by people against the paper's claims.
Family 271: Bloch's law, its lattice correction, and the spherical magnetization law
Proves Bloch's T3/2 law with its exact coefficient for three-dimensional quantum Heisenberg ferromagnets at every positive quantum spin, allowing nonnegative symmetric finite-range couplings whose support generates ℤ3. The thermodynamic limit precedes the zero-field derivative and low-temperature limit. The family also proves spontaneous magnetization for nearest-neighbor models in every dimension d ≥ 3 and determines the first lattice correction for three-dimensional nearest-neighbor couplings.
Other papers in this family:
- Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions · October 5, 2026
- The first lattice correction to Bloch's law · October 5, 2026
- The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet · October 5, 2026