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The first lattice correction to Bloch's law

OpenAI · October 5, 2026

Lean

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Abstract

We prove the first lattice correction to Bloch's law for the three-dimensional nearest-neighbor quantum Heisenberg ferromagnet at every fixed spin $S=\tfrac12,1,\tfrac32,\ldots$. The spontaneous-magnetization deficit agrees with the full ideal-magnon density up to $o(\beta^{-5/2})$. In addition to the leading Bloch term, this gives the correction $3\zeta(5/2)(\beta S)^{-5/2}/(128\pi^{3/2})$. The magnetization is the right derivative at zero field of the thermodynamic pressure; the volume limit precedes the field derivative, and the low-temperature limit is taken last.

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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 stated scope covers nearest-neighbor spontaneous magnetization in dimensions $d\ge3$, with a translation-invariant zero-field KMS state of magnetization at least $S/4$, and convergence of the dynamics; the lattice-correction asymptotic of this paper is not in its scope. [family-level]

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Cite

@misc{OAI:The-first-lattice-correction-to-Blochs-law-October-5-2026,
  author = {{OpenAI}},
  title = {{The first lattice correction to Bloch's law}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026/first-lattice-correction-bloch-law.pdf}{OAI:The-first-lattice-correction-to-Blochs-law-October-5-2026}},
  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.

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