Browse › Mathematical physics › 271 Bloch's law, its lattice correction, and the spherical magnetization law
Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
We prove Bloch's T3/2 law for the spontaneous magnetization of three-dimensional quantum Heisenberg ferromagnets at every fixed positive quantum spin. The result holds for every nonnegative symmetric finite-range interaction whose support generates ℤ3, including spatially anisotropic couplings. The leading magnetization deficit has the exact coefficient determined by the determinant of the quadratic one-magnon dispersion. The thermodynamic limit is taken before the right field derivative at zero, and the low-temperature limit is taken last.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
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 Bloch-law asymptotic of this paper is not in its scope. [family-level]
Possibly relevant literature (found by the agent, not checked)
- M. Correggi and A. Giuliani, The Free Energy of the Quantum Heisenberg Ferromagnet at Large Spin (J. Stat. Phys. 149 (2012), 234–245). Proves the spin-wave free-energy asymptotic in 3D in the limit S → ∞ with βS fixed, a rigorous precursor to the fixed-spin results surveyed in this paper's historical section.
- P. Federbush, For the Quantum Heisenberg Ferromagnet, Tao to the Proof of a Phase Transition (arXiv:math-ph/0202044, 2002). Proposes a permutation-cycle approach with a random-walk recurrence heuristic in 3D versus 2D, related in spirit to the return-probability mechanism that determines the Bloch coefficient here.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 271
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026,
author = {{OpenAI}},
title = {{Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026/bloch-law-heisenberg.pdf}{OAI:Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-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.
Other papers in this family:
- 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
- Spontaneous magnetization in the quantum Heisenberg ferromagnet · September 24, 2026