Tóth's phase-transition conjecture for spontaneous magnetization

About 16 years old · traced to 2010

Let $m(\beta)$ denote the spontaneous magnetization of the quantum Heisenberg ferromagnet, represented through the interchange process on $[-r,r]^3$ at inverse temperature $\beta$ . Let $c_\beta(0)$ be the size of the cycle containing $0$ at time $\beta$ , and let $s_k(\beta)$ be the number of cycles of length $k$ . Tóth's conjecture. The function $m(\beta)$ admits a phase transition: there exists some $\beta_c$ such that

m(\beta)=0\quad\text{for }\beta<\beta_c,

and

m(\beta)>0\quad\text{for }\beta>\beta_c.

This conjecture concerns the existence of an ordered, or high- $\beta$ , phase in the quantum Heisenberg ferromagnet. The existence of that phase remains open; the main unresolved issue is not sharpness or uniqueness of the transition, but whether the high- $\beta$ phase exists at all.

References

Primary source

Gil Alon and Gady Kozma, “The probability of long cycles in interchange processes”, arXiv:1009.3723 (2012).

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RemarkAI-assistedClaimed by OpenAI. Claims Bloch's low-temperature T^(3/2) spontaneous-magnetization law for every fixed positive quantum spin in three-dimensional Heisenberg ferromagnets with nonnegative symmetric finite-range generating interactions, including anisotropic couplings, with the exact dispersion-dependent coefficient.See full solutionHide full solution

Claimed by OpenAI. Claims Bloch's low-temperature T^(3/2) spontaneous-magnetization law for every fixed positive quantum spin in three-dimensional Heisenberg ferromagnets with nonnegative symmetric finite-range generating interactions, including anisotropic couplings, with the exact dispersion-dependent coefficient.

Scope relative to this problem: This claims the leading Bloch T^(3/2) low-temperature magnetization law for each fixed positive spin in three dimensions, for nonnegative symmetric finite-range generating interactions, including anisotropic couplings. It gives the dispersion-dependent coefficient. It is ordered-phase observable progress and does not by itself determine a sharp critical temperature or identify every target interchange-process convention.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026/bloch-law-heisenberg.pdf

OpenAI-271-01-Bloch-s-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions.pdf616,873 bytesOpen

RemarkAI-assistedClaimed by OpenAI. For every fixed positive quantum spin in the three-dimensional nearest-neighbor Heisenberg ferromagnet, claims that the low-temperature magnetization deficit agrees with the ideal-magnon density through order beta^(-5/2), giving the first lattice correction to Bloch’s law.See full solutionHide full solution

Claimed by OpenAI. For every fixed positive quantum spin in the three-dimensional nearest-neighbor Heisenberg ferromagnet, claims that the low-temperature magnetization deficit agrees with the ideal-magnon density through order beta^(-5/2), giving the first lattice correction to Bloch’s law.

Scope relative to this problem: This is the first lattice correction for the three-dimensional nearest-neighbor quantum Heisenberg ferromagnet at each fixed positive spin: the magnetization deficit agrees with ideal-magnon density through beta^(-5/2). It is low-temperature refinement, not a complete all-temperature phase diagram or an inferred interchange-process identity.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-first-lattice-correction-to-Blochs-law-October-5-2026/first-lattice-correction-bloch-law.pdf

OpenAI-271-02-The-first-lattice-correction-to-Bloch-s-law.pdf526,174 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Claims a uniform-direction, deterministic-positive-radius limiting magnetization law for the three-dimensional nearest-neighbor isotropic quantum Heisenberg ferromagnet at every fixed positive quantum spin and sufficiently low positive temperature, taking growing even periodic cubes.See full solutionHide full solution

Claimed by OpenAI. Claims a uniform-direction, deterministic-positive-radius limiting magnetization law for the three-dimensional nearest-neighbor isotropic quantum Heisenberg ferromagnet at every fixed positive quantum spin and sufficiently low positive temperature, taking growing even periodic cubes.

Scope relative to this problem: This claims a deterministic positive-radius, uniform-direction limiting magnetization distribution in growing even periodic three-dimensional cubes, at fixed positive spin and sufficiently low positive temperature. It is ordered-phase progress with these boundary/limit conventions; it does not supply a sharp beta_c value or every all-temperature assertion in the target interchange representation.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet-October-5-2026/spherical-magnetization.pdf

OpenAI-271-03-The-spherical-magnetization-law-for-the-three-dimensional-quantum-Heisenberg-ferromagnet.pdf660,549 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Claims spontaneous magnetization in the nearest-neighbor isotropic quantum Heisenberg ferromagnet in every dimension at least three and every positive quantum spin, with a translation-invariant zero-field KMS equilibrium state of magnetization at least one quarter of the spin at sufficiently low positive temperature.See full solutionHide full solution

Claimed by OpenAI. Claims spontaneous magnetization in the nearest-neighbor isotropic quantum Heisenberg ferromagnet in every dimension at least three and every positive quantum spin, with a translation-invariant zero-field KMS equilibrium state of magnetization at least one quarter of the spin at sufficiently low positive temperature.

Scope relative to this problem: This claims a translation-invariant zero-field KMS equilibrium state with magnetization at least one quarter of the spin, for the nearest-neighbor isotropic model in every dimension at least three and each positive spin at sufficiently low positive temperature. It gives an ordered low-temperature phase in that state sense; no sharp beta_c or identification with all finite-box interchange/boundary conventions is inferred.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026/paper.pdf

OpenAI-271-04-Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet.pdf775,556 bytesOpen