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Bloch's law, its lattice correction, and the spherical magnetization law
Proves Bloch's T 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 ℤ. 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.
Manuscripts in this family · 4
Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions
Oct 5, 2026·bloch-law-heisenberg.pdf
We prove Bloch's T 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 ℤ, 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.
The first lattice correction to Bloch's law
Oct 5, 2026·first-lattice-correction-bloch-law.pdf
We prove the first lattice correction to Bloch's law for the three-dimensional nearest-neighbor quantum Heisenberg ferromagnet at every fixed spin . The spontaneous-magnetization deficit agrees with the full ideal-magnon density up to . In addition to the leading Bloch term, this gives the correction . 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.
The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet
Oct 5, 2026·spherical-magnetization.pdf
We prove the spherical magnetization law for the three-dimensional nearest-neighbor isotropic quantum Heisenberg ferromagnet at every fixed positive quantum spin and every sufficiently low fixed positive temperature. As the even periodic cubes grow, the symmetric zero-field magnetization converges in moments to a uniform direction with a deterministic positive magnitude. This magnitude equals the right derivative at zero field of the infinite-volume pressure. The law is expressed through self-adjoint linear combinations of the spin components and requires no joint measurement of noncommuting observables.
Spontaneous magnetization in the quantum Heisenberg ferromagnet
Sep 24, 2026·paper.pdf
For every dimension d ≥ 3 and every spin , 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 . This resolves the low-temperature ordering problem in the spontaneous-magnetization formulation.
OpenAI also released the model's reasoning trace for this result: reasoning trace (PDF).