Bloch's law, its lattice correction, and the spherical magnetization law

We prove Bloch's T^3/2 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.

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.

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.

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.