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Spontaneous magnetization in the quantum Heisenberg ferromagnet

OpenAI

Published Oct 6, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.83295/trf3-p8g6

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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.
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.
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.
We rigorously prove that in three or more dimensions, the nearest-neighbor, simple-cubic, ferromagnetic, quantum Heisenberg model of spin S(=12,1,…)S(=\frac{1}{2}, 1, \dots{})S(=21​,1,…) has a phase transition at nonzero temperature.
We prove the existence of spontaneous symmetry breaking in suitably low-energy eigenstates of certain gapless and frustrated many-body quantum systems, namely symmetric quantum perturbations to classical models which exhibit spontaneous symmetry breaking of a finite group at some positive temperature. Additionally, the classical model need not be local in space, as long as it satisfies a quantum analogue of the Peierls condition. As an example of our technique, we establish robust ferromagnetism in random-bond Ising models in d=2d= 2d=2 dimensions with sufficiently biased random couplings, with weak transverse field. Our mathematical technique is based on establishing quantum bottlenecks, similar to a "many-body WKB" method for evaluating tunneling rates. Using these same methods, we provide new proofs of metastability and the slow decay of the false vacuum, applicable to gapless metastable states. Our work represents a first step towards a rigorous classification of stable gapless quantum phases.