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

Proves 3D quantum Heisenberg ferromagnets magnetize at low temperature; claims Bloch's T^(3/2) law and its correction.

4 papers September 24, 2026 Explained by AI

A ferromagnet like iron stays magnetized with no outside field, below some temperature. For the standard quantum model of this, the Heisenberg ferromagnet, no one had proved that this ordering happens at any positive temperature. One paper shows it does in three or more dimensions, for every spin, and Lean checks this. Three more papers claim the exact low-temperature law for how magnetization falls, first found by Bloch in 1930, and its next correction.

First step
1930
Felix Bloch, spin-wave prediction that magnetization falls like T^(3/2)
Significance
Major
Spontaneous magnetization of the quantum Heisenberg ferromagnet at positive temperature is a long-standing open problem in mathematical physics.
Trust
Formal proof listed
Field
Mathematical physics
4 papers

The explanation

Picture a cubic grid with a tiny quantum magnet, a spin, at each point. Neighbouring spins lower their energy by lining up. At zero temperature all spins point the same way. Each spin-1/2 site then contributes its full value 1/2. Heat stirs up spin waves, ripples called magnons, and the average drops a little. Bloch predicted the drop is proportional to T^(3/2), where T is temperature. Take spin S = 1/2 and β = 1/T = 20, so βS = 10. The predicted drop is ζ(3/2)/(8π^(3/2)) × 10^(−3/2), about 0.00185. Here ζ is the Riemann zeta function, with ζ(3/2) about 2.612. The magnetization is then about 0.4981 instead of 0.5.

That is a prediction. A proof must show order survives at a fixed positive temperature, with no outside field. The model, the Heisenberg ferromagnet, gives a quantum mechanism for ferromagnetism. Its energy does not change if all spins rotate together. So a magnetized state must pick a direction on its own. That is spontaneous magnetization. In one and two dimensions, the Mermin–Wagner theorem rules it out. In three dimensions, Dyson, Lieb and Simon proved ordering for the quantum antiferromagnet at spin one and up. There, neighbours prefer to point opposite ways. Their method does not give the needed bound in the ferromagnetic case. That gap was the obstacle.

The fourth paper settles existence. In every dimension d ≥ 3 and for every spin S, at low enough temperature there is an equilibrium state with magnetization at least S/4. Equilibrium here means the KMS condition, the standard definition for infinite quantum systems. The paper says this resolves the low-temperature ordering problem in this formulation.

The other three papers claim exact laws in three dimensions. The first claims Bloch's T^(3/2) law at every spin. Its coefficient comes from the spin-wave energy near zero momentum. The second claims the next term, 3ζ(5/2)(βS)^(−5/2)/(128π^(3/2)). Dyson had traced this term to the grid shape of independent magnons. In our example, βS = 10, it adds only about 0.0000179. The third claims that, on large cubes, the zero-field magnetization settles to a uniform direction with a fixed positive size.

How sure is this? Lean, a computer proof checker, proves the existence result. For every d ≥ 3 and every spin, it builds the magnetized equilibrium state with magnetization at least S/4. Bloch's law, its correction and the spherical law are not in Lean. They rest on three long papers, and no specialist review of them has been published.

What it doesn't show. This does not cover one or two dimensions, where ordering is ruled out, or find the temperature where order disappears.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds. Spontaneous magnetization has a Lean proof; Bloch's law, its correction and the spherical law rest on unreviewed papers.

Words used

Heisenberg ferromagnet
A grid of quantum spins where each neighbouring pair lowers its energy by pointing the same way.
Spin S
The size of each quantum magnet: 1/2, 1, 3/2 and so on.
Spontaneous magnetization
A net magnetization that survives with no outside magnetic field.
Magnon
A spin wave: a ripple in the spin directions that carries a small unit of lost magnetization.
Bloch's law
The prediction that magnetization falls below its maximum in proportion to T^(3/2) at low temperature.
KMS state
The standard mathematical definition of a thermal equilibrium state for an infinite quantum system.
β
Inverse temperature, 1/T; large β means low temperature.

The original papers (4)

Lean notes Model's reasoning summary

All quoted lines (11)
  1. The spin-wave description of a ferromagnet begins with Bloch's prediction of a magnetization deficit proportional to T3/2T^{3/2}, where T=β−1T=\beta^{-1}.
    The first lattice correction to Bloch's law Back to text
  2. Heisenberg's exchange model provided a quantum mechanism for ferromagnetism.
    Spontaneous magnetization in the quantum Heisenberg ferromagnet Back to text
  3. The Mermin–Wagner theorem excludes positive-temperature ferromagnetism in one and two dimensions for finite-range isotropic Heisenberg models.
    Spontaneous magnetization in the quantum Heisenberg ferromagnet Back to text
  4. Dyson, Lieb and Simon proved quantum antiferromagnetic ordering for spin at least one in those dimensions, and for spin one half in sufficiently large dimension.
    Spontaneous magnetization in the quantum Heisenberg ferromagnet Back to text
  5. The corresponding ferromagnetic problem resisted this approach: the reflection-positivity argument does not provide the needed ferromagnetic Duhamel infrared bound, as explained by Dyson, Lieb and Simon.
    Spontaneous magnetization in the quantum Heisenberg ferromagnet Back to text
  6. The same state satisfies the KMS condition for the zero-field dynamics and has magnetization at least S/4S/4.
    Spontaneous magnetization in the quantum Heisenberg ferromagnet Back to text
  7. This resolves the low-temperature ordering problem in the spontaneous-magnetization formulation.
    Spontaneous magnetization in the quantum Heisenberg ferromagnet Back to text
  8. We prove Bloch's T3/2T^{3/2} law for the spontaneous magnetization of three-dimensional quantum Heisenberg ferromagnets at every fixed positive quantum spin.
    Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions Back to text
  9. In addition to the leading Bloch term, this gives the correction 3ζ(5/2)(βS)−5/2/(128π3/2)3\zeta(5/2)(\beta S)^{-5/2}/(128\pi^{3/2}).
    The first lattice correction to Bloch's law Back to text
  10. In that expansion, the terms of orders T5/2T^{5/2} and T7/2T^{7/2} in the magnetization already arise from the lattice dispersion of independent magnons; the leading interaction contribution is of order T4T^4.
    The first lattice correction to Bloch's law Back to text
  11. As the even periodic cubes grow, the symmetric zero-field magnetization converges in moments to a uniform direction with a deterministic positive magnitude.
    The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet Back to text

Other sources

  1. openai/math, Lean scope note for family 271 At every sufficiently low positive temperature, it constructs a translation-invariant equilibrium state satisfying the KMS condition for the zero-field dynamics and having magnetization at least S/4S/4.
  2. openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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