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Low-temperature magnetization in quantum Heisenberg ferromagnets

The question is whether for the nearest-neighbor isotropic quantum Heisenberg ferromagnet on ℤᵈ with d≥3 and every positive half-integer spin S, a translation-invariant zero-field KMS equilibrium state exists at sufficiently low positive temperature with magnetization at least S/4.

Why it matters and what counts as progress

Why it matters: It asks for spontaneous positive-temperature magnetization in the standard isotropic quantum ferromagnet throughout dimensions at least three.

What counts as progress: Construct a magnetized zero-field equilibrium state at positive temperature for the isotropic model, uniformly through every fixed dimension d≥3 and spin S.

How the problem developed

  1. 27 Jan 1999 · Open problem [1]

    Elliott Lieb Recorded the open problem of proving positive-temperature long-range order for the quantum Heisenberg ferromagnet in dimension at least three.

  2. 2015 · Partial result [2]

    Michele Correggi, Alessandro Giuliani and Robert Seiringer Proved low-temperature spin-wave free-energy asymptotics, while noting that mathematical proof of spontaneous magnetization remained open.

  3. 24 Sept 2026 · Theorem claim [3]

    OpenAI Claimed a translation-invariant zero-field KMS state with magnetization at least S/4 for the nearest-neighbor isotropic quantum Heisenberg ferromagnet, every d≥3 and every positive half-integer spin, at sufficiently low positive temperature.

AI contribution

Sources

  1. Elliott Lieb, Long range order for the quantum Heisenberg model
  2. Michele Correggi, Alessandro Giuliani and Robert Seiringer, Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet
  3. Spontaneous-magnetization paper
  4. Spontaneous magnetization in the quantum Heisenberg ferromagnet
  5. Lean ordered-state scope
  6. Lean ordered-state theorem
  7. OpenAI research release
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