Statement copy in the gpt 6 astra pro collection (Phys-13)

Long-Range Order in the Quantum Heisenberg Ferromagnet

Definitions and mathematical statement

At each site of ${\mathbb{Z}}^d$, spin-$S$ operators act on ${\mathbb{C}}^{2S+1}$, satisfy the angular-momentum commutation relations, and have $\mathbf S_x^2=S(S+1)I$. The finite-volume Hamiltonian is $H_\Lambda=-\sum_{\langle x,y\rangle\subset\Lambda}\mathbf S_x\cdot\mathbf S_y$. An infinite-volume equilibrium state is understood in the quantum Gibbs/KMS sense of the cited model. For each $d\ge3$ and positive half-integer $S$, the assertion is that some finite $\beta_0(d,S)$ satisfies \[ \forall\beta\ge\beta_0\text{ finite},\quad \exists\text{ translation-invariant zero-field equilibrium state }\omega_\beta, \quad\omega_\beta(\mathbf S_0)\ne0. \] The state is allowed to break rotational symmetry; its rotation-invariant mixture need not have nonzero magnetization. The coupling is fixed to one.

Formulation and concept references: [S1].

Short English statement

At sufficiently low positive temperature, must the quantum Heisenberg ferromagnet in every dimension at least three possess an equilibrium state with spontaneous magnetization?

Sources

• S1 Long range order for the quantum Heisenberg model, contributed27January1999. https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9901.HeisenbergFerr.html. Catalogue formulation source.
• E1 M. Correggi, A. Giuliani, and R. Seiringer, Validity of the spin-wave approximation for the free energy of the Heisenberg ferromagnet, Communications in Mathematical Physics 339 $2015$, 279--307, Theorem 2.1, Section 3, and Proposition 5.2. https://arxiv.org/pdf/1312.7873v2; https://doi.org/10.1007/s00220-015-2402-0. Published result; arXiv version 2 is dated 10 August 2019. Consulted 24 September 2026.
• E2 E. R. Speer, Failure of reflection positivity in the quantum Heisenberg ferromagnet, Letters in Mathematical Physics 10 $1985$, 41--47, abstract, Introduction, and the finite-volume counterexamples. https://sites.math.rutgers.edu/~speer/homepage/pubs/LMP_10_41.pdf. Its specified regimes are not replaced here by a claim about every temperature or every possible reflection. Consulted 24 September 2026.
• E3 D. Ueltschi, Random loop representations for quantum spin systems, Journal of Mathematical Physics 54 $2013$, 083301; author manuscript arXiv:1301.0811, Theorem 3.2, the spin/loop correspondence of Section 3, and Section 7.1, especially equation $7.1$. https://arxiv.org/pdf/1301.0811. The spin-$1/2$ ferromagnetic representation is used, not the different models for which this paper proves long-range order. Consulted 24 September 2026.
• E4 B. Nachtergaele and R. Sims, Lieb--Robinson bounds in quantum many-body physics, in R. Sims and D. Ueltschi (eds.), Entropy and the Quantum, Contemporary Mathematics 529, American Mathematical Society $2010$, 141--176, Section 3.1, Theorem 3.1 and its proof (norm convergence of bounded-interaction dynamics on local observables). https://arxiv.org/pdf/1004.2086v1. Locality input for the thermodynamic-limit argument; the Gaussian-smearing passage of the KMS identity is given in the text. Consulted 24 September 2026.
• E5 D. Heson, S. Starr, and J. Thornton, Violation of ferromagnetic ordering of energy levels in spin rings for the singlet, arXiv:2307.12773v5, 7 January 2025, abstract and Section 2. https://arxiv.org/html/2307.12773v5. The manuscript distinguishes its largely numerical ordering claim from a rigorous singlet-sector result; neither is used as a proof of the lattice conjecture here. Consulted 24 September 2026.