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Low-Temperature Order in Quantum Heisenberg Ferromagnets

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For the short-range quantum Heisenberg ferromagnet in dimension $d\geq 3$, prove that sufficiently low temperature produces long-range order; specifically, prove that $\lim_{\Lambda\nearrow\mathbb{Z}^d}|\Lambda|^{-2}\sum_{x,y\in\Lambda}\langle\widetilde S_x\cdot\widetilde S_y\rangle_{T,\Lambda}>0$ for all sufficiently small $T>0$.

Formulation references: [S1], [S2].

Short English statement

For the short-range quantum Heisenberg ferromagnet in dimension $d\geq 3$, prove that sufficiently low temperature produces long-range order; specifically, prove that $\lim_{\Lambda\nearrow\mathbb{Z}^d}|\Lambda|^{-2}\sum_{x,y\in\Lambda}\langle\widetilde S_x\cdot\widetilde S_y\rangle_{T,\Lambda}>0$ for all sufficiently small $T>0$.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Phys-40-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-40

The d=3 Bloch law establishes low-temperature order for finite-range quantum Heisenberg ferromagnets, but the problem statement quantifies every d>=3 and an explicit zero-field averaged-correlation thermodynamic limit. The theorem constructs a magnetized translation-invariant zero-field KMS state for all d>=3, but does not itself assert the problem statement exact thermodynamic limit of averaged finite-volume zero-field correlations. The latter is separately quantified, so it is not silently identified with state existence. The d=3 nearest-neighbor low-temperature expansion strengthens positive magnetization in that case, without providing the problem statement in every d>=3 or its full averaged-limit formulation. The theorem proves the positive zero-field averaged-correlation limit m^2 for all spins at sufficiently low positive temperature in d=3. The problem statement requires the corresponding assertion in every dimension d>=3. This entry records partial support; complete coverage of the question is not established by this catalogue.

Low-Temperature Order in Quantum Heisenberg Ferromagnets

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Finite-range three-dimensional Heisenberg ferromagnets are claimed to obey the exact $T^{3/2}$ spontaneous-magnetization deficit law, with the dispersion determinant fixing the coefficient; For every positive quantum spin in dimensions $d\geq3$, low-temperature nearest-neighbor Heisenberg ferromagnets are claimed to admit zero-field KMS states with magnetization at least $S/4$; For the nearest-neighbor model, magnetization deficit is claimed equal to ideal-magnon density up to $o(\beta^{-5/2})$, including the stated $\zeta(5/2)$ lattice correction; At low fixed temperature, symmetric zero-field magnetization is claimed to approach a uniformly oriented vector of deterministic positive magnitude equal to the right pressure derivative. Together they provide the following support within the scopes described below.

Mathematical statement of the claimed result

Result 1.
Fix $S\in\{1/2,1,3/2,\ldots\}$. Let $J:\mathbb Z^3\to[0,\infty)$ have finite support, $J(0)=0$, $J(z)=J(-z)$, and support generating $\mathbb Z^3$. For the free-boundary Hamiltonian $H=-\tfrac12\sum_{x,y}J(y-x)\boldsymbol S_x\cdot\boldsymbol S_y$, let $m_{S,J}(\beta)$ be obtained by differentiating the limiting pressure from positive fields and evaluating at field zero. With \[ D_{S,J}=\frac S2\sum_zJ(z)zz^{\mathsf T}, \] the claimed low-temperature law is \[ \lim_{\beta\to\infty}\beta^{3/2}(S-m_{S,J}(\beta)) =\frac{\zeta(3/2)}{8\pi^{3/2}\sqrt{\det D_{S,J}}}. \] The thermodynamic limit precedes removal of the positive field.
Result 2.
For the coupling-one nearest-neighbor spin-$S$ quantum ferromagnet on $\mathbb Z^d$, with \[ H_\Lambda=-\sum_{\{x,y\}\subset\Lambda,\ |x-y|_1=1}\boldsymbol S_x\cdot\boldsymbol S_y, \] the source claims that every integer $d\ge3$ and $S\in\{1/2,1,3/2,\ldots\}$ admit a finite $\beta_0(d,S)>0$. At every finite $\beta\ge\beta_0$, the induced infinite-volume dynamics has a translation-invariant $\beta$-KMS state $\omega$ satisfying \[ \omega(S_0^z)\ge S/4. \] This chooses an ordered equilibrium state; it does not assign a nonzero one-site mean to the rotation-invariant state.
Result 3.
For the three-dimensional coupling-one nearest-neighbor spin-$S$ quantum ferromagnet, define spontaneous magnetization $m_S(\beta)$ by the thermodynamic pressure's right field derivative at zero. Put \[ \varepsilon_S(k)=2S\sum_{j=1}^3(1-\cos k_j),\qquad n_S^{\mathrm{sw}}(\beta)=\int_{[-\pi,\pi]^3} \frac{d^3k}{(2\pi)^3(e^{\beta\varepsilon_S(k)}-1)}. \] For each fixed $S\in\{1/2,1,3/2,\ldots\}$, the assertion is \[ \lim_{\beta\to\infty}(\beta S)^{5/2} [S-m_S(\beta)-n_S^{\mathrm{sw}}(\beta)]=0. \] Thus the full lattice spin-wave integral captures the magnetization deficit through the stated scale.
Result 4.
On the even-side torus $\Lambda_L=(\mathbb Z/L\mathbb Z)^3$, set $V=L^3$, $H_{S,L}=-\sum_{\{x,y\}\text{ nearest neighbors}}\boldsymbol S_x\cdot\boldsymbol S_y$, and $M_L^a=\sum_xS_x^a$. For each fixed $S\in\{1/2,1,3/2,\ldots\}$, the claim provides finite $\beta_0(S)$ so that any finite $\beta\ge\beta_0$ has positive spontaneous magnetization $m=p'_{S,\beta}(0+)$. With zero-field Gibbs expectation, \[ \lim_{L\to\infty,\ L\text{ even}} \left\langle\exp\!\left(V^{-1}\sum_{a=1}^3t_aM_L^a\right)\right\rangle_{\beta,L} =\frac1{4\pi}\int_{S^2}e^{m t\cdot n}\,d\sigma(n) =\frac{\sinh(m|t|)}{m|t|}\qquad(t\in\mathbb R^3). \] The value at $t=0$ is one. The same limit gives $V^{-2}\sum_{x,y}\langle\boldsymbol S_x\cdot\boldsymbol S_y\rangle\to m^2$.

Outline of the argument

Pinned exchange returns are compared with the one-magnon walk. Uniform sparse-pin and long-return remainder bounds isolate finite returns, whose small-momentum asymptotics produce the Bloch coefficient; volume, zero-field derivative, and temperature limits are taken in that order. Spin slots and exchange loops translate a simultaneous pin test into return-path estimates. Sparse obstacles are bypassed by local diffusion flows, and heat-flow leakage bounds close a pinning bootstrap, producing a magnetized thermodynamic equilibrium state.

Capacity comparison controls pinned return cycles on a period. Sparse-obstacle propagation and an ideal-potential profile identify comparison sources whose cancellation improves the error beyond the leading Bloch term, allowing the next dispersion coefficient to survive. Exchange-cycle pins and zero-free traces control torus comparisons at vanishing field. Removing the field across spatial scales and suppressing replica cross terms identifies the moment law; pressure tangents determine its radius without assigning joint measurements to noncommuting components.

Formal verification

For family 271, the selected theorem constructs low-temperature zero-field magnetized KMS states for all dimensions at least three and all positive spins. The Bloch expansion, lattice correction, and spherical moment law are separate unselected companions. The listed Comparator files are Heisenberg.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

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