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Phys-57 · physics · mathematics · Physics

The exact Bloch coefficient for three-dimensional finite-range quantum ferromagnets

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every positive half-integer spin S and each admissible nonnegative finite-range translation-invariant coupling on $Z^3$ with positive spin-stiffness matrix $D_{S,J}$, does $\beta ^{3/2}(S-m_{S,J}(\beta ))$ tend as beta$\to$infinity to $\zeta (3/2)/(8\pi ^{3/2}\operatorname{sqrt}(det$ $D_{S,J}))$?

Short English statement

For every positive half-integer spin S and each admissible nonnegative finite-range translation-invariant coupling on $Z^3$ with positive spin-stiffness matrix $D_{S,J}$, does $\beta ^{3/2}(S-m_{S,J}(\beta ))$ tend as beta$\to$infinity to $\zeta (3/2)/(8\pi ^{3/2}\operatorname{sqrt}(det$ $D_{S,J}))$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-57-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-57

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. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The exact Bloch coefficient for three-dimensional finite-range quantum ferromagnets

OpenAI's claim

OpenAI presents a proof in the selected manuscript. 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.

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.

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.

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. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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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