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Phys-63 · physics · mathematics · Physics

A spherical distribution of the low-temperature total quantum spin

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every positive half-integer spin S and every sufficiently large fixed finite beta, do zero-field even periodic $Z^3$ Heisenberg cubes have volume-normalized total spin whose moment generating function tends to sinh(m|t|)/(m|t|) for some $m>0$ and every $t\in R^3$? Require the averaged finite-volume spin-dot-spin sum divided by volume squared to tend to $m^2$.

Short English statement

For every positive half-integer spin S and every sufficiently large fixed finite beta, do zero-field even periodic $Z^3$ Heisenberg cubes have volume-normalized total spin whose moment generating function tends to sinh(m|t|)/(m|t|) for some $m>0$ and every $t\in R^3$? Require the averaged finite-volume spin-dot-spin sum divided by volume squared to tend to $m^2$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-63-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-63

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

A spherical distribution of the low-temperature total quantum spin

OpenAI's claim

OpenAI presents a proof in the selected manuscript. 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.

Mathematical statement of the claimed result

Result 1.
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

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. 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.

Sources

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