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Soft Phases in Two-Dimensional O(N) Models

Status Partially solved · unsolvedmath.comLLM status Open unverified

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.
Do spin correlations in the classical Heisenberg model, and other $O(N)$ models with $N>2$, decay exponentially at every nonzero temperature, or do these models have soft phases like the XY model? Prove or disprove that the low-temperature perturbative expansion is asymptotically correct, and determine whether there is a nonzero-temperature critical point and, if so, its critical indices.

Formulation references: [S1], [S2].

Short English statement

Do spin correlations in the classical Heisenberg model, and other $O(N)$ models with $N>2$, decay exponentially at every nonzero temperature, or do these models have soft phases like the XY model? Prove or disprove that the low-temperature perturbative expansion is asymptotically correct, and determine whether there is a nonzero-temperature critical point and, if so, its critical indices.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-25-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-25

The main bound and plane-limit corollary establish exponential two-point decay at every finite beta for every n>=3, answering the first question and ruling out a positive-temperature critical point in that free-boundary convention. The bundled low-temperature perturbative-expansion assertion is not proved in this paper. This entry records partial support; complete coverage of the question is not established by this catalogue.

Soft Phases in Two-Dimensional O(N) Models

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. Square-lattice classical $O(n)$ spin correlations for $n\geq3$ are claimed to decay exponentially at every finite positive inverse temperature.

Mathematical statement of the claimed result

Result 1.
On any finite nearest-neighbor square-lattice subgraph $G=(V,E)$, put spins $\sigma_x\in S^{n-1}$ and give their free-boundary law density proportional to \[ \exp\!\left(\sum_{\{x,y\}\in E}b_{xy}\sigma_x\cdot\sigma_y\right) \prod_{x\in V}d\omega_{n-1}(\sigma_x),\qquad 0\le b_e\le\beta. \] For every integer $n\ge3$ and finite $\beta>0$, the asserted constants $A(n,\beta)<\infty$, $m(n,\beta)>0$ work for every such graph, coupling array, and pair of vertices: \[ 0\le\langle\sigma_x\cdot\sigma_y\rangle\le A e^{-m\|x-y\|_2}. \] There is no fixed boundary spin, applied field, or added interaction.

Outline of the argument

Rotation identities and multiscale estimates force a quantitative loss of effective stiffness. Once a suitable coarse scale is reached, block estimates give exponential transmission bounds uniformly for free finite graphs and bounded nonnegative couplings.

Formal verification

For family 215, the selected theorem gives exponential finite-free-graph $O(n)$ spin decay for $n\geq3$ and bounded ferromagnetic edge strengths. Plane limits, full $O(4)$ spectral gaps, and continuum $O(3)$ results are beyond that entry. The listed Comparator files are ClassicalON.lean. Named declarations include OAI.ClassicalON.main. 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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