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Phys-47 · physics · mathematics · Physics

The first lattice correction to quantum Bloch magnetization

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every fixed positive half-integer spin S in the nearest-neighbor $Z^3$ quantum Heisenberg ferromagnet, does (beta $S)^{5/2}[S-m_S(\beta )-n_S^sw(\beta )]\to 0$ as beta$\to$infinity, where $n_S^sw$ is the full lattice spin-wave Bose occupation integral?

Short English statement

For every fixed positive half-integer spin S in the nearest-neighbor $Z^3$ quantum Heisenberg ferromagnet, does (beta $S)^{5/2}[S-m_S(\beta )-n_S^sw(\beta )]\to 0$ as beta$\to$infinity, where $n_S^sw$ is the full lattice spin-wave Bose occupation integral?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-47-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-47

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

The first lattice correction to quantum Bloch magnetization

OpenAI's claim

OpenAI presents a proof in the selected manuscript. 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.

Mathematical statement of the claimed result

Result 1.
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.

Outline of the argument

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.

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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