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A uniform gap in the periodic spin-one antiferromagnetic chain

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For the standard spin-one periodic Hamiltonian $H_L=sum_j$ $S_j\cdot S_{j+1}$ and even lengths L, is the ground state unique for all sufficiently large L and is the spectral gap bounded below by a strictly positive constant independent of L? Use $[S^a,S^b]=i$ $\epsilon _abc$ $S^c$ and $sum_a(S^a)^2=2$.

Short English statement

For the standard spin-one periodic Hamiltonian $H_L=sum_j$ $S_j\cdot S_{j+1}$ and even lengths L, is the ground state unique for all sufficiently large L and is the spectral gap bounded below by a strictly positive constant independent of L? Use $[S^a,S^b]=i$ $\epsilon _abc$ $S^c$ and $sum_a(S^a)^2=2$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-77-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-77

The pure spin-one antiferromagnetic Heisenberg chain is claimed to have a size-independent positive gap on even periodic rings. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

A uniform gap in the periodic spin-one antiferromagnetic chain

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The pure spin-one antiferromagnetic Heisenberg chain is claimed to have a size-independent positive gap on even periodic rings.

Mathematical statement of the claimed result

Result 1.
For spin-one matrices satisfying $[S^a,S^b]=i\varepsilon_{abc}S^c$, $\sum_a(S^a)^2=2I$, use the even periodic ring \[ H_L=\sum_{j=0}^{L-1}\boldsymbol S_j\cdot\boldsymbol S_{j+1},\qquad\boldsymbol S_L=\boldsymbol S_0. \] Let $\gamma_L$ be the difference between its first two distinct energies. The source claims a unique ground state for even $L\ge60$, and \[ \gamma_L>\frac4{105}\log\frac{80}{79}\quad(L\ge60),\qquad \gamma_L>\frac{\log20}{784}\quad(L\ge2304), \] for even lengths. Therefore $\liminf_{L\to\infty,\ L\text{ even}}\gamma_L\ge\log20/784>0$. Simultaneously similar realizations with the same spin normalization have the same spectral conclusion.

Outline of the argument

Auxiliary twists express partition functions as spatial moments. Two purity comparisons yield a gap criterion, and two finite initializations start an iterative improvement that makes the criterion uniform in ring size.

Formal verification

Family 268 has no Lean scope page in this release. Its selected result is not presented as formalized. 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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