On the uniqueness of translation-invariant ground states of the S=1 Heisenberg and J--D antiferromagnetic chains
Abstract
Using the analytic machinery and numerical inputs of OpenAI's proofs of the Haldane gap, we prove uniqueness of the translation-invariant ground state of the infinite S=1 Heisenberg antiferromagnetic chain and of the J-D model for sufficiently small |D|. Uniqueness is among all translation-invariant ground states in the operator-algebraic sense, without assuming a gap or spin-rotation symmetry for a competing state. The proof combines concentration of the spatial transfer spectrum, an exponentially small low-temperature free-energy correction, and a Golden--Thompson comparison for separated local observables. The unique translation-invariant state is pure, locally unique and gapped, and continuous in D. The ground states of even periodic chains and odd open chains with equal positive endpoint fields converge to this state along their full sequences, uniformly in D, with a stretched-exponential local rate. Every choice of ground states of open chains without endpoint magnetic fields has the same local limit. These conclusions do not establish uniqueness among arbitrary nontranslation-invariant ground states of the infinite chain. The author formulated the problems and proposed the core strategies. ChatGPT Astra made substantial contributions to developing the proofs and preparing the manuscript.
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BibTeX
@misc{tasaki2026uniqueness,
title = {{On the uniqueness of translation-invariant ground states of the $S=1$ Heisenberg and $J$--$D$ antiferromagnetic chains}},
author = {Hal Tasaki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34330},
eprint = {2610.34330},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34330},
}
RIS
TY - UNPB
TI - On the uniqueness of translation-invariant ground states of the S=1 Heisenberg and J--D antiferromagnetic chains
AU - Hal Tasaki
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.34330
UR - https://arxiv.org/abs/2610.34330
AB - Using the analytic machinery and numerical inputs of OpenAI's proofs of the Haldane gap, we prove uniqueness of the translation-invariant ground state of the infinite S=1 Heisenberg antiferromagnetic chain and of the J-D model for sufficiently small |D|. Uniqueness is among all translation-invariant ground states in the operator-algebraic sense, without assuming a gap or spin-rotation symmetry for a competing state. The proof combines concentration of the spatial transfer spectrum, an exponentially small low-temperature free-energy correction, and a Golden--Thompson comparison for separated local observables. The unique translation-invariant state is pure, locally unique and gapped, and continuous in D. The ground states of even periodic chains and odd open chains with equal positive endpoint fields converge to this state along their full sequences, uniformly in D, with a stretched-exponential local rate. Every choice of ground states of open chains without endpoint magnetic fields has the same local limit. These conclusions do not establish uniqueness among arbitrary nontranslation-invariant ground states of the infinite chain. The author formulated the problems and proposed the core strategies. ChatGPT Astra made substantial contributions to developing the proofs and preparing the manuscript.
ER -