arXiv 论文用 OpenAI 证明 Haldane 能隙的方法证明 S=1 链基态唯一性
On the uniqueness of translation-invariant ground states of the $S=1$ Heisenberg and $J$--$D$ antiferromagnetic chains
用 OpenAI 证明 Haldane 能隙那套方法，把 S=1 海森堡链基态唯一性证出来了，还坦承 ChatGPT 参与写证明，数学和 AI 协作两个角度都挺有意思。
一篇 arXiv 论文证明无限 S=1 Heisenberg 反铁磁链在平移不变基态中的唯一性，并推广到 J-D 模型中 |D| 足够小的情形。证明借助 OpenAI 此前 Haldane 能隙证明的分析工具与数值输入，结合空间转移谱集中、指数小的低温自由能修正和 Golden-Thompson 比较，不假设能隙或自旋旋转对称。作者指出唯一纯基态具有局域唯一性和能隙，且对 D 连续。作者说明 ChatGPT Astra 在证明构建和论文撰写上作出实质贡献。
On the uniqueness of translation-invariant ground states of the $S=1$ Heisenberg and $J$--$D$ antiferromagnetic chains
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.