AI 中文总结
研究有理$\mathfrak{gl}_\ell$自旋链，从量子谱曲线等出发推导无嵌套贝塞耳方程组，推广到任意秩，识别递归层次结构，给出转移矩阵本征值表达式，揭示相关联系并研究准经典极限。
AI 中文摘要
我们开发了一种向量表示下有理$\mathfrak{gl}_\ell$自旋链的无嵌套贝塞耳假设描述。从量子谱曲线和变量分离框架出发，我们推导了仅涉及携带动量的贝塞耳根的封闭贝塞耳方程组。该构造在$\mathfrak{gl}_3$和$\mathfrak{gl}_4$自旋链中明确给出，然后推广到任意秩。这项工作的一个核心结果是识别了与基本转移矩阵相关的递归层次结构。该层次结构由较低转移矩阵的正则性条件生成，并通过通用的$\ell$阶方程$\mathcal{R}_{\ell}=0$封闭。此方程取代了传统嵌套贝塞耳假设的最后一级，并消除了所有辅助贝塞耳根。因此，本征态的完整谱数据仅编码在第一个巴克斯特多项式$Q_{1}(u)$中。我们还仅根据携带动量的根得到了所有基本转移矩阵本征值的显式表达式。所得公式提供了有理$\mathfrak{gl}_\ell$自旋链谱的紧凑表征，并揭示了量子谱曲线、转移矩阵融合关系以及无嵌套描述背后的截断$Q$系统之间的直接联系。最后，我们研究了无嵌套贝塞耳方程的准经典（高丁）极限。对于$\mathfrak{gl}_3$自旋链，我们表明主导的非平凡贡献产生了高丁方程，其无极点形式自然地定义了一个标量三阶$\mathfrak{gl}_3$算子。
英文摘要
We develop a non-nested Bethe ansatz description of rational $\mathfrak{gl}_\ell$ spin chains in the vector representation. Starting from the quantum spectral curve and the separation-of-variables framework, we derive closed systems of Bethe equations involving only the momentum-carrying Bethe roots. The construction is worked out explicitly for the $\mathfrak{gl}_3$ and $\mathfrak{gl}_4$ spin chains and then generalized to arbitrary rank. A central result of this work is the identification of a recursive hierarchy associated with the fundamental transfer matrices. The hierarchy is generated by regularity conditions of the lower transfer matrices and closes through a universal rank-$\ell$ equation $\mathcal{R}_{\ell}=0$. This equation replaces the final level of the conventional nested Bethe ansatz and eliminates all auxiliary Bethe roots. Consequently, the complete spectral data of an eigenstate are encoded solely in the first Baxter polynomial $Q_{1}(u)$. We further obtain explicit expressions for the eigenvalues of all fundamental transfer matrices in terms of the momentum-carrying roots alone. The resulting formulation provides a compact characterization of the spectrum of rational $\mathfrak{gl}_\ell$ spin chains and reveals a direct connection between the quantum spectral curve, transfer-matrix fusion relations, and a truncated $Q$-system underlying the non-nested description. Finally, we investigate the quasi-classical (Gaudin) limit of the non-nested Bethe equations. For the $\mathfrak{gl}_3$ spin chain, we show that the leading non-trivial contribution gives rise to Gaudin equations whose pole-free form naturally defines a scalar third-order $\mathfrak{gl}_3$ oper.
Comments35 pages
