Auxiliary matrices for the six-vertex model at roots of 1 and a geometric interpretation of its symmetries
Auxiliary matrices for the six-vertex model at roots of 1 and a geometric interpretation of its symmetries
复制标题
根为 1 的六顶点模型的辅助矩阵及其对称性的几何解释
DOI：
10.1088/0305-4470/36/19/305
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发表时间：
2003
期刊：
影响因子：
--
通讯作者：
C. Korff
中科院分区：
文献类型：
--
作者：
C. Korff
从量子群论的角度研究了单位根六顶点模型的辅助矩阵的构造问题。利用量子环代数$U_q({sl}_2)$在$q^N=1时的交织概念，构造了一个三参数辅助矩阵族。这一族的元素满足与传递矩阵的函数关系，使得人们可以求解模型的特征值问题，并推导出Bethe ansatz方程。这种函数关系是通过评价表示的张量积的分解得到的，并且涉及具有不同参数的辅助矩阵。由于这种对附加参数的依赖，辅助矩阵一般破坏了六顶点模型的有限对称性，例如自旋反转或自旋守恒。更重要的是，它们还提升了传输矩阵的额外简并，这是由于在有理耦合值处存在回路对称性。证明了辅助矩阵中的额外参数与量子环代数$U_q({sl}_2)$在$q^N=1$时的扩大中心中的元素直接相关。这种联系提供了有理耦合条件下六顶点模型对称性增强的几何解释。标记辅助矩阵的参数可以解释为三维复杂超曲面上的坐标，该坐标在无限维解析变换组的作用下保持不变，称为量子共伴作用量。
The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra $U_q(\tilde{sl}_2)$ at $q^N=1$ a three parameter family of auxiliary matrices is constructed. The elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the Bethe ansatz equations. This functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters. Because of this dependence on additional parameters the auxiliary matrices break in general the finite symmetries of the six-vertex model, such as spin-reversal or spin conservation. More importantly, they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values. The extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged center of the quantum loop algebra $U_q(\tilde{sl}_2)$ at $q^N=1$. This connection provides a geometric interpretation of the enhanced symmetry of the six-vertex model at rational coupling. The parameters labelling the auxiliary matrices can be interpreted as coordinates on a three-dimensional complex hypersurface which remains invariant under the action of an infinite-dimensional group of analytic transformations, called the quantum coadjoint action.