The open XXZ spin chain with the anisotropy parameter $\Delta=-\frac12$ and diagonal boundary magnetic fields that depend on a parameter $x$ is studied. For real $x>0$, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in $x$ with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik-Zamolodchikov equations associated with the $R$-matrix and diagonal $K$-matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.

中文翻译：

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Δ = -1/2 处的开放 XXZ 链和边界量子 Knizhnik-Zamolodchikov 方程

研究了具有各向异性参数 $\Delta=-\frac12$ 和依赖于参数 $x$ 的对角边界磁场的开放 XXZ 自旋链。对于实数 $x>0$，明确计算自旋链哈密顿量的精确有限大小基态特征值。在适当的归一化中，基态分量被表征为具有整数系数的 $x$ 中的多项式。该特征向量的线性求和规则和特殊分量是根据行列式公式明确计算的。这些结果来自于边界量子 Knizhnik-Zamolodchikov 方程的轮廓积分解的构建，该方程与六顶点模型的 $R$-矩阵和对角线 $K$-矩阵相关。推测此解与全对称交替符号矩阵的加权枚举之间的关系。