论文标题
与XYZ自旋链有关的某些多项式的组合描述
A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain
论文作者
论文摘要
我们研究了Bazhanov和Mangazeev的三色模型与多项式$ q_n（z）$，它们出现在XYZ Spin链的哈密顿特征向量中。通过使用DWBC和反射端的8VSO模型的分区函数中的参数，我们在具有相同边界条件的三色模型的分区函数中找到了$ q_n（z）$的显式组合表达式。 Bazhanov和Mangazeev猜想$ q_n（z）$具有正整数系数。我们证明了$ q_n（z+1）$和$（z+1）^{n（n+1）} q_n（1/（z+1））$具有正整数系数。此外，对于三色模型，我们在每种颜色的面孔数量的状态数量上找到了一些结果，并为每种颜色的可能面数量计算严格的边界。
We study the connection between the three-color model and the polynomials $q_n(z)$ of Bazhanov and Mangazeev, which appear in the eigenvectors of the Hamiltonian of the XYZ spin chain. By specializing the parameters in the partition function of the 8VSOS model with DWBC and reflecting end, we find an explicit combinatorial expression for $q_n(z)$ in terms of the partition function of the three-color model with the same boundary conditions. Bazhanov and Mangazeev conjectured that $q_n(z)$ has positive integer coefficients. We prove the weaker statement that $q_n(z+1)$ and $(z+1)^{n(n+1)}q_n(1/(z+1))$ have positive integer coefficients. Furthermore, for the three-color model, we find some results on the number of states with a given number of faces of each color, and we compute strict bounds for the possible number of faces of each color.
