A one-dimensional cluster model with next-nearest-neighbor interactions and two additional composite interactions is solved; the free energy is obtained and a correlation function is derived exactly. The model is diagonalized by a transformation obtained automatically from its interactions, which is an algebraic generalization of the Jordan-Wigner transformation. The gapless condition is expressed as a condition on the roots of a cubic equation, and the phase diagram is obtained exactly. We also find that the distribution of roots for this algebraic equation determines the existence of long-range order, and we again obtain the ground-state phase diagram. Finally, we note that our results are universally valid for an infinite number of solvable spin chains whose interactions obey the same algebraic relations.

中文翻译：

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具有下近邻交互的集群模型的精确解

求解具有次最近邻相互作用和两个附加复合相互作用的一维聚类模型；获得自由能并精确导出相关函数。该模型通过从其相互作用中自动获得的变换对角化，这是 Jordan-Wigner 变换的代数推广。无间隙条件表示为三次方程根上的条件，精确得到相图。我们还发现，这个代数方程的根分布决定了长程有序的存在，我们又得到了基态相图。最后，我们注意到我们的结果对无数可解自旋链普遍有效，它们的相互作用遵循相同的代数关系。