树上的链接簇扩展
Linked cluster expansion on trees
摘要：
已被证明是非常有效的，在计算平衡和非平衡性质的各种一维和二维的经典和量子晶格模型的连接集团展开。在这篇文章中，我们扩展的链接集群方法的凯莱树和它的无边界的表弟Bethe格。我们的目标是（a）开发这些格的连接簇扩展，一个新的应用，和（b）进一步了解连接簇方法令人惊讶的收敛效率，以及它的局限性。我们得到几个关键结果。首先，我们证明了对于一个特定形式的最近邻哈密顿算子，所有的有限树状簇都可以映射到一维有限链上。然后，我们表明，凯莱树和贝特格之间的定性区别出现由于不同的晶格常数，这是一个结果的贝特格是无边界的。我们使用这些结果得到明确的封闭形式的零场磁化率公式为整个无序相的临界点任意程度的贝特晶格;值得注意的是，只有一维链状集群的贡献。我们还获得了精确的零场配分函数的伊辛模型上的两棵树，只有两个最小的集群，类似于1D链。最后，这些结果实现了一个不可忽略的边界和一个没有任何边界的无限晶格之间的直接比较，使我们能够表明，链接集群扩展消除边界项在每个顺序的扩展，回答了关于其惊人的收敛效率的问题。我们得出结论，这些结果的一些分支，以及可能的推广和应用。
英文摘要：
The linked cluster expansion has been shown to be highly efficient in calculating equilibrium and nonequilibrium properties of a variety of 1D and 2D classical and quantum lattice models. In this article, we extend the linked cluster method to the Cayley tree and its boundaryless cousin the Bethe lattice. We aim to (a) develop the linked cluster expansion for these lattices, a novel application, and (b) to further understand the surprising convergence efficiency of the linked cluster method, as well as its limitations. We obtain several key results. First, we show that for nearest-neighbor Hamiltonians of a specific form, all finite treelike clusters can be mapped to one dimensional finite chains. We then show that the qualitative distinction between the Cayley tree and Bethe lattice appears due to differing lattice constants that is a result of the Bethe lattice being boundaryless. We use these results to obtain the explicit closed-form formula for the zero-field susceptibility for the entire disordered phase up to the critical point for Bethe lattices of arbitrary degree; remarkably, only 1D chainlike clusters contribute. We also obtain the exact zero field partition function for the Ising model on both trees with only the two smallest clusters, similar to the 1D chain. Finally, these results achieve a direct comparison between an infinite lattice with a nonnegligible boundary and one without any boundary, allowing us to show that the linked cluster expansion eliminates boundary terms at each order of the expansion, answering the question about its surprising convergence efficiency. We conclude with some ramifications of these results, and possible generalizations and applications.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
