Department of Physics & Astronomy, Bucknell University, 1 Dent Dr, Lewisburg, Pennsylvania 17837, USA.
Department of Physics and Astronomy, University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame, Indiana 46556, USA.
Phys Rev E. 2023 Feb;107(2-1):024108. doi: 10.1103/PhysRevE.107.024108.
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.
链接群展开已被证明在计算各种一维和二维经典和量子格模型的平衡和非平衡性质方面非常高效。在本文中，我们将链接群方法扩展到 Cayley 树及其无界的近亲 Bethe 晶格。我们的目标是：（a）为这些晶格开发链接群展开，这是一个新的应用；（b）进一步理解链接群方法令人惊讶的收敛效率及其局限性。我们得到了几个关键结果。首先，我们表明对于特定形式的最近邻哈密顿量，所有有限的树状簇都可以映射到一维有限链。然后，我们表明 Cayley 树和 Bethe 晶格之间的定性区别是由于晶格常数的不同，这是由于 Bethe 晶格是无界的。我们利用这些结果获得了任意度的 Bethe 晶格整个无序相的零场磁化率的显式闭式公式，直到临界点；值得注意的是，只有一维链状簇才有贡献。我们还获得了两种树状晶格上的 Ising 模型的精确零场配分函数，仅使用两个最小的簇，类似于一维链。最后，这些结果实现了具有不可忽略边界的无限晶格和没有任何边界的晶格之间的直接比较，使我们能够证明链接群展开在每次展开的阶数上消除了边界项，回答了其令人惊讶的收敛效率的问题。我们以这些结果的一些推论、可能的推广和应用结束。
