论文标题
ISING模型的系列扩展中的Bell多项式
Bell polynomials in the series expansions of the Ising model
论文作者
论文摘要
通过将钟形多项式应用于三角形和六角形晶格的Ising模型的自由能的积分表示，我们获得了在给定能量下自旋构型数量的确切组合公式（即，分区功能的低温序列膨胀或其他状态的数量）。我们还将这种方法推广到（方格）Utiyama图的更广泛类别。除了提出的精确公式外，我们的技术还可以在晶状体的完美气体与伊辛模型之间建立对应关系，这些晶格在低温膨胀中具有正系数（例如平方晶格，六边形晶格）。但是，并非总是如此 - 我们提出，对于三角形晶格，系数可能为负，并且簇解释的完美气体是有问题的。
Through applying Bell polynomials to the integral representation of the free energy of the Ising model for the triangular and hexagonal lattices we obtain the exact combinatorial formulas for the number of spin configurations at a given energy (i.e. low-temperature series expansion of the partition function or, alternatively, the number of states). We also generalize this approach to the wider class of the (chequered) Utiyama graphs. Apart from the presented exact formulas, our technique allows one to establish the correspondence between the perfect gas of clusters and the Ising model on the lattices which have positive coefficients in the low-temperature expansion (e.g. square lattice, hexagonal lattice). However it is not always the case -- we present that for the triangular lattice the coefficients could be negative and the perfect gas of clusters interpretation is problematic.
