THEORY OF MONOMER-DIMER SYSTEMS
THEORY OF MONOMER-DIMER SYSTEMS
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DOI：
10.1007/bf01877590
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发表时间：
1972-01-01
影响因子：
2.4
通讯作者：
LIEB, EH
中科院分区：
文献类型：
--
作者：
HEILMANN, OJ;LIEB, EH
我们研究了一般的单体-二聚体配分函数P(x)，它是单体活度x的多项式，其系数取决于二聚体活度。我们的主要结果是，当二聚体活度是非负时，p (x)在虚轴上有零。因此，除非单体密度极小（即x=0），否则任何单体-二聚体体系都不可能发生作为单体密度函数的相变。在此基础上，我们证明了在热力学极限下相关函数（远离x=0）的存在性和可解析性。除此之外，我们还得到了可压缩性的界限，并导出了一个新的变量，在这个变量中，自由能的扩展收敛到最小单体密度。我们还将单体-二聚体问题与磁铁的Heisenberg和Ising模型联系起来，并推导了单体-二聚体和Ising模型配分函数的Christoffell-Darboux公式。这使Ising模型有了新的认识，并为Lee-Yang圆定理提供了另一种证明。我们还推导了单体和二聚体活性的联合复分析结构域。我们的考虑与几何无关，因此对任何维度都有效。
We investigate the general monomer-dimer partition function,P(x), which is a polynomial in the monomer activity,x, with coefficients depending on the dimer activities. Our main result is thatP(x) has its zeros on the imaginary axis when the dimer activities are nonnegative. Therefore, no monomer-dimer system can have a phase transition as a function of monomer density except, possibly, when the monomer density is minimal (i.e.x=0). Elaborating on this theme we prove the existence and analyticity of correlation functions (away fromx=0) in the thermodynamic limit. Among other things we obtain bounds on the compressibility and derive a new variable in which to make an expansion of the free energy that converges down to the minimal monomer density. We also relate the monomer-dimer problem to the Heisenberg and Ising models of a magnet and derive Christoffell-Darboux formulas for the monomer-dimer and Ising model partition functions. This casts the Ising model in a new light and provides an alternative proof of the Lee-Yang circle theorem. We also derive joint complex analyticity domains in the monomer and dimer activities. Our considerations are independent of geometry and hence are valid for any dimensionality.