易辛模型之配分函數零點分布的演化

Evolutions of The Distributions of Partition Function Zeros for Ising Model on Decorated Lattices

周彥良(Yen-Liang Chou)

指導教授 ： 黃敏章

中原大學/理學院/應用物理研究所/碩士(2002年)

https://doi.org/10.6840/cycu200200506

摘要

在此篇論文中，我們研究了二維修飾晶格(decoration lattice)的易辛模型。首先我們找出任意兩個連續修飾層次(decoration level)之間配分函數(partition function zeros)的遞回關係式。藉由遞回關係式我們可以求得任何修飾層次晶格的配分函數，並且討論相變溫度與配分函數零點分布如何隨著修飾層次的演化。我們討論了兩種不同類型的修飾晶格，分別為三角形核修飾晶格(triangular type lattice with cell-decoration)與四角形鍵結修飾晶格(rectangular lattice with bond-decoration)。
關於相變溫度方面：當修飾層次遞增時，相變溫度隨之遞減。在修飾層次趨近於無限大時，三角形核修飾晶格的相變溫度趨近於零，而四角形鍵結修飾晶格則是趨近於一有限值。
關於配分函數零點分布方面：對於三角形核修飾晶格，當修飾層次趨近於無限大時，晶格本身便成為了由希爾彬斯基襯墊(Sierpiński gasket)所構成的平面無限大晶格。而配分函數零點分布則是一些散開的點與一個糾等曲線(Jordan curve)。相較於單一個希爾彬斯基襯墊的碎形晶格，此兩晶格的配分函數零點分布是相互重疊的。對於四角形鍵結修飾晶格，當修飾成次趨近於無限大時，配分函數零點分布演化成一些散開的點與一個朱利雅集合(Julia set)。朱利雅集合是一個複合碎形(multifractal)，我們作了複合碎形的分析。雖然此朱利雅集合與單一鍵結修飾晶格的配分函數零點分布的形狀不同，但是他們的複合碎形分析的結果是相同的。

關鍵字

配分函數 ； 修飾晶格 ； 易辛模型 ； 朱利雅集合 ； 費雪零點

並列摘要

In this thesis, the distribution of partition function zeros of the Two-dimensional Ising model with bond and cell decoration in the complex temperature plane is studied. By constructing exact recursion relations for the zeros between two successive decoration levels, we are able to exhibit the changes of the critical temperatures and the distribution patterns on the complex plane when the decoration level changes. We study two kinds of decoration lattice: One is the triangular type lattice with cell-decoration and, the other is the rectangular lattice with bond-decoration.
In general, the critical temperatures will decrease as the decoration level increasing for both lattices. The difference is that when the decoration level approaches to infinite the critical temperature of the triangular type lattice approaches to a finite value but the rectangular lattice is at zero temperature.
For the triangular type lattice, in the limit of infinite decoration level, the decorated lattices essentially possess the Sierpiński gasket or its triangle-star transformation as the inherent structure. The positions of the zeros for the infinite decorated lattices are shown to coincide with the ones for the Sierpiński gasket or its triangle-star transformation, and the distributions of zeros all appear to be an union of infinite scattered points and a Jordan curve which is the limit of the scattered points. For the rectangular lattice, the Julia sets with multifractal structure in the distribution arises when the decoration level approaches the limit of infinity, and the road to these Julia sets subject to the change of the decoration level is described. The global scaling properties of the Julia sets are studied by calculating the singularity spectrum and the generalized dimension.

並列關鍵字

Julia set ； Fisher zeros ； partition function ； hierarchical lattice ； Ising model

參考文獻

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