【作者】 李佳；

【导师】 何文辰；

【作者基本信息】 河北工业大学 ， 理论物理， 2004， 硕士

【摘要】 在相变和临界现象的研究中，重整化群方法是一种重要的方法，用它计算出的晶格的临界指数和临界点比平均场理论的结果更接近实验值。晶格的研究对象一般有两种，分形晶格和平移对称晶格。这两种晶格的特点也就决定了在重整化群计算时选取什么样的粗粒化方法，平移对称晶格一般采用自旋—元块法，分形晶格采用格点消元法。在前人的文章中平移对称晶格都是选三角或六角格子作为研究对象，本文采用苯型烃晶格作为研究对象。在Kadanoff集团的选取上本文提出了一个新观点，即不但要保持选取前和选取后晶格的对称性不变，还要保持格点的配位数不变。本文在计算过程中仍采用重整化群的自旋—元块法，并采用Ising模型，计算出的结果比三角和六角格子更接近Ising模型的严格解。所以本文还提出假设，只要是保持对称性和配位数不变，用这种方法计算出的任何二维晶格的临界行为都是相同的，这表明它们是同一普适类。无分支科赫曲线是一种典型的分形，前人的研究都局限于N=4(N是用线元，面元，或体元覆盖分形系统所需要的覆盖次数，确切的应该写成N=4~n，其中n为科赫曲线的级，但我们在重整化群计算时只考虑一个生成元，所以简化为N=4)情况，这种科赫曲线的相变点为零，是一种零温相变，这也是有限分岔系统的相变特征。本文对其进行一种推广，推广后N≥4，对于取不同N值的科赫曲线应用重整化群的格点消元法，仍采用Ising模型，结果得到同样的相变点，但是临界指数不完全相同，其中α，β，γ，δ相同，而υ，η不同。说明这些推广后的无分支科赫曲线并不属于同一个普适类。更多还原

【Abstract】 Renormalization group is an important method on phase transition and critical phenomena. The critical point and exponents of lattice by renormalization group are closer to the experimental values than by mean-field theory. Usually there are two kinds of lattice with different symmetry being studied, i.e., the fractals and transitionally invariant lattice. The traits of the two kinds of lattice determine which method we use to study it. The site-block method is often for transitionally invariant lattice and decimation for fractals. In previous papers the triangular and the hexagonal lattices are often the study objects, but benzenoid lattice is the study object in this paper. In the selection of Kadanoff cells a new idea is given, i.e., not only the symmetry of lattice before and after selection must be kept unchanged, but also the coordination number must be kept unchanged. The results of benzenoid lattice by site-block based on Ising model are closer to the exact values of Ising model than the triangular and hexagonal lattices. So a hypothesis is proposed, i.e., as long as the symmetry and coordination number are kept unchanged the critical behavior of any two-dimensional lattice is same by this method. That indicate those two-dimensional lattices belong to the same universal class. Nonbranching Koch curve is one typical fractal and the former work on it confine to N=4(N is the times that we use line, area or body unit to cover the fractal system. The exact expression is N = 4", in which n is the stage of Koch curve. Since we only consider one generator in computation process, so we simplify it as N=4). The critical point of this kind of Koch curve is zero, also called zero temperature phase transition, and this is the character of all the limited branching systems. A generalization, N> 4, is given in this paper. In result the critical points of these Koch curve with different values of TV by decimation based on Ising model are same, but the critical exponents are different. The four ones α, β, γ, δ are same, and the other two v, η are different. That indicates these generalized Koch curves don’t belong to the same universal class.更多还原