| Summary: | 博士 === 中原大學 === 物理學系 === 87 === We shall study the critical behavior of the Ising model on hierarchical lattices. We first obtain the renormalization maps for various hierarchical lattices. Based on the renormalization maps we obtain critical exponents, reduced free energy per bond, and zeros of partition functions. Then we analyze the effects of the geometric structure of an MKH (Migdal-Kadanoff Hierarchy) on the Ising model. The geometric structure of an MKH is described by two factors in a basic cell: one is the number of strings, M, and the other is the number of vertices, A, in a string. We first study the variation of the critical behaviors, including the location of the critical temperature and the critical exponents, in terms of these two geometric factors, and then we study the distribution of complex zeros closest to the real temperature axis and observe their finite-size scaling behavior for various MKHs. Finally, we analyze the geometric effect on the global scaling properties of the Julia sets of the Fisher zeros of the partition function by studying the variations of generalized fractal dimensions and the singularity spectrum. We want to assess what degree of metric universality is there. There are two tools performed to analyze the metric properties of the Julia set. The first one is the famous thermodynamic formalism of multifractal analysis. The analysis is done on the basis of knowledge of the recursion relation of the multifractal invariant measure by calculating the generalized fractal dimensions and the singularity spectrum. The second one is the wavelet transform which has recently been developed. The basic idea is to calculate the Hölder exponent h of the measure and its D(h) singularity spectrum that the Hausdorff dimension of the set where the Hölder exponent is equal to h. From the analysis of our results, we find that the scaling law holds for hierarchical lattices as for regular lattices, but the universality concept is violated. The claim that the fractal dimension and the connectivity serve as criteria for universality of critical exponents on hierarchical lattices is not enough. It therefore seems that one has to introduce more parameters to distinguish completely these MKHs. On the other hand, it is found that there is no metric universality, although there are some aspects of metric universality shown in the multifractal behavior of Fisher zeros.
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