| Phase transitions and critical phenomena in statistical physics are a quite important field of inquiry. As is well known to us, the conversion of iron from paramagnetic to ferromagnetic form, the transitions from conductor to superconductor and from normal fluid to superfluid are examples of critical phenomena. The fractal is a geometrical figure with self-similar symmetry, and it is an important tool for characterizing irregular structures in nature that are self-similar on certain length scales. For example, the Koch curves can be viewed as a mathematical model for coastlines, random walks can be used to mimic protein molecular chains, and self-avoiding walks can serve as a model for linear polymers, and so on. In order to describe explicitly the physical properties and laws of irregular structure systems, especially magnetic systems, it is no doubt necessary for us to study phase transitions of fractal lattices.In the early 1980s Gefen et al have presented their series work of critical phenomena on fractals. Since then, an investigation into phase transitions of spin models on fractals aroused people's great interest and developed gradually an important research subject. The usual ways to study the subject are the transfer-matrix method, combination solution, the renormalization-group technique, and graphic expansion, and so forth. But due to the fact that the fractal has the property of self-similarity, the technique of real-space renormalization-group is proved to a comparatively powerful means. In this thesis, some theoretical studies of phase transition of ferromagnetic systems on three fractal lattices, namely, the Sierpinski Gasket, the X fractal lattice and the Diamond-type Hierarchial lattice are performed in the external magnetic field by means of real-space renormalization group. We calculate the critical points and critical exponents, and compare the results obtained from different models on thesame fractal lattices. Furthermore, we discuss two fundamental questions involved in phase transitions ?universality and scaling property. The thesis's main contents are composed of five parts below:1. We studied the critical properties of the Ising model on Sierpinski Gasket lattices in the presence of external field by using the real-space renormalizaion-group method. The results show that even in external field, the phase transition still occurs at zero-temperature, which can be known to be the first-order rather than second-order phase transition in terms of the feature of zero-temperature phase transition.2. A technique of spin-rescaling is introduced in the procedure of renormalizaion-group transformation. With such two methods, we investigate the critical properties of the Gauss model on Sierpinski Gasket lattices in the presence of external field. It is found that, at the critical point, the nearest interaction parameter associated with temperature is K'=b/4 and the external field h'=0. The result is much different from that of the Ising model.3. In the same way as that used above, the critical behavior of the Gauss model on X fractal lattices is investigated in the case of external field. It is shown that the X fractal lattice, an inhomogeneous fractal lattice, has the critical point K' = bifJq, ,h' =0, where qt and 6(/ are the coordination number and Gaussian distribution constant of the site /. In addition, we find that the critical points do not vary with the space dimensionality of the fractal lattices but the critical exponents reveal certain tendency in change as the space dimensionality increases.4. With the renormalization group method and the technique of cumulate expansion, we studied the critical behavior of the S4 model on Diamond-type Hierarchial lattices. In the course of studies, we make a modification on the form of the S4 model, in which it is assumed that the Gaussian distribution constant b, four spins interaction parameter u andthe external magnetic field h depend on the coordination number q1, of the... |
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