| In this thesis, we mainly studied phase transitions of the Ashkin-Teller model on hierachical lattices with irreducible generators. There are three parts in this thesis. In first chapter, we introduced the phase transition and renormalization group theory. In second chapter, we introduced the hierarchical lattices and Ashkin-Teller model. And in third chapter, we studied the phase transitions of the Ashkin-Teller model on hierarchical lattices with irreducible generators.In the first chapter, we introduced the theory of phase transition and renormalization group. Firstly, we introduced some concepts, including phase transition, phase transition classification, order paremeter, critical exponents and universality. Secondly, we introduced the theory of renormalization group transformation, And in order to understand renormalization group transformation further.,we used the Ising model. At last, on the base of the characteristics of hierarchical lattice self-similar, we emphasized the importance of renormalization group transformation methods for the statistical study of level lattice spin. And these theories are the basis of the following parts.In the second chapter, we introduced the hierarchical lattice and Ashkin-Teller model. We introduced the formation and classification of hierarchical lattice. By comparing with translation symmetry lattice, we emphasized the meaning of studying Ashkin-Teller model phase transition which have irreducible generators hierarchical lattice. Besides, we reviewed the Ashkin-Teller model, and introduce some results of Ashkin-Teller model in domestic and foreign.In the third chapter, we studied phase transition of five kinds of Ashkin-Teller models, which have irreducible generators hierarchical lattice. Taking use of the method of grid eliminate renormalization group transformation, we studied phase transition of five kinds of Ashkin-Teller models, which have irreducible generator hierarchical lattice, and calculated the correlation length critical exponents and cross index. And then by analyzing how the hierarchical lattice geometric parameters influence on phase transition critical exponents, we find the question that the universality of fractal phase transition is still to be solved. |
