The Renormalization Group Analysis Of Spin Models On Fractal Lattices

Much attention has been paid to phase transition and critical phenomena which is an interdisciplinary field . Phase transition on fractal lattices is also an important field in physics.In the study of fractal lattices, the renormalization group approach is proved to be the most powerful means. This method is one of the most widely applied theoretical approach. This way don't solve partition function directly, but study the transformation that makes partition function unchanged. These transformation are made up of so-called renormalization group. Then find the fixed points of transformation, among which those unstable ones or saddle points are critical points of phase transition. It is known that system' critical character through calculating critical exponents and fractal dimension of system.Quasicrystal is a multiple fractional dimension pattern. Many models have been proposed to conveniently study the property of quasi-lattice system. Among which one dimension Fibonacci model is the main one. Binary Fibonacci model had been studied previously. In this thesis, the Ising model on one dimensional ternary generalized Fibonacci model generated bythe substitution A→ ABC,B → A,C → B is studied by renormalization transformation. Weuse the model which is made up of three kinds of atomic separation, i.e. long middle and short to show the transformation of Fibonacci chain. Those three kinds of atomic separation satisfythe substitution rule on ternary sequence. Critical temperature T_c and critical exponent vofcorrelation length which are calculated do not differ from those for a regular periodic system. So the conclusion that one dimensional Ising did not has phase transition also apply to ternary generalized Fibonacci chain.In the article, phase transition and critical behavior for BEG model is studied by renormalization transformation on particular diamond-type hierarchical lattice that is one kind of nonuniform lattice. The temperature of phase transition occurs for BEG is infinity on particular diamond-type hierarchical lattice. But the result is different obviously, when there is crystal field A or not. When crystal field strength A does not equal zero, the eigenvalues for matrix of a linear transformation which come from renormalization transformation don t greater than numerical value one, that is to say, there is not unstable fixed point. So critical exponents can t be calculated and critical character can t be analyzed further. When A equal zero, critical exponent v of correlation length can be calculated. It is shown that the critical behavior of this lattice differ from that of general diamond-type hierarchical lattice, moreover, critical exponent v is different. It indicates these two kinds of lattice don t belong to the same universal class.