Sub-lattices Spin Model Phase Transition Of Research

Phase transitions and critical phenomena in condensed matter physics and statistical physics have been a quite important field of inquiry, since T. Andrews discovered the critical point in 1869. In 1970s B. Mandelbrot named the self-similar structure with scaling invariant for fractal and presented detailed discuss of in his treatise, which expanded phase transitions into another research subject.With the development and application of fractal, phase transitions and critical phenomena on fractal lattices have aroused people's great interest, and many significant results have been obtained in the last two decades. By placing Ising-models on the sites of fractal lattices, Genfen et al have studied critical phenomena of Ising-model on fractals and it's universality in the early 1980s, obtained the representative results. Recently, much attention has been paid to the study of phase transitions of the continuous spin models on fractal lattices, e.g. Gauss model, S4 model.The usual ways to study the subject are the transfer-matrix method, combination solution, the graphic expansion and the renormalization -group technique, and so on. The renormalization-group technique is proved to a comparatively powerful means, because it avoids calculating partition function directly.In this thesis some theoretical studies of phase transitions and critical phenomena of ferromagnetic systems on fractal lattices, namely, the Gauss model, S4 model on Sierpinski-gasket and S4 model on non-branching Koch curve are performed without the external magnetic field by the deci -mation real-apace renormalization-group with spin-rescaling technique. We calculated the critical points and critical exponents, and discussed one of the fundamental questions involved in phase transition-universality. The paper's main contents are composed of three parts below:1. Decimation renormalization-group with spin-rescaling techniques are applied to Gaussian model with triplet interaction on the sites of Sierpinski -gasket fractal lattices without external magnetic field. Fixed points and critical exponent are got. The results show that the fixed points and critical exponent changed because of the perturbation.2. In the same way, the critical behavior of the S4 model on SG fractal lattices is investigated without external field. We obtained Gaussian and W.F. fixed points. Comparing the result with that of Gaussian model and Ising model obtained recently, we propose that the phase transition of Ising model on Sierpinski-gaskit occur at zero-temperature, but the phase of Gauss model occur at finitude-temperature, and that the critical exponent of S4 depend on the fractal dimensionality, while the critical exponent of Ising model has nothing to do with the fractal dimensionality, v = 1. These show S model has some similar property with Gaussian model.3. We studied the critical behavior of the S4 model on non-branching Koch curve and only obtained the Gaussian fixed point. The result reflects the S4 model and the Gaussian model belongs to the same universality class. In comparison with the results on Sierpinski-gasket, it shows that this model on Koch has the same properties as on translation symmetric lattices.