The Study Of Phase Transitions Of S～4 Spin Systems On Fractal Lattices

In this thesis, phase transitions of spin systems on the Sierpinski gasket, diamond-type hierarchical lattices and Sierpinski carpet are investigated by means of real-space renormalization group. The main results of this thesis are as follows:1. The Ising system with two-spin interactions and triplet-spin interactions on the Sierpinski gasket is studied, and its critical point and critical exponents are obtained. The results indicate that there is only zero temperature phase transition in the system.2. Using the renormalization-group transformation and cumulative expansion technique, the phase transitions and critical properties of the S⁴ model on a special diamond-type hierarchical lattice are investigated. The results show that there exists a Wilson-Fisher fixed point and a Gaussian fixed point. Compared with the Gaussian model on the same hierarchical lattice, the critical exponents of two systems are different.3. The phase transitions and critical properties of the S⁴ model on a family of diamond-type hierarchical lattices (m branches) are studied. For the case of 3 ≤m ≤ 12, there exists a Wilson-Fisher fixed point and a Gaussian fixed point, and the former has a decisive effect on the critical properties of the system. For the case of m>12 and m ≤ 2, there only exists a Gaussian fixed point (K^* =b₂/2, u₂^* =0, h₂^* = 0), and the critical exponents of S⁴ system are identical with those of the corresponding Gaussian system on these lattices.4. Using the bond-moving renormalization-group method, the phase transitions and critical properties of S⁴ system on Sierpinski carpet are investigated, and its recursion relations and fixed points are obtained. For the case of Gaussian distribution constant b_w = 0, there exists a Wilson-Fisherfixed point and a Gaussian fixed point. For the case of Gaussian distribution constant 6 = 0, there only exists a Gaussian fixed point.