The Limited Size Cayley Trees Spin System Phase Transition Study

In the recent years, spin models on the hierarchical lattice have received much more attention. In this thesis, fixing a single spin, we study the phase transitions and critical properties of the Ising model and the Blume-Capel model on the Cayley tree of the finite size by using the recursion relations. The main results are as follows:1. It is known that there is no spontaneous magnetizations for the Cayley tree in the thermodynamic limit. For the Ising model on the Cayley tree of the finite size, we get the spontaneous magnetizations for the different branching numbers B of 3 and 6 of the Cayley tree. Phase diagrams are plotted. Symmetry breaking is found for the finite systems from the diagrams. All these results show that the thermodynamic limit is a questionable approximation for the Ising model on the Cayley tree of the finite size. Our results are similar to the earlier results.2. In the thermodynamic limit condition, no spontaneous magnetizations is found by using the recursion method of the Cayley tree. For the Blume-Capel model on the Cayley tree of the finite size, we obtain the expression of the spontaneous magnetizations. Phase diagrams are plotted. Symmetry breaking is found for the finite system from the diagrams. It is thought that the symmetry breaking is due to the finite size and the structure of the Cayley tree. We also study the influences of the single spin anisotropy parameter on the finite magnetization of the Blume-Capel model on the Cayley tree.