Research On Phase Transition And Critical Phenomena Of Two Lattice With Different Symmetry Using Renormalization Group

Renormalization group is an important method on phase transition and critical phenomena. The critical point and exponents of lattice by renormalization group are closer to the experimental values than by mean-field theory. Usually there are two kinds of lattice with different symmetry being studied, i.e., the fractals and transitionally invariant lattice. The traits of the two kinds of lattice determine which method we use to study it. The site-block method is often for transitionally invariant lattice and decimation for fractals. In previous papers the triangular and the hexagonal lattices are often the study objects, but benzenoid lattice is the study object in this paper. In the selection of Kadanoff cells a new idea is given, i.e., not only the symmetry of lattice before and after selection must be kept unchanged, but also the coordination number must be kept unchanged. The results of benzenoid lattice by site-block based on Ising model are closer to the exact values of Ising model than the triangular and hexagonal lattices. So a hypothesis is proposed, i.e., as long as the symmetry and coordination number are kept unchanged the critical behavior of any two-dimensional lattice is same by this method. That indicate those two-dimensional lattices belong to the same universal class. Nonbranching Koch curve is one typical fractal and the former work on it confine to N=4(N is the times that we use line, area or body unit to cover the fractal system. The exact expression is N = 4", in which n is the stage of Koch curve. Since we only consider one generator in computation process, so we simplify it as N=4). The critical point of this kind of Koch curve is zero, also called zero temperature phase transition, and this is the character of all the limited branching systems. A generalization, N> 4, is given in this paper. In result the critical points of these Koch curve with different values of TV by decimation based on Ising model are same, but the critical exponents are different. The four ones α, β, γ, δ are same, and the other two v, η are different. That indicates these generalized Koch curves don't belong to the same universal class.