Renormalization Method And Numerical Simulation Analysis Of 2D Ising Model

Phase transition, which exists in nature widely, is a sort of break phenomenon and transition process which substance system changes from one steady state to another. Phase transition research plays a great role in the fields of physics, chemistry, biology and even sociology and it is a highlight theme in academia for a long time. The commonest phase transitions are the paramagnetic-ferromagnetic phase transition, the liquid-gas phase transition and the superconductive-normality phase transition in the metals. Nowadays people mostly apply renormalization group theory and numerical simulation methods to deal with critical phase transition problem. From 1970s, it is the renormalization group method got a breakthrough and became important implement and means for phase transition. But there are still some disadvantages, and then numerical simulation methods turned to effective means, especially the Monte-Carlo and Cellular Automata simulation. In this article we took the example of the two-dimensional Ising model, applied renormalization group method and numerical simulation, made a deep research and study on the paramagnetic-ferromagnetic critical phase transition problem. First of all, we compared and analyzed the results of the renormalization group transition with different size groups. Based on the literature [1] , we selected the enneahedral crystal lattice and hexagonal crystal lattice as Kadanoff groups and got the critical points and exponents by renormalization group transition and compared with precise results. Secondly, considering the detailed balance effect we simulated the critical temperature of the two-dimensional Ising spin dynamic model by Monte-Carlo method. As for the two dimension spin square-lattice, the array size selected 100 ′100, 200 ′200, 250 ′250, we got the critical temperature which is about 0.33—0.35 and inosculation with the rigorous results. Finally, as energy the final parameter, we construct a microcanonical ensemble and find that spin lattice system appears symmetry damaged with the energy increase, this shows magnetization phenomena will be occur in microcanonical ensemble system. Here we describe and discuss Ising model by Cellular Automata Q2R rule. Compared with the renormalization transition method, numerical simulation has an obvious advantage of solving Ising model, it not only can gain exact results, but also can describe the idiographic image and fluctuating of substance microcosmic states. So numerical simulation method would be applied widely in critical phase transition problem in the future.