Study Of Percolation Properties,Spiral Order And Phase Transition Dynamics Of Low Dimensional Magnetic Systems

Monte Carlo method is a powerful simulation tool with an extended application. It has been used in many fields of science research and every aspect of life. In this paper, we use this method to study the problems of thermodynamics and phase transition in low dimensional magnetic systems.The magnetic dilution was found to be existed universally in many materials. The dilutions of magnetic materials have an important effect on the properties of system. It has a valuable and practical application to find the phase transition properties as well as study the percolation properties and thermodynamic behaviors. In this paper, using a useful combined Monte Carlo algorithm, we simulate the dilute planar rotator model and generalized XY model to study the properties of thermodynamics and phase transition. During the simulation, two cases, dilution and no dilution, are considered to study respectively. Though the existence of dilutions reduces the correlations between spins and adds the effects of finite size scaling, it does not change the quasi-long-range order of system when the dilution density is above the percolation threshold. This is to say, it doesn't change the nature of phase transition. We get the BKT transition temperatures of different dilution densities by three effective methods: Binder's fourth order cumulant, finite-size scaling of magnetic susceptibility and helicity modulus. A linear relation between the transition temperature and occupied density is found, and the transition temperature vanishes at the point that the occupied density equals to the site percolation threshold. We conclude a regularity that the thermodynamic quantities and transition temperatures vary with the generalized parameter. The simulation results will give a significant reference to the observation of experiment.As an important type of anisotropic interaction, DM interaction was found to exist extensively in many materials. The study of the effects of helical order that deduced by DM interaction on the thermodynamic quantities and critical properties, as well as the examination of the universality for finite-size scaling under the chiral DM interaction, have an important meaning to understand the general regularity of magnetic systems. In this paper, we develop a type of Swendsen-Wang cluster algorithm and firstly use it to simulate the planar rotator model with DM interaction. The validity of this algorithm is proved. We study the properties of ground state with DM interaction and find the periodic arrangement of spins on ground states. Meanwhile, we get the periodicity of spin arrangement from theoretical derivation and simulation. We find that the classical order parameter is out of work and can't be used to attribute the BKT phase transition in this model. The helicity modulus has a negative value at this time. Based on the detailed analysis of boundary conditions and spin structures, we explain the reasons that cause these unusual phenomena, that is, the mismatch among the system size, the boundary conditions and the periodicity of spin arrangement. We discuss the effects of mismatch on the results of finite-size system. Also, we discuss the universality of finite-size scaling with the DM interactions. This phase transition is proved to be similar to the BKT phase transition of XY model, but their critical exponents are different. Based on the general regularity of spin arrangement, we give a new method to solve this mismatch. By the method of helicity modulus, the transition temperatures of different DM interactions are obtained and the linear relation between transition temperature and DM interaction is found at the same time. All the results give a new theoretical foundation and research method for the study of low dimensional magnetic systems with different anisotropic interactions.