Gaussian approach for Ising model

Magnetization phenomenon has always been a subject of interest in physics. For this reason, one of the most studied topics in physics even today, the phase transition concept of magnetization, was described in 1925 by a model created by Ising and exactly solved in one dimension. In two dimensions, this model has been expanded and solved for some special cases. Since there is no exact solution for the rest of the cases such as three dimensions and higher dimensions, the mean-field theories have been developed to explain this phenomenon. In this thesis, solutions in one dimension and correlation functions are demonstrated after introducing Ising model. Later the mean-field theories are presented and the configurational mean-field transfer matrix method is introduced. In this method, it is aimed to combine L. Kadanoff's mean-field approach and T. Kaya's mean-field approach. The mean-field approximation is based on the assumption that a particular configurational average magnetic moment is substituted for the central magnetic moment in Ising Hamiltonian. Two different mean-field approaches have been used in this approach. The first one is the pure mean-field type approach and the second one is the reduced transfer matrix method. In both of these approaches, the critical coupling strengths are calculated both numerically and analytically using the saddle point approximation. In this study, it is considered that the adoption of configurational averages instead of the ensemble averages provides a more appropriate physical framework than the self-consistent mean-field theories since it does not assume the existence of phase transition from the outset.