Studies of Isotropic Ising Systems with Competing Interactions

In this thesis we examine the phase diagrams of isotropic Ising models on the square and simple cubic lattices with ferromagnetic coupling J_1 between nearest neighbours and additional interactions J_2 between diagonal neighbours and J_3 between second axial neighbours (the I123 model). The methods used are high temperature susceptibility series expansions and Monte Carlo simulations. We have extended the multigraph expansion method up to 13th order for the susceptibility. We also investigate the application of Schwinger-Dyson equations to these models. The critical properties of the models are obtained through: (1) analysis of the series by using the ratio method, the Pade approximant method and in some cases the more general differential approximant method, (2) Monte Carlo simulations; by observations of the specific heat peaks in the specific heat-temperature diagrams and by integrating the energy to obtain the entropy for cases where the energy-temperature diagrams show strong hysteresis. We have obtained the phase diagrams for special cases of the models: (1) the J_1 - J_2 model (J_3 = 0), (2) the BNNNI model (J_2 = 0) and (3) the Widom model of microemulsions (J_2/J_3 = 2). Phase diagrams for more general cases ( J_2/J_3 = 2.5, 1.0 and J _3/J_1 = -1.0 in two dimensions were also obtained. Our work confirms previous results for the breakdown of universality in the superantiferromagnetic region for the J_1 - J_2 model, and gives more accurate estimates for the critical parameters. For the BNNNI model, and for the I123 model in the region J_2/J_3 < 2, we find complex behaviour showing a possible sequence of transitions in the modulated region. For the Widom model in two dimensions we find that the critical temperature in the region of strong antiferromagnetic J_3 is very small or perhaps zero. Overall this studies enable us to better understand the critical behaviour of the isotropic Ising models with competing interactions.