Inhomogeneous Ising model on the Cayley tree of 2nd and 3rd order and two-dimensional integer lattice

Abstract

The phenomenon of phase transition occurs in our daily life. However among models of phase transition in magnetic system, Ising model is the most studied model. In this research, we investigate the Ising model on the semi-infinite Cayley tree of second order with left and right interactions. We describe phase diagram of the model and show that it consists of two phases, namely ferromagnetic and antiferromagnetic. We also produce the recurrence equations of the model and proved that for two different interaction parameters the phase transition will occur for both phases. Similarly, for the Ising model on the Cayley tree of third order with inhomogeneous interactions we establish necessary and sufficient conditions to reach phase transition. Besides, we also study the Ising model on two-dimensional integer lattice with inhomogeneous interactions. In this model, we consider two different interaction parameters which are the horizontal interactions parameter and also the vertical interactions parameter. We proved that this model can reach a phase transition. Previously, Onsager considered the case where both horizontal interactions parameter and vertical interactions parameter are different. For any fixed horizontal interactions parameter and vertical interactions parameter, he showed that below a critical temperature which depends on horizontal interactions parameter and vertical interactions parameter, phase transition occurs using some transfer matrix method. However, we verify the existence of phase transition using contours methods introduced by Sinai. We showed that there exist at least two limits Gibbs distribution which leads to the phenomena of phase transition.