Ising model with competing interactions on Cayley tree of order four : an analytic solution

Abstract

In this research, we investigate an Ising model with two restricted competing interactions (nearest neighbors, and one-level neighbors) on the Cayley tree of order four. We derive recurrent equation for the Cayley tree of order k. We found an analytic solution for a given interactions in the case of order 4. Our result of the critical curve shows the existence of the phase transition occurs in this model. This is a continuous study from the paper of (Ganikhodjaev N.N., 2002; Ganikhodjaev N.N., C H Pah, and M R B Wahiddin, 2003; Farrukh, M., Rozikov, U.A., 2005). Our investigation in this case is based on the recurrence equation of the partition function. We also give the calculation of the free energy from the description of Gibbs measure of the given Hamiltonian on Cayley tree of order four.