Ising model with competing interactions on Cayley tree of Order 4: an analytic solution

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 a recurrent equation for the Cayley tree of order k. We found an analytic solution for the 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. 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