Phase transitions for lattice models on Cayley trees

Abstract

In this research, we investigate phase transitions phenomenon for two lattice models, i.e., Ising model and λ-model on Cayley trees. For the first result, we prove the existence of phase transitions by analyzing the recurrent equation that derived from the Ising model with competing interactions (nearest neighbors, and one-level neighbors) on the Cayley tree of order five. We found an exact solution for the given interactions in the case of order 5. We confirm a special case of the conjecture for the critical curve that separate the region of single fixed point and multiple fixed points that have been proposed by Pah and Ali (2013). For the second result, we study the λ-model with spin values {1, 2, 3} on the Cayley tree of order two. We calculated the ground states energy of the λ-model. We proved that for some cases for the ground states, there exist three translation-invariant, periodic and uncountable number of ground states. Further, we described all translation-invariant Gibb measures for λ-model. Lastly, we proved the existence of 2-periodic Gibbs measures for λ-model by considering some special cases.