q-state modified Potts model on a Cayley tree and its phase transition in antiferromagnetic region

Abstract

We introduce a modified version of the Potts model, characterized by a new Hamiltonian that assigns energy $+J$ when two nearest neighboring spins are identical, and $-J$ when interacting spins differ. This research initializes the $q$-state modified Potts model on a semi-infinite Cayley tree of order $k$, utilizing a newly proposed Hamiltonian that promotes dissimilar neighboring spins. This modification, which diverges from the traditional Potts model, addresses the influence of competing interactions pertinent to the antiferromagnetic phase transition regime. Using the cavity method, we construct limiting Gibbs measures by analyzing the associated recurrence equations. The existence of translation-invariant solutions to these relations are further explored using Preston’s approach. Our results demonstrate the existence of phase transitions exclusively in the antiferromagnetic region. Furthermore, through a stability analysis of the dynamical system, we uncover both chaotic and periodic behaviors, highlighting the rich complexity induced by the interplay of non-trivial interactions and the non-amenable geometry of the Cayley tree.