Phase transition for lattice models with restricted competing interactions on cayley tree

Abstract

We investigate the phenomenon of phase transition on Ising model with restricted competing interactions on Cayley tree. We first consider an Ising model with four competing interactions (external field, nearest neighbor, second neighbors and triples of neighbors) on the Cayley tree of order two. We found the analytic solution of the problem of phase transition for the case absent of external field and for the case absent of ternary interaction. Our main result is the critical curve of phase transition where for the condition satisfied, a phase transition occurs. This result is the generalization of ordinary Ising model and also other results in (Ganikhodjaev & Pah, 2003; Ganikhodjaev, 2002; Mukhamedov & Rozikov, 2004). Our investigation is based on two methods: Markov random field and recurrent equation of partition function. For general case with absent of ternary interaction, we extend the result from Ising model to Potts model. Based on the recurrent equation of the partition function derived, employing numerical method, a phase diagram is plotted with four regions i.e. Paramagnetic(P), Ferromagnetic(F), Modulated (M) and Anti-phase(< 2 >) similar as in (Vannimenus, 1981), (Inawashiro et al., 1983) and (Mariz et al., 1985). A new region is found, we called it quasi-paramagnetic, slightly different from the ordinary paramagnetic case. Our result is different from (Vannimenus, 1981; Inawashiro et al., 1983; Mariz et al., 1985). Also using another approach, namely contour method, we show that phase transition exist in 2 component model and we describe the phase diagram of the ground state of 3 component model. The 2 component and 3 component model are generalization of the Ising model and Potts model respectively. This method was recently introduced for binary interaction on the Cayley tree (Rozikov, 2005), while our method is carried out after revising and developed from the results of Minlos’s(Minlos, 2000).