The phase diagram of a three-state potts model with competing binary interaction on a cayley tree up to the third nearest-neighbour generations

Abstract

This research is an extension of Ganikhodjaev et al. (2008), where they had generated and analysed the phase diagram consisting of the first and second nearest-neighbour binary interaction on the three-state Potts model on a Cayley tree. Therefore, in continuing the research, it is in our interest to investigate the effect of the third nearest-neighbour binary interaction to the phase diagrams of the three-state Potts model on a Cayley tree. Therefore, we generate and analyse the phase diagrams of the three-state Potts model, considering prolonged competing binary interaction "J" _"2" and〖" J" 〗_"3" on the same branch of the Cayley tree up to the third nearest-neighbour generations. We derive the recurrence system of equations while considering some ranges of competing parameters. We carry out a numerical procedure by applying several stability conditions and characteristic points into the iteration scheme. For some non-zero parameter "J" _"3" , we found the additional phases of period 5, 6, 9, and 11, with the ferromagnetic, antiphase, paramagnetic, antiferromagnetic and modulated phase. For the modulated phase, we further study the existence of phases with period larger than 12 by conducting a numerical analysis on the variation of wavevector and Lyapunov exponent. This results in the discovery of some phases with period larger than 12, which are the phases of period 13, 16, 23, 26 and 49. From the results obtained as presented in this thesis, it is clear that the third nearest-neighbour binary interaction on the Cayley tree, considering the three-state Potts model, gives significant effect to the generation of the phase diagram.