Potts Model with Competing Interaction Up to the Third Nearest-neighbour Generation on a Cayley Tree

It is in our interest to investigate the effect of the third nearest-neighbour binary interaction to the phase diagrams of the Potts model on a Cayley tree. Therefore, we generate and analyse the phase diagrams of the 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 taking into account 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.