Potts Model with the Simplest Modulated Phase

We study the phase diagrams for the Potts model on the Bethe lattice with competing nearest-neighbor interactions J(1) and ternary interactions J(p) and J(t). At vanishing temperature, the phase diagram is fully determined for all values and signs of J(p)/J(1) and J(t)/J(1); in particular, we show that the phase diagrams contain ferromagnetic, paramagnetic and < 2 > phases only; i.e., the set of modulated phases consists of the < 2 > phase only for J(1) > 0 and J(p)/J(1) < 0. We verify that values of T/J(1) in the very narrow strip -0.11 < T/J(1) < 0 with -J(p)/J(1) < 0 favor the modulated phase, which is either a commensurate phase with a very large period or an incommensurate phase. The transition lines are obtained from stability conditions, and characteristic points in the phase diagram are analyzed by using numerical iterations. Also the wavevectors versus temperature are plotted for some critical points in the modulated phases.