Modulated Phase of a Potts Model with Competing Binary Interactions on a Cayley Tree

We study the phase diagram for Potts model on a Cayley tree with competing nearest-neighbor interactions J ₁, prolonged next-nearest-neighbor interactions J _p and one-level next-nearest-neighbor interactions J _o . Vannimenus proved that the phase diagram of Ising model with J _o =0 contains a modulated phase, as found for similar models on periodic lattices, but the multicritical Lifshitz point is at zero temperature. Later Mariz et al. generalized this result for Ising model with J _o ≠0 and recently Ganikhodjaev et al. proved similar result for the three-state Potts model with J _o =0. We consider Potts model with J _o ≠0 and show that for some values of J _o the multicritical Lifshitz point be at non-zero temperature. We also prove that as soon as the same-level interactionJ _o is nonzero, the paramagnetic phase found at high temperatures for J _o =0 disappears, while Ising model does not obtain such property. To perform this study, an iterative scheme similar to that appearing in real space renormalization group frameworks is established; it recovers, as particular case, previous work by Ganikhodjaev et al. for J _o =0. At vanishing temperature, the phase diagram is fully determined for all values and signs of J ₁,J _p and J _o . At finite temperatures several interesting features are exhibited for typical values of J _o /J ₁.