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 J1, prolonged next-nearest-neighbor interactions Jp and one-level next-nearest-neighbor interactions Jo. Vannimenus proved that the phase diagram of Ising model with Jo=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 Jo≠0 and recently Ganikhodjaev et al. proved similar result for the three-state Potts model with Jo=0. We consider Potts model with Jo≠0 and show that for some values of Jo the multicritical Lifshitz point be at non-zero temperature. We also prove that as soon as the same-level interactionJo is nonzero, the paramagnetic phase found at high temperatures for Jo=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 Jo=0. At vanishing temperature, the phase diagram is fully determined for all values and signs of J1,Jp and Jo. At finite temperatures several interesting features are exhibited for typical values of Jo/J1