Potts model with the competing interactions on a Cayley tree-like lattice: Quadrangle Chandelier

We study the phase of diagrams for the Potts model on a Cayley tree-like lattice, called Quadrangle Chandelier, with competing nearest-neighbour interactions J1,prolonged next nearest-neighbour interactions Jp and one-level quadrangle nearest-neighbour interactions Jo. The phase diagram is studied for several ranges of the competing parameters and its show the appareance of several features and modulated phrase arising from the frustation effects introduced by the one-level quadrangle nearest-neighbour interactions Jo. The result gives the different with the phase studied in Cayley tree order 2. To perform this study, an iterative scheme similar to that appearing in real space renomalization group frame works is established, it recovers as particular case, previous work by Ganikhodjaev et.al (2008) for k=