Abstract
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.
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Ganikhodjaev, N., Temir, S. & Akin, H. Modulated Phase of a Potts Model with Competing Binary Interactions on a Cayley Tree. J Stat Phys 137, 701–715 (2009). https://doi.org/10.1007/s10955-009-9869-z
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DOI: https://doi.org/10.1007/s10955-009-9869-z