Phase transitions of the Ashkin-Teller model including antiferromagnetic interactions on a type of diamond hierarchical lattice

Using the real-space renormalization-group transformation, we study the phase transitions of the Ashkin-Teller model including the antiferromagnetic interactions on a type of diamond hierarchical lattices, of which the number of bonds per branch of the generator is odd. The isotropic Ashkin-Teller model and the anisotropic one are, respectively, investigated. We find that the phase diagram, for the isotropic Ashkin-Teller model, consists of five phases, two of which are associated with the partially antiferromagnetic ordering of the system, while the phase diagram, for the anisotropic Ashkin-Teller model, contains 11 phases, six of which are related to the partially antiferromagnetic ordering of the system. The correlation length critical exponents and the crossover exponents are also calculated.