Criticality of two- and three-spin Ising model in an external field on a fractal family

We study the Ising model with pair and alternate triplet interactions subjected to an external magnetic field on a family of infinitely ramified fractal lattices with a triangular topology. The three-dimensional phase diagram and correlation length critical exponents are calculated within an exact real-space renormalization group framework. The zero-field results for the ferromagnetic model show that, although the pure triplet case and the pure nearest-neighbor pair interaction model are in different universality classes, there is no crossover phenomenon since the system becomes paramagnetic in the mixed case. In the pure nearest-neighbor antiferromagnetic model, the appearance of an unusual Berker and Kadanoff's-phase type (with a power-law decay of correlations) when the fractal dimension is sufficiently high is destroyed by the application of a magnetic field or a triplet interaction field.