First order phase transitions of the Potts model in fractal dimensions

The phase diagram of the q-state Potts model in fractal dimensions is studied with the help of Wang-Landau Monte Carlo simulations on Sierpinski and Menger fractal structures. A particular attention is paid to first order transitions just above the border separating the second order phase transition regime from the first order one. Although the translation invariance is strongly broken in deterministic fractals, evidence is given that such a deviation from the translational symmetry is not able to induce second order transitions for large values of q when the dimension lies between 1.9746 and 3. Moreover, the occurrence of second order transitions for very large values of q in the case of hierarchically weakly connected systems, that is when the fractal dimension is significantly smaller than 2, is pointed out. At last, the evolution of first order physical averages such as the latent heats and the interfacial free energies with the space dimensionality and the number of spin states is discussed.