Critical behavior of the ferromagnetic q-state Potts model in fractal dimensions:: Monte Carlo simulations on Sierpinski and Menger fractal structures

The extension of the phase diagram of the q-state Potts model to noninteger dimension is investigated by means of Monte Carlo simulations on Sierpinski and Menger fractal structures. Both multicanonical and canonical simulations have been carried out with the help of the Wang-Landau and the Wolff cluster algorithms. Lower bounds are provided for the critical values q(c) of q where a first-order transition is expected in the cases of two structures whose fractal dimension is smaller than 2: The transitions associated with the seven-state and ten-state Potts models on Sierpinski carpets with fractal dimensions d(f)similar or equal to 1.8928 and d(f)similar or equal to 1.7925, respectively, are shown to be second-order ones, the renormalization eigenvalue exponents y(h) are calculated, and bounds are provided for the renormalization eigenvalue exponents y(t) and the critical temperatures. Moreover, the results suggest that second-order transitions are expected to occur for very large values of q when the fractal dimension is lowered below 2-that is, in the case of hierarchically weakly connected systems with an infinite ramification order. At last, the transition associated with the four-state Potts model on a fractal structure with a dimension d(f)similar or equal to 2.631 is shown to be a weakly first-order one.