Potts model with q=3 and 4 states on directed small-world network

Monte Carlo simulations are performed to study the two-dimensional Potts models with q=3 and 4 states on directed small-world network. The disordered system is simulated applying the heat bath Monte Carlo update algorithm. The first-order and second-order phase transitions are found for q=3 depending on the rewiring probability p, but for q=4 the system presents only a first-order phase transition for any value p. This critical behavior is different from the Potts model on a square lattice, where the second-order phase transition is present for q≤4 and a first-order phase transition is present for q>4.