6 Potts model with q states on directed Barabási - Albert networks

On directed Barabási-Albert networks with two and seven neighbours selected by each added site, the Ising model with spin S = 1/2 was seen not to show a spontaneous magnetisation. Instead, the decay time for flipping of the magnetisation followed an Arrhenius law for Metropolis and Glauber algorithms, but for Wolff cluster flipping the magnetisation decayed exponentially with time. However, on these networks the Ising model spin S = 1 was seen to show a spontaneous magnetisation. In this model with spin S = 1 a first-order phase transition for values of connectivity z = 2 and z = 7 is well defined. On these same networks the Potts model with q = 3 and 8 states is now studied through Monte Carlo simulations. We have obtained also for q = 3 and 8 states a first-order phase transition for values of connectivity z = 2 and z = 7 of the directed Barabási-Albert network. Theses results are different from the results obtained for same model on two-dimensional lattices, where for q = 3 the phase transition is of second order, while for q = 8 the phase transition is first-order. Introduction Sumour and Shabat [1, 2] investigated Ising models with spin S = 1/2 on directed Barabási-Albert networks [3] with the usual Glauber dynamics. No spontaneous magnetisation was found, in contrast to the case of undi-rected Barabási-Albert networks [4, 5, 6] where a spontaneous magnetisation was found lower a critical temperature which increases logarithmically with system size. In S=1/2 systems on undirected, scale-free hierarchical-lattice small-world networks [7], conventional and algebraic (Berezinskii-Kosterlitz-Thouless) ordering, with finite transition temperatures, have been found. Lima and Stauffer [8] simulated directed square, cubic and hypercubic lattices in two to five dimensions with heat bath dynamics in order to separate the network effects form the effects of directedness. They also compared different spin flip algorithms, including cluster flips [9], for Ising-Barabási-Albert networks. They found a freezing-in of the magnetisation similar to