Ising Model Spin S = 1 on Directed Barabási–albert Networks

Abstract

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. On these networks the Ising model spin S = 1 is now studied through Monte Carlo simulations. However, in this model, the order-disorder phase transition is well defined in this system. We have obtained a first-order phase transition for values of connectivity m = 2 and m = 7 of the directed Barabási-Albert network.Keywords:Monte Carlo simulation, Ising , networks, desorden. IntroductionSumour 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 (and we now confirmed this effect), in contrast to the case of undirected Barabási-Albert networks [4,5,6] where a spontaneous magnetisation was found lower a critical temperature which increases logarithmically with system size. More recently, Lima and Stauffer [7] 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 [8], for Ising-Barabási-Albert networks. They found a freezing-in of the magnetisation similar to [1,2], following an Arrhenius law at least in low dimensions. This lack of a spontaneous magnetisation (in the usual sense) is consistent with the fact that if on a directed lattice a spin S j influences spin S i , then spin S i in turn does not influence S j , and there may be no well-defined total energy. Thus, they show that for the same scale-free networks, different algorithms give different results. Now we study the Ising model for spin S = 1 on directed Barabási-Albert network and different from the Ising model for spin S = 1/2, the order-disorder phase transition of order parameter is well defined in this system. We have obtained a first-order

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