Critical behavior of the XY-rotor model on regular and small-world networks

We study the XY rotors model on small networks whose number of links scales with the system size N-links similar to N-gamma, where 1 <= gamma <= 2. We first focus on regular one-dimensional rings in the microcanonical ensemble. For gamma < 1.5 the model behaves like a short-range one and no phase transition occurs. For gamma > 1.5, the system equilibrium properties are found to be identical to the mean field, which displays a second-order phase transition at a critical energy density epsilon = E/N, epsilon(c) = 0.75. Moreover, for gamma(c) similar or equal to 1.5 we find that a nontrivial state emerges, characterized by an infinite susceptibility. We then consider small-world networks, using the Watts-Strogatz mechanism on the regular networks parametrized by gamma. We first analyze the topology and find that the small-world regime appears for rewiring probabilities which scale as p(SW) proportional to 1/N-gamma. Then considering the XY-rotors model on these networks, we find that a second-order phase transition occurs at a critical energy ec which logarithmically depends on the topological parameters p and gamma. We also define a critical probability p(MF), corresponding to the probability beyond which the mean field is quantitatively recovered, and we analyze its dependence on gamma.