Critical behaviour of the XY -rotors model on regular and small world networks

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We study the XY-rotors model on small networks whose number of links scales with the system size Nlinks Nγ, where 1≤γ≤2. We first focus on regular one dimensional rings in the microcanonical ensemble. For γ<1.5 the model behaves like short-range one and no phase transition occurs. For γ>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 ε=E/N, εc=0.75. Moreover for γc≃1.5 we find that a non trivial state emerges, characterized by an infinite susceptibility. We then consider small world networks, using the Watts-Strogatz mechanism on the regular networks parametrized by γ. We first analyze the topology and find that the small world regime appears for rewiring probabilities which scale as pSW∝1/Nγ. Then considering the XY-rotors model on these networks, we find that a second order phase transition occurs at a critical energy εc which logarithmically depends on the topological parameters p and γ. We also define a critical probability pMF, corresponding to the probability beyond which the mean field is quantitatively recovered, and we analyze its dependence on γ.