Critical behaviour of the XY -rotors 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 $N_{links}\sim N^{\gamma}$, where $1\le\gamma\le2$. We first focus on regular one dimensional rings in the microcanonical ensemble. For $\gamma<1.5$ the model behaves like 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 $\varepsilon=E/N, \varepsilon_{c}=0.75$. Moreover for $\gamma_{c}\simeq1.5$ 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 $\gamma$. We first analyze the topology and find that the small world regime appears for rewiring probabilities which scale as $p_{SW}\propto1/N^{\gamma}$. Then considering the XY-rotors model on these networks, we find that a second order phase transition occurs at a critical energy $\varepsilon_{c}$ 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 $\gamma$.