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Article Dans Une Revue Physical Review E : Statistical, Nonlinear, and Soft Matter Physics Année : 2013

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

Résumé

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 $\gamma$.
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Dates et versions

hal-00814719 , version 1 (17-04-2013)
hal-00814719 , version 2 (02-09-2013)

Identifiants

Citer

Sarah de Nigris, Xavier Leoncini. Critical behaviour of the XY -rotors model on regular and small world networks. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, 2013, 88 (1), pp.012131. ⟨10.1103/PhysRevE.88.012131⟩. ⟨hal-00814719v2⟩
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