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

Abstract

We study the XY-rotors model on small networks whose number of links scales with the system size $N_{links}\sim N^{\gamma }$, where $1\le \gamma \le 2$. 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 $\epsilon =E/N,{\epsilon }_c=0.75$. Moreover for ${\gamma }_c\simeq 1.5$ 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 $\gamma$. We first analyze the topology and find that the small world regime appears for rewiring probabilities which scale as $p_{SW}\propto 1/N^{\gamma }$. Then considering the XY-rotors model on these networks, we find that a second order phase transition occurs at a critical energy ${\epsilon }_c$ 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 $\gamma$.

References in corpus (9)

• Critical phenomena in complex networks
• Statistical mechanics and dynamics of solvable models with long-range interactions
• A maximum entropy principle explains quasi-stationary states in systems with long-range interactions: the example of the Hamiltonian Mean Field model
• Core-halo distribution in the Hamiltonian Mean-Field Model
• Exact Solution of Ising Model on a Small-World Network
• Ergodicity Breaking and Parametric Resonances in Systems with Long-Range Interactions
• Action diffusion and lifetimes of quasistationary states in the Hamiltonian Mean Field model
• Emergence of a non trivial fluctuating phase in the XY model on regular networks
• Linear theory and violent relaxation in long-range systems: a test case

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• Winding number excitation detects phase transition in one-dimensional XY model with variable interaction range
• Quantum magnetism on small-world networks
• Networks of planar Hamiltonian systems
• The free and freer XY models