Effective field theory for models defined over small-world networks. First and second order phase transitions

Abstract

We present an effective field theory method to analyze, in a very general way, models defined over small-world networks. Even if the exactness of the method is limited to the paramagnetic regions and to some special limits, it provides, yielding a clear and immediate (also in terms of calculation) physical insight, the exact critical behavior and the exact critical surfaces and percolation thresholds. The underlying structure of the non random part of the model, i.e., the set of spins filling up a given lattice L_0 of dimension d_0 and interacting through a fixed coupling J_0, is exactly taken into account. When J_0\geq 0, the small-world effect gives rise, as is known, to a second-order phase transition that takes place independently of the dimension d_0 and of the added random connectivity c. When J_0<0, a different and novel scenario emerges in which, besides a spin glass transition, multiple first- and second-order phase transitions may take place. As immediate analytical applications we analyze the Viana-Bray model (d_0=0), the one dimensional chain (d_0=1), and the spherical model for arbitrary d_0.

28 pages, 18 figures; merged version of the manuscripts arXiv:0801.3454 and arXiv:0801.3563 conform to the published version

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• 1D Three-state mean-field Potts model with first- and second-order phase transitions
• Mean-field models with short-range correlations
• Absence of small-world effects at the quantum level and stability of the quantum critical point
• Critical phenomena on heterogeneous small-world networks
• Long-time tails in sheared fluids
• Small-world of communities: communication and correlation of the meta-network
• Exact results for the Ising model on a small-world network
• Forest expansion of two-body partition functions for sparse interaction graphs