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

TL;DR

This paper develops an effective field theory approach to analyze phase transitions in models on small-world networks, capturing critical behavior and thresholds, including novel scenarios with multiple phase transitions.

Contribution

It introduces a general effective field theory method that accounts for the underlying lattice structure and predicts critical phenomena in small-world network models.

Findings

Second-order phase transitions occur independently of lattice dimension and connectivity.

Novel scenarios with multiple first- and second-order phase transitions emerge for negative coupling.

Analytical analysis of specific models like Viana-Bray and spherical models demonstrates the method's applicability.

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.