Exact results and new insights for models defined over small-world networks. First and second order phase transitions. I: General result

TL;DR

This paper introduces a general effective field theory for models on small-world networks, accurately predicting critical behavior and phase transitions, including novel insights into first- and second-order transitions under various conditions.

Contribution

It provides a new, general method to analyze phase transitions in small-world network models, capturing critical surfaces and thresholds with exactness in certain regions.

Findings

Second order phase transition occurs for J_0 ≥ 0 regardless of dimension or connectivity

Negative J_0 leads to complex phase behavior including spin glass and multiple first- and second-order transitions

The method offers clear insights into the physics of models on small-world networks.

Abstract

We present, as a very general method, an effective field theory to analyze 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 gives the exact critical behavior and the exact critical surfaces and percolation thresholds, and provide a clear and immediate (also in terms of calculation) insight of the physics. The underlying structure of the non random part of the model, i.e., the set of spins staying in 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 to the known fact that a second order phase transition takes place, independently of the dimension d_0 and of the added random connectivity c. However, when J_0<0, a completely different scenario emerges where, besides a spin glass transition, multiple first- and second-order phase transitions may take place.