Phase transitions in the Potts model on complex networks

TL;DR

This paper investigates phase transitions in the q-state Potts model on uncorrelated scale-free networks, revealing how the nature of the transition depends on network topology and model parameters, with specific results for percolation.

Contribution

It extends the analysis of the Potts model to complex networks, showing how phase transition types depend on the degree distribution exponent and the number of states.

Findings

Identifies conditions for first- or second-order phase transitions based on q and λ.

Establishes a correspondence between Potts model exponents and percolation on scale-free networks.

Finds logarithmic corrections to scaling at λ=4 for the q=1 case.

Abstract

The Potts model is one of the most popular spin models of statistical physics. The prevailing majority of work done so far corresponds to the lattice version of the model. However, many natural or man-made systems are much better described by the topology of a network. We consider the q-state Potts model on an uncorrelated scale-free network for which the node-degree distribution manifests a power-law decay governed by the exponent \lambda. We work within the mean-field approximation, since for systems on random uncorrelated scale-free networks this method is known to often give asymptotically exact results. Depending on particular values of q and \lambda one observes either a first-order or a second-order phase transition or the system is ordered at any finite temperature. In a case study, we consider the limit q=1 (percolation) and find a correspondence between the magnetic exponents and those describing percolation on a scale-free network. Interestingly, logarithmic corrections to scaling appear at \lambda=4 in this case.