Critical phenomena on scale-free networks: logarithmic corrections and scaling functions

TL;DR

This paper investigates logarithmic corrections to phase transition scaling laws on scale-free networks, deriving new relations and analyzing how network properties influence critical behavior and scaling functions.

Contribution

It introduces new scaling relations for logarithmic correction exponents and analyzes their emergence due to network structure rather than spin fluctuations.

Findings

Logarithmic corrections appear at marginal node degree exponents.

Scaling functions depend nontrivially on network properties.

Thermodynamic scaling relations are fulfilled by network-induced exponents.

Abstract

In this paper, we address the logarithmic corrections to the leading power laws that govern thermodynamic quantities as a second-order phase transition point is approached. For phase transitions of spin systems on d-dimensional lattices, such corrections appear at some marginal values of the order parameter or space dimension. We present new scaling relations for these exponents. We also consider a spin system on a scale-free network which exhibits logarithmic corrections due to the specific network properties. To this end, we analyze the phase behavior of a model with coupled order parameters on a scale-free network and extract leading and logarithmic correction-to-scaling exponents that determine its field- and temperature behavior. Although both non-trivial sets of exponents emerge from the correlations in the network structure rather than from the spin fluctuations they fulfil the respective thermodynamic scaling relations. For the scale-free networks the logarithmic corrections appear at marginal values of the node degree distribution exponent. In addition we calculate scaling functions, which also exhibit nontrivial dependence on intrinsic network properties.