Magnetic models on Apollonian networks

TL;DR

This paper investigates the thermodynamic and magnetic properties of Ising models on the triangular Apollonian network, revealing long-range magnetic order and providing exact thermodynamic calculations through transfer matrix methods.

Contribution

It introduces a transfer matrix approach to exactly compute thermodynamic functions of Ising models on Apollonian networks, considering various coupling constants.

Findings

Long-range magnetic order observed for ferromagnetic and antiferromagnetic couplings.

Exact thermodynamic properties obtained via iterative transfer matrix maps.

No evidence of asymptotic criticality, except for size-dependent effects.

Abstract

Thermodynamic and magnetic properties of Ising models defined on the triangular Apollonian network are investigated. This and other similar networks are inspired by the problem of covering an Euclidian domain with circles of maximal radii. Maps for the thermodynamic functions in two subsequent generations of the construction of the network are obtained by formulating the problem in terms of transfer matrices. Numerical iteration of this set of maps leads to exact values for the thermodynamic properties of the model. Different choices for the coupling constants between only nearest neighbors along the lattice are taken into account. For both ferromagnetic and anti-ferromagnetic constants, long range magnetic ordering is obtained. With exception of a size dependent effective critical behavior of the correlation length, no evidence of asymptotic criticality was detected.