Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks

TL;DR

This paper investigates the zeros of the partition function of the Ising model on complete graphs and annealed scale-free networks, revealing how the zeros' behavior depends on network structure and degree distribution exponent.

Contribution

It derives the integral representation of the partition function for annealed scale-free networks and analyzes how the zeros' properties vary with the degree exponent <<5, including violations of the Lee-Yang circle theorem.

Findings

Zeros on complete graphs are purely imaginary.

Zeros on scale-free networks have non-zero real parts for 3<<5.

Critical exponents depend on the degree distribution exponent <<5.

Abstract

We analyze the partition function of the Ising model on graphs of two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as $P(k)\sim k^{-\lambda}$. We are interested in zeros of the partition function in the cases of complex temperature or complex external field (Fisher and Lee-Yang zeros respectively). For the model on an annealed scale-free network, we find an integral representation for the partition function which, in the case $\lambda > 5$, reproduces the zeros for the Ising model on a complete graph. For $3<\lambda < 5$ we derive the $\lambda$-dependent angle at which the Fisher zeros impact onto the real temperature axis. This, in turn, gives access to the $\lambda$-dependent universal values of the critical exponents and critical amplitudes ratios. Our analysis of the Lee-Yang zeros reveals a difference in their behaviour for the Ising model on a complete graph and on an annealed scale-free network when $3<\lambda <5$. Whereas in the former case the zeros are purely imaginary, they have a non zero real part in latter case, so that the celebrated Lee-Yang circle theorem is violated.