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

M. Krasnytska, B. Berche, Y. Holovatch, Ralph Kenna

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    Abstract

    We analyze the partition function of the Ising model on graphs of twodifferent types: complete graphs, wherein all nodes are mutually linked and annealedscale-free networks for which the degree distribution decays as P(k) ∼ k−λ. We areinterested in zeros of the partition function in the cases of complex temperature orcomplex external field (Fisher and Lee-Yang zeros respectively). For the model on anannealed scale-free network, we find an integral representation for the partition functionwhich, in the case λ > 5, reproduces the zeros for the Ising model on a completegraph. For 3 <λ <5 we derive the λ-dependent angle at which the Fisher zerosimpact onto the real temperature axis. This, in turn, gives access to the λ-dependentuniversal values of the critical exponents and critical amplitudes ratios. Our analysisof the Lee-Yang zeros reveals a difference in their behaviour for the Ising model on acomplete graph and on an annealed scale-free network when 3 <λ <5. Whereas inthe former case the zeros are purely imaginary, they have a non zero real part in lattercase, so that the celebrated Lee-Yang circle theorem is violated.
    Original languageEnglish
    Article number135001
    JournalJournal of Physics A: Mathematical and Theoretical
    Volume49
    Issue number13
    Early online date17 Feb 2016
    DOIs
    Publication statusPublished - 1 Apr 2016

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