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

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.