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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 two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as $P(k)\sim k^{-λ}$. 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 $λ> 5$, reproduces the zeros for the Ising model on a complete graph. For $3<λ< 5$ we derive the $λ$-dependent angle at which the Fisher zeros impact onto the real temperature axis. This, in turn, gives access to the $λ$-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 $3<λ<5$. 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.