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

We analyse 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) similar to 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.