Violation of Lee-Yang circle theorem for Ising phase transitions on complex networks

The Ising model on annealed complex networks with degree distribution decaying algebraically as $p(K)\sim K^{-\lambda }$ has a second-order phase transition at finite temperature if $\lambda >3$. In the absence of space dimensionality, $\lambda$ controls the transition strength; mean-field theory applies for $\lambda >5$ but critical exponents are $\lambda$-dependent if $\lambda <5$. Here we show that, as for regular lattices, the celebrated Lee-Yang circle theorem is obeyed for the former case. However, unlike on regular lattices where it is independent of dimensionality, the circle theorem fails on complex networks when $\lambda <5$. We discuss the importance of this result for both theory and experiments on phase transitions and critical phenomena. We also investigate the finite-size scaling of Lee-Yang zeros in both regimes as well as the multiplicative logarithmic corrections which occur at $\lambda =5$.