Griffiths singularities and algebraic order in the exact solution of an Ising model on a fractal modular network.

We use an exact renormalization-group transformation to study the Ising model on a complex network composed of tightly knit communities nested hierarchically with the fractal scaling recently discovered in a variety of real-world networks. Varying the ratio KJ of intercommunity to intracommunity couplings, we obtain an unusual phase diagram: at high temperatures or small KJ we have a disordered phase with a Griffiths singularity in the free energy, due to the presence of rare large clusters, which we analyze through the Yang-Lee zeros in the complex magnetic field plane. As the temperature is lowered, true long-range order is not seen, but there is a transition to algebraic order, where pair correlations have power-law decay with distance, reminiscent of the XY model. The transition is infinite order at small KJ and becomes second order above a threshold value (KJ)_{m} . The existence of such slowly decaying correlations is unexpected in a fat-tailed scale-free network, where correlations longer than nearest neighbor are typically suppressed.