We have obtained exact results for the Ising model on a hierarchical lattice incorporating three key features characterizing many real-world networks—a scale-free degree distribution, a high clustering coefficient, and the small-world effect. By varying the probability $p$ of long-range bonds, the entire spectrum from an unclustered, non-small-world network to a highly clustered, small-world system is studied. Using the self-similar structure of the network, we obtain analytic expressions for the degree distribution $P\left(k\right)$ and clustering coefficient $C$ for all $p$, as well as the average path length $\ell$ for $p=0$ and $1$. The ferromagnetic Ising model on this network is studied through an exact renormalization-group transformation of the quenched bond probability distribution, using up to $562500$ renormalized probability bins to represent the distribution. For $p<0.494$, we find power-law critical behavior of the magnetization and susceptibility, with critical exponents continuously varying with $p$, and exponential decay of correlations away from $T_c$. For $p\ge 0.494$, in fact where the network exhibits small-world character, the critical behavior radically changes: We find a highly unusual phase transition, namely an inverted Berezinskii-Kosterlitz-Thouless singularity, between a low-temperature phase with nonzero magnetization and finite correlation length and a high-temperature phase with zero magnetization and infinite correlation length, with power-law decay of correlations throughout the phase. Approaching $T_c$ from below, the magnetization and the susceptibility, respectively, exhibit the singularities of $\exp (-C∕\sqrt{T_c-T})$ and $\exp (D∕\sqrt{T_c-T})$, with $C$ and $D$ positive constants. With long-range bond strengths decaying with distance, we see a phase transition with power-law critical singularities for all $p$, and evaluate an unusually narrow critical region and important corrections to power-law behavior that depend on the exponent characterizing the decay of long-range interactions.

• Received 12 December 2005

DOI:https://doi.org/10.1103/PhysRevE.73.066126