We have obtained exact results for the Ising model on a novel 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-distance 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 analytical expressions for the degree distribution $P(k)$ and clustering coefficient $C$ for all $p$, as well as the average path length $ell$ in the limiting cases $p=0$ 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 562,500 renormalized probability bins to represent the distribution. For $p lesssim 0.5$, we find critical behavior similar to that of a regular lattice, with power-law scaling of the magnetization and susceptibility, and exponential damping of correlations away from $T_c$. For $p gtrsim 0.5$, where the network exhibits a 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 non-zero magnetization and finite correlation length and a high-temperature phase with zero magnetization and infinite correlation length, with power-law decay of correlations. Approaching $T_c$ from below, the magnetization and the susceptibility respectively exhibit the singularities of $exp(-C/sqrt{T_c-T})$ and $exp(D/sqrt{T_c-T})$, where $C$ and $D$ are positive constants. With long-distance bond strengths decaying with distance, we see a regular phase transition for all $p$, with a critical region and corrections to power-law scaling that depend on the exponent characterizing the decay.

(This research has been performed in collaboration with A. Nihat Berker.)

Feza Gürsey Research Institute Bosphorus University Turkey