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We have obtained exact results for the Ising model on a hierarchical latticewith a scale-free degree distribution, high clustering coefficient, andsmall-world behavior. By varying the probability p of long-range bonds, theentire spectrum from an unclustered, non-small-world network to ahighly-clustered, small-world system is studied. We obtain analyticalexpressions for the degree distribution P(k) and clustering coefficient C forall p, as well as the average path length l for p=0 and 1. The Ising model onthis network is studied through an exact renormalization-group transformationof the quenched bond probability distribution, using up to 562,500 probabilitybins to represent the distribution. For p < 0.494, we find power-law criticalbehavior of the magnetization and susceptibility, with critical exponentscontinuously varying with p, and exponential decay of correlations away fromT_c. For p >= 0.494, where the network exhibits small-world character, thecritical behavior radically changes: We find a highly unusual phase transition,namely an inverted Berezinskii-Kosterlitz-Thouless singularity, between alow-temperature phase with non-zero magnetization and finite correlation lengthand a high-temperature phase with zero magnetization and infinite correlationlength. Approaching T_c from below, the magnetization and the susceptibilityrespectively exhibit the singularities of exp(-C/sqrt(T_c-T)) andexp(D/sqrt(T_c-T)), with C and D positive constants. With long-range bondstrengths decaying with distance, we see a phase transition with power-lawcritical singularities for all p, an unusually narrow critical region andimportant corrections to power-law behavior that depend on the exponentcharacterizing the decay of long-range interactions.Comment: 22 pages, 19 figures; replaced with published versio

physical review. e, statistical, nonlinear and soft matter physics

Inverted Berezinskii-Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network