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

Abstract

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 K/J of of inter- to intra-community coupling, we obtain an unusual phase diagram: at high temperatures or small K/J 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 K/J, and becomes second-order above a threshold value (K/J)_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.

7 pages, 4 figures; replaced with final version to appear in Phys. Rev. E

References in corpus (6)

• Self-similar disk packings as model spatial scale-free networks
• A deterministic small-world network created by edge iterations
• Critical behavior of Griffiths ferromagnets
• Ising model on two connected Barabasi-Albert networks
• Criticality on networks with topology-dependent interactions
• Multicritical Point Relations in Three Dual Pairs of Hierarchical-Lattice Ising Spin-Glasses

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