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

TL;DR

This paper presents an exact solution of the Ising model on a fractal modular network, revealing a phase diagram with Griffiths singularities, algebraic order, and unusual critical behavior influenced by network structure.

Contribution

It introduces an exact renormalization-group analysis of the Ising model on a fractal, modular network, uncovering novel phase transitions and correlation properties.

Findings

Disordered phase exhibits Griffiths singularity due to rare large clusters.

Transition to algebraic order with power-law decay of correlations.

Transition is infinite-order at low K/J and second-order above a threshold.

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.