Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network

TL;DR

This paper investigates the Ising model on a hierarchical small-world network, revealing a unique phase transition driven by stable fixed points, with peculiar critical behaviors including logarithmic singularities.

Contribution

It uncovers a novel phase transition mechanism governed by stable renormalization fixed points, contrasting with traditional unstable fixed point transitions.

Findings

Identifies a phase transition between ordered and critical phases driven by shortcut edge coupling.

Shows the critical phase is governed by a nontrivial fixed point connected to the ordered phase.

Reveals the existence of a finite correlation-length-like quantity diverging at the transition.

Abstract

We study the Ising model in a hierarchical small-world network by renormalization group analysis, and find a phase transition between an ordered phase and a critical phase, which is driven by the coupling strength of the shortcut edges. Unlike ordinary phase transitions, which are related to unstable renormalization fixed points (FPs), the singularity in the ordered phase of the present model is governed by the FP that coincides with the stable FP of the ordered phase. The weak stability of the FP yields peculiar criticalities including logarithmic behavior. On the other hand, the critical phase is related to a nontrivial FP, which depends on the coupling strength and is continuously connected to the ordered FP at the transition point. We show that this continuity indicates the existence of a finite correlation-length-like quantity inside the critical phase, which diverges upon approaching the transition point.