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

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.

5 pages, 5 figures

References in corpus (9)

• Critical phenomena in complex networks
• Ordinary Percolation with Discontinuous Transitions
• Percolation on hyperbolic lattices
• Monte-Carlo simulation study of the two-stage percolation transition in enhanced binary trees
• Fixed Point Properties of the Ising Ferromagnet on the Hanoi Networks
• Generating-function approach for bond percolations in hierarchical networks
• Generalized scaling theory for critical phenomena including essential singularity and infinite dimensionality
• Critical Phase of Bond Percolations on Growing Networks
• Reply to Comment on `Monte-Carlo simulation study of the two-stage percolation transition in enhanced binary trees'

Cited by in corpus (7)

• Transition-type change between an inverted Berezinskii-Kosterlitz-Thouless transition and an abrupt transition in the bond percolation on a random hierarchical small-world network
• Scaling of Clusters near Discontinuous Percolation Transitions in Hyperbolic Networks
• Classification of critical phenomena in hierarchical small-world networks
• Profile and scaling of the fractal exponent of percolations in complex networks
• From explosive to infinite-order transitions on a hyperbolic network
• Absence of the non-percolating phase for percolation on the non-planar Hanoi network
• Renormalization-group theory of the abnormal singularities at the critical-order transition in bond percolation on pointed hierarchical graphs