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. Recently, various physical phenomena in non-Euclidean graphs have been studied, especially in the context of complex networks, and their properties have been found to be beyond the scope of the conventional theory for Euclidean graphs [1]. Of particular interest are systems regarded as infinitedimensional in the sense that the equidistant surface S r of radius r grows exponentially as S r ∝ e y d r with a positive constant y d , which is faster than any power function r d−1 as in d-dimensional Euclidean graphs. Typical examples are trees and hyperbolic lattices [2]. Remarkably, such infinitedimensional systems often exhibit the critical phases in which the (nonlinear) susceptibility diverges [3][4][5]. Although the critical phase is also observed in some Euclidean systems, e.g., the quasi-long-range ordered phase in the two-dimensional XY model [6,7], the critical phases in infinite-dimensional systems are considered to be due to rather geometrical effects. Indeed, the exponential growth of a graph admits the divergence of the susceptibility χ , which is calculated by the integral of the two-point correlation function [8], even with finite correlation lengthξ [9] The property of the phase transitions between a critical phase and an ordered phase is an interesting issue. Quite recently, some models in the simple hierarchical network shown in Fig. 1(a) were investigated to examine these transitions. This network is very useful because rigorous real-space renormalization is possible for various models in the simplest way. Furthermore various types of phase transition are observed depending on the model used, e.g., a discontinuous transition of the bond percolation model [12], equivalent to the one-state Potts model [14,15], and continuous transitions with a power-law singularity (PLS) or an essential singularity (ES) for the q-state Potts model with q 3 [16]. These are observed in other graphs [3][4][5]17,18]. Thus this hierarchical network is a good stage to investigate what determines the type of phase transitions in a systematic way. In particular the two-state Potts model, equivalent to the Ising model, ...

Absence of the nonpercolating phase for percolation on the nonplanar Hanoi network

“…Our results give the two critical probabilities as p c1 = 0(< p BCZ c1 ) and p c2 = p BCZ c2 , implying that the system has only a critical phase and a percolating phase, and does not have a non-percolating phase for p > 0. As far as we know, all complex network models with a critical phase have only the critical phase and the percolating phase ( [8-11, 13-15, 19] for percolation and [12,19,[22][23][24][25][26][27] for spin systems). It will be challenging to clarify the origin of such universal behavior.…”

Classification of critical phenomena in hierarchical small-world networks

“…In particular, the iterative structure of hierarchical networks may facilitate their realization in engineered devices to unlock and control their unconven-tional behaviors. Work on percolation [15][16][17][18][19][20], the Ising model [21][22][23][24][25], and the q-state Potts model [26][27][28] have shown that critical behavior, once thought to be exotic and model-specific [4], can be categorized with the renormalization group [26] for a large class of hierarchical networks with a hyperbolic structure.…”

From explosive to infinite-order transitions on a hyperbolic network

“…[24] (see also Ref. [26]), we introduce the following couplings. All variables x i interact with their nearest-neighbors along the backbone with a coupling K 0 (solid links in Fig.…”

“…These recursively defined structures provide deeper insights into small-world effects compared to random networks that otherwise require approximate or numerical methods. Work on percolation [6,17,[20][21][22], the Ising model [15,[23][24][25], and the q-state Potts model [19,26] have shown that critical behavior, once thought to be exotic and model-specific [3], can be universally categorized near the transition point [26,27] for a large class of hyperbolic networks, such as those discontinuous percolation transitions described in Refs. [6,14].…”