Phase transition without global ordering in a hierarchical scale-free network

TL;DR

This paper investigates percolation on a hierarchical scale-free network, revealing a unique phase transition between critical states without global order, driven by site occupancy probability.

Contribution

It uncovers a novel phase transition between critical phases in a hierarchical network, characterized by changes in the fractal exponent, without the emergence of a percolating phase.

Findings

Existence of critical phases without global order.

Transition characterized by fractal exponent change.

Phase boundary depends on site occupancy fraction.

Abstract

We study the site-bond percolation on a hierarchical scale-free network, namely, the decorated (2,2)-flower, by using the renormalization group technique. The phase diagram essentially depends on the fraction of occupied sites. Surprisingly, when each site is unoccupied even with a small probability, the system permits neither the percolating phase nor the nonpercolating phase, but rather only critical phases. Although the order parameter always remains zero, a transition still exists between the critical phases that is characterized by the value of the fractal exponent, which measures the degree of criticality; the system changes from one critical state to another with the jump of the fractal exponent at the transition point. The phase boundary depends on the fraction of occupied sites. When the fraction of unoccupied sites exceeds a certain value, the transition line between the critical phases disappears, and a unique critical phase remains.