Critical Percolation Phase and Thermal BKT Transition in a Scale-Free Network with Short-Range and Long-Range Random Bonds

TL;DR

This paper exactly solves percolation in a scale-free hierarchical network, revealing a critical phase with algebraic order and contrasting behaviors of geometric and thermal BKT transitions, highlighting complex disorder effects.

Contribution

It introduces an exact renormalization-group solution for percolation in a scale-free network with mixed bonds, identifying a novel critical BKT phase and contrasting it with thermal transitions.

Findings

Discovery of a critical percolation phase with BKT order

No direct link between geometric and thermal BKT onsets

Inverted BKT behavior at low bond probability and high temperature

Abstract

Percolation in a scale-free hierarchical network is solved exactly by renormalization-group theory, in terms of the different probabilities of short-range and long-range bonds. A phase of critical percolation, with algebraic (Berezinskii-Kosterlitz-Thouless) geometric order, occurs in the phase diagram, in addition to the ordinary (compact) percolating phase and the non-percolating phase. It is found that no connection exists between, on the one hand, the onset of this geometric BKT behavior and, on the other hand, the onsets of the highly clustered small-world character of the network and of the thermal BKT transition of the Ising model on this network. Nevertheless, both geometric and thermal BKT behaviors have inverted characters, occurring where disorder is expected, namely at low bond probability and high temperature, respectively. This may be a general property of long-range networks.