Phase transitions in Ising models on directed networks

TL;DR

This paper investigates phase transitions of Ising models on various directed networks, revealing that lattice structure and connectivity thresholds critically influence magnetic ordering and critical temperatures.

Contribution

It demonstrates that directed networks exhibit unique phase transition behaviors, including the impact of percolation thresholds on ferromagnetic ordering and the development of an approximate scheme for analysis.

Findings

Ising models on directed triangular and cubic lattices undergo phase transitions in the Ising universality class.

On directed square lattices, the model remains paramagnetic at all positive temperatures.

Finite-temperature ferromagnetic order on directed random graphs depends on surpassing a connectivity threshold.

Abstract

We examine Ising models with heat-bath dynamics on directed networks. Our simulations show that Ising models on directed triangular and simple cubic lattices undergo a phase transition that most likely belongs to the Ising universality class. On the directed square lattice the model remains paramagnetic at any positive temperature as already reported in some previous studies. We also examine random directed graphs and show that contrary to undirected ones, percolation of directed bonds does not guarantee ferromagnetic ordering. Only above a certain threshold a random directed graph can support finite-temperature ferromagnetic ordering. Such behaviour is found also for out-homogeneous random graphs, but in this case the analysis of magnetic and percolative properties can be done exactly. Directed random graphs also differ from undirected ones with respect to zero-temperature freezing. Only at low connectivity they remain trapped in a disordered configuration. Above a certain threshold, however, the zero-temperature dynamics quickly drives the model toward a broken symmetry (magnetized) state. Only above this threshold, which is almost twice as large as the percolation threshold, we expect the Ising model to have a positive critical temperature. With a very good accuracy, the behaviour on directed random graphs is reproduced within a certain approximate scheme.