Phase transitions in Ising models on directed networks

Adam Lipowski, António Luis Ferreira, Dorota Lipowska, and Krzysztof Gontarek
Phys. Rev. E 92, 052811 – Published 23 November 2015

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 can a random directed graph support finite-temperature ferromagnetic ordering. Such behavior 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 do 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, do we expect the Ising model to have a positive critical temperature. With a very good accuracy, the behavior on directed random graphs is reproduced within a certain approximate scheme.

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  • Received 1 October 2015

DOI:https://doi.org/10.1103/PhysRevE.92.052811

©2015 American Physical Society

Authors & Affiliations

Adam Lipowski1, António Luis Ferreira2, Dorota Lipowska3, and Krzysztof Gontarek1

  • 1Faculty of Physics, Adam Mickiewicz University, Poznań, Poland
  • 2Departamento de Física, I3N, Universidade de Aveiro, Aveiro, Portugal
  • 3Faculty of Modern Languages and Literature, Adam Mickiewicz University, Poznań, Poland

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Vol. 92, Iss. 5 — November 2015

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