Disordered configurations of the Glauber model in two-dimensional networks

TL;DR

This study investigates how disordered configurations emerge in the zero-temperature Glauber model on Watts-Strogatz networks, revealing that small-world effects hinder full ordering and lead to complex multi-clustered states.

Contribution

It provides new insights into the impact of network topology and rewiring probability on the ordering dynamics of the Glauber model in two-dimensional networks.

Findings

Small-world regime prevents reaching ordered states in large systems.

Disordered states feature multi-clustered domains and isolated spin droplets.

Final states depend on inter-layer connectivity in multilayer networks.

Abstract

We analyze the ordering efficiency and the structure of disordered configurations for the zero-temperature Glauber model on Watts-Strogatz networks obtained by rewiring 2D regular square lattices. In the small-world regime, the dynamics fails to reach the ordered state in the thermodynamic limit. Due to interplay of the perturbed regular topology and the energy neutral stochastic state transitions, the stationary state consists of two intertwined domains, manifested as multi-clustered states on the original lattice. Moreover, for intermediate rewiring probabilities, one finds an additional source of disorder due to the low connectivity degree, which gives rise to small isolated droplets of spins. We also examine the ordering process in paradigmatic two-layer networks with heterogeneous rewiring probabilities. Comparing the cases of a multiplex network and the corresponding network with random inter-layer connectivity, we demonstrate that the character of the final state qualitatively depends on the type of inter-layer connections.