Zero temperature dynamics of Ising model on a densely connected small world network

TL;DR

This study investigates the zero temperature dynamics of the Ising model on a densely connected small world network, revealing rapid relaxation to equilibrium without freezing, due to the network's unique topological properties.

Contribution

It demonstrates that the Ising model on a densely connected small world network reaches equilibrium quickly, contrasting with behavior on sparse networks, and links this to the network's topological features.

Findings

No freezing occurs for any non-zero probability p.

Residual energy and spin flips decay exponentially to equilibrium.

Persistence probability saturates at 0.5 in the thermodynamic limit.

Abstract

The zero temperature quenching dynamics of the ferromagnetic Ising model on a densely connected small world network is studied where long range bonds are added randomly with a finite probability $p$. We find that in contrast to the sparsely connected networks and random graph, there is no freezing and an initial random configuration of the spins reaches the equilibrium configuration within a very few Monte Carlo time steps in the thermodynamic limit for any $p \ne 0$. The residual energy and the number of spins flipped at any time shows an exponential relaxation to equilibrium. The persistence probability is also studied and it shows a saturation within a few time steps, the saturation value being 0.5 in the thermodynamic limit. These results are explained in the light of the topological properties of the network which is highly clustered and has a novel small world behaviour.