Driven Random Field Ising Model: some exactly solved examples in threshold activated kinetics

TL;DR

This paper analyzes exactly solvable examples of the driven random field Ising model on a Bethe lattice, highlighting differences in behavior between ferromagnetic and anti-ferromagnetic interactions under threshold activated dynamics.

Contribution

It provides analytical solutions for the non-equilibrium response of the driven RFIM, distinguishing ferromagnetic and anti-ferromagnetic cases and exploring their critical behaviors.

Findings

Ferromagnetic RFIM exhibits avalanches and critical behavior.

Anti-ferromagnetic RFIM lacks avalanches and criticality.

The ferromagnetic model is Abelian, anti-ferromagnetic is non-Abelian.

Abstract

The random field Ising model driven by a slowly varying uniform external field at zero temperature provides a caricature of several threshold activated systems. In this model, the non-equilibrium response of the system can be obtained analytically on a Bethe lattice if the initial state of the system has all spins aligned parallel to each other. We consider ferromagnetic as well as anti-ferromagnetic interactions. The ferromagnetic model exhibits avalanches and non-equilibrium critical behavior. The anti-ferromagnetic model is marked by the absence of these features. The ferromagnetic model is Abelian, and the anti-ferromagnetic model is non-Abelian. Theoretical approaches based on the probabilistic method are discussed in the two cases, and illustrated by deriving some basic results.