Hysteresis and Avalanches in the Random Field Ising Model

TL;DR

This paper investigates the hysteresis and avalanche phenomena in the zero-temperature random field Ising model on Bethe lattices, providing analytical solutions for avalanche distributions and insights into stable configurations and coercive fields.

Contribution

It introduces explicit calculations of avalanche distributions for Bethe lattices with specific coordination numbers and develops self-consistent equations for magnetization and hysteresis loops.

Findings

Explicit avalanche distribution formulas for $z=2$ and 3 Bethe lattices.

Self-consistent equations for magnetization on minor hysteresis loops.

Dependence of coercive field on lattice coordination number.

Abstract

In this thesis, we discuss nonequilibrium ferromagnetic random field Ising model (RFIM) with zero temperature Glauber single spin flip dynamics. We briefly review the hysteresis in ferromagnets and Barkhausen effect. We discuss some earlier results on the zero temperature RFIM. We also discuss some of the equilibrium properties of RFIM. We setup the generating function for the avalanche distribution for arbitrary distribution of the quenched random field on a Bethe lattice. We explicitly calculate the probability distribution of avalanches, for the for Bethe lattices with coordination numbers $z=2$ and 3, for the special case of a rectangular distribution of the random field. We analyse the self-consistent equations to determine the form of the avalanche distribution for some general unimodal continuous distributions of the random field. We derive the self-consistent equations for the magnetization on minor hysteresis loops on a Bethe lattice, when the external field is varying cyclically with decreasing magnitudes. We also discuss some properties of stable configurations, when the external field is varying. We study the model with an asymmetric distribution of quenched fields, in the limit of low disorder in two and three dimensions. We relate the spin flip process to bootstrap percolation, and find nontrivial dependence of the coercive field on the coordination number of the lattice.