Hysteresis in the Random Field Ising Model and Bootstrap Percolation

TL;DR

This paper investigates hysteresis phenomena in the random-field Ising model, revealing how characteristic length scales and coercive fields depend on disorder and system size, with connections to bootstrap percolation.

Contribution

It establishes a novel relationship between hysteresis in the RFIM and bootstrap percolation, deriving explicit scaling laws for characteristic lengths and coercive fields in low-disorder regimes.

Findings

Characteristic length scales grow double-exponentially with inverse disorder in 2D and 3D.

Coercive field decreases logarithmically with system size, tending to zero as size increases.

Limiting magnetization behavior varies with lattice coordination number.

Abstract

We study hysteresis in the random-field Ising 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 show that the characteristic length for self-averaging $L^*$ increases as $exp(exp (J/\Delta))$ in 2d, and as $exp(exp(exp(J/\Delta)))$ in 3d, for disorder strength $\Delta$ much less than the exchange coupling J. For system size $1 << L < L^*$, the coercive field $h_{coer}$ varies as $2J - \Delta \ln \ln L$ for the square lattice, and as $2J - \Delta \ln \ln \ln L$ on the cubic lattice. Its limiting value is 0 for L tending to infinity, both for square and cubic lattices. For lattices with coordination number 3, the limiting magnetization shows no jump, and $h_{coer}$ tends to J.