Hysteresis in the Random-Field Ising Model and Bootstrap Percolation

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/Δ)] in 2D, and as expexp[exp(J/Δ)] in 3D, for disorder strength Δ much less than the exchange coupling J. For system size 1L<L^, the coercive field hcoer varies as 2J-ΔlnlnL for the square lattice, and as 2J-ΔlnlnlnL on the cubic lattice. Its limiting value is 0 for L→∞ for both square and cubic lattices. For lattices with coordination number 3, the limiting magnetization shows no jump, and hcoer tends to J.