Hysteresis in One-Dimensional Anti-Ferromagnetic Random-Field Ising Model at Zero-Temperature

We analyse hysteresis in a one-dimensional anti-ferromagnetic random field Ising model at zero-temperature. The random field is taken to have a rectangular distribution of width $2 \Delta$ centered about the origin. A uniform applied field is varied slowly from $-\infty$ to $+\infty$ and back. Analytic expression for the hysteresis loop is obtained in the case $\Delta \le |J|$, where $|J|$ is the strength of the nearest neighbor interaction.