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 $2\mathrm{\Delta }$ centered about the origin. A uniform applied field is varied slowly from $-\mathrm{\infty }$ to $+\mathrm{\infty }$ 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.