Glauber Dynamics for One-dimensional Spin Models with Random Fields

We present an exact solution of the long-time relaxational behavior of the magnetization in the Ising and $\mathrm{XY}$ chains in a quenched random field. The random field is assumed to be of infinite strength but present at only a fraction of the spin sites. We use Glauber dynamics for the Ising case, and a suitably generalized master equation for the $\mathrm{XY}$ case. In both models we find that for $T=0\mathrm{}$, as time $t\to \infty$ the magnetization decays as exp (${-Ct}^{\frac{1}{3}}$), where $C$ is a constant. At finite temperatures the ultimate asymptotic behavior is purely exponential.