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 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 XY case. In both models we find that for $T=0$, as time t→∞ the magnetization decays as exp (-Ct1/3), where C is a constant. At finite temperatures the ultimate asymptotic behavior is purely exponential.