Exact solution of the Heisenberg equation of motion for the surface spin in a semi-infinite S=1/2 XY chain at infinite temperatures

The Heisenberg equation of motion for the surface spin operator in a semi-infinite S=1/2 XY chain is exactly solved at infinite temperatures via the recurrence-relations method of Lee. It is found that the time evolution of the surface spin shows Bessel-like on-site relaxation that is drastically different from the corresponding Gaussian relaxation of a bulk spin. This difference is shown to be related to broken translational symmetry in the former case. The memory function and the random force for the surface spin is exactly determined. It is found that the surface spin dynamics is, in some sense, "harmonic" in nature.