Dynamicsof a Local Perturbationin the X - V Model

In a previo]ls_paper we considered the magnetization of a single spin embedded in the XY chain after the external magnetic field is switched off. We have shown that this spin thermalizes [1]. This problem was further studied by Emch and Radin [2] from a C*-algebra point of view. In this paper we continue our investigation by considering the reverse question. If at time t = 0, we turn on a magnetic field which approaches a limit as t --t00, does the magnetization of this spin also thermalize and approach a limit? The answer here turns out to be no: we find that the magnetization does not approach equilibrium, no matter how slowly the external field reaches its limit. We shall also study the effect of an oscillatory field on the isolated spin. Because of technical difficulties we study this by perturbation theory, and find a resonance of the driving field with the external points of the one particle spectrum. We have limited ourselves to the isotropic chain. In I, we showed that the anisotropic chain can be studied along the same lines as the isotropic chain, and does not reveal more information to the questions asked. In the case of local oscillatory excitations, we conjecture that the magnetization of a spin in the anisotropic chain with a constant field in the z direction will resonate with one of the global frequencies found earlier [3] for the tanslationally invariant system.