On Random Field Induced Ordering in the Classical XY Model

Consider the classical XY model in a weak random external field pointing along the Y axis with strength epsilon. We study the behavior of this model as the range of the interaction is varied. We prove that in any dimension d >= 2 and for all epsilon sufficiently small, there is a range L = L(epsilon) so that whenever the inverse temperature beta is larger than some beta(epsilon), there is strong residual ordering along the X direction.