On Random Field Induced Ordering in the Classical XY

Abstract: Consider the classical XY model in a weak random external field pointing along the Y axis with strength . 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 sufficiently small, there is a range L = L() so that whenever the inverse temperature β is larger than some β(), there is strong residual ordering along the X direction. 1