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 $ε$. We study the behavior of this model as the range of the interaction is varied. We prove that in any dimension $d \geq 2$ and for all $ε$ sufficiently small, there is a range $L=L(ε)$ so that whenever the inverse temperature $β$ is larger than some $β(ε)$, there is strong residual ordering along the $X$ direction.

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