Random Field Induced Order in Low Dimension

Abstract

Consider the behavior of a classical O(n) model in a weak random external field acting along some $k$-dimensional subspace in ${\mathbb{R}}^n$ with $k<n$. We show rigorously that if $k=n-1$, for the model defined on ${\mathbb{Z}}^d$, $d=2,3$ there is residual magnetic order perpendicular to the subspace which supports the distribution of the random field only. Furthermore, when $k\le n-1$ and $d\ge 2$ we show in general that the magnitude of spin projections onto this subspace are small.

5 pages. Summary of forthcoming work

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