Random Field Induced Order in Low Dimension

TL;DR

This paper rigorously demonstrates that in low-dimensional classical O(n) models with a weak random external field, residual magnetic order persists perpendicular to the field's subspace in certain dimensions, revealing a phenomenon called random field induced order.

Contribution

The paper provides a rigorous proof of residual magnetic order in low-dimensional O(n) models under weak random fields, extending understanding of disorder effects in statistical physics.

Findings

Residual magnetic order exists perpendicular to the random field in 2D and 3D models.

Spin projections onto the random field subspace are small when the subspace dimension is less than n.

Order persists even with weak random external fields in low-dimensional systems.

Abstract

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