On Random Field Induced Ordering in the Classical XY Model

TL;DR

This paper proves that in the classical XY model with a weak random external field, residual ordering persists along a specific direction when the temperature is sufficiently low and the interaction range is appropriately chosen.

Contribution

It demonstrates residual ordering in the XY model under weak random fields for various interaction ranges, extending understanding of phase behavior in disordered systems.

Findings

Residual ordering along the X direction for small epsilon

Existence of a critical interaction range L(epsilon)

Order persists at low temperatures for all dimensions d ≥ 2

Abstract

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 $d \geq 2$ 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.