Random Field Induced Order in Low Dimension I

TL;DR

This paper proves that the classical XY model in three dimensions maintains residual magnetic order in the horizontal direction even under weak random external fields, depending on temperature and field strength.

Contribution

It demonstrates residual magnetic order in the 3D XY model under weak random fields, extending understanding of disorder effects in low-dimensional systems.

Findings

Residual magnetic order persists in 3D XY model under weak random fields

Order depends on temperature and field strength

Order exists for arbitrarily weak fields at low enough temperatures

Abstract

Consider the classical XY model in a weak random external field pointing along the Y axis with strength $\epsilon$. We prove that the model defined on $\Z^3$ with nearest neighbor coupling exhibits residual magnetic order in the horizontal direction for arbitrarily weak random field strengths and, depending on field strength, sufficiently low temperature.