Random field induced order in two dimensions

TL;DR

This paper proves that in two-dimensional classical XY models with a weak fixed-direction random field, residual magnetic order persists perpendicular to the field, extending previous three-dimensional results using a multi-scale Peierls contour approach.

Contribution

It establishes residual magnetic order in 2D XY models under weak random fields, adapting a multi-scale contour method to overcome two-dimensional technical challenges.

Findings

Residual magnetic order exists in 2D XY models with weak random fields.

The order aligns perpendicular to the random field direction.

The approach extends previous 3D results using a multi-scale Peierls contour argument.

Abstract

In this article we prove that a classical $XY$ model subjected to weak i.i.d. random field pointing in a fixed direction exhibits residual magnetic order in $\mathbb{Z}^2$ and aligns perpendicular to the random field direction. The paper is a sequel to \cite{NC} where the three-dimensional case was treated. Our approach is based on a multi-scale Peierls contour argument developed in \cite{NC}. On the microscopic scale we extract energetic costs from the occurrence of contours, which themselves are defined on a macroscopic scale. The technical challenges in $\mathbb{Z}^2$ stem from difficulties controlling the size and roughness of the fluctuation fields which model the short length-scale oscillations of near-optimizers of the random field Hamiltonian.