The continuous spin random field model: Ferromagnetic ordering in d>=3

TL;DR

This paper studies the behavior of continuous spin models with ferromagnetic interactions under random fields, demonstrating ferromagnetic order in three or more dimensions through a novel coarse-graining and contour representation approach.

Contribution

It introduces a new coarse-graining method and contour representation for continuous spin systems with randomness, proving ferromagnetic ordering in high dimensions.

Findings

Ferromagnetic order established in d≥3 for weak disorder.

Development of a positive contour activity representation.

Application of discrete-spin renormalization group to continuous spins.

Abstract

We investigate the Gibbs-measures of ferromagnetically coupled continuous spins in double-well potentials subjected to a random field (our specific example being the $\phi^4$ theory), showing ferromagnetic ordering in $d\geq 3$ dimensions for weak disorder and large energy barriers. We map the random continuous spin distributions to distributions for an Ising-spin system by means of a single-site coarse-graining method described by local transition kernels. We derive a contour- representation for them with notably positive contour activities and prove their Gibbsianness. This representation is shown to allow for application of the discrete-spin renormalization group developed by Bricmont/Kupiainen implying the result in $d\geq 3$.