Two-dimensional Ferronematics, Canonical Harmonic Maps and Minimal Connections

TL;DR

This paper analyzes a variational model for two-dimensional ferronematics, showing that in a certain regime, the nematic order converges to a harmonic map with defects, while the magnetisation forms line defects connecting these points.

Contribution

It establishes the asymptotic behavior of ferronematic models, linking defect structures to harmonic maps and minimal connections, which is a novel insight in the field.

Findings

Nematic order converges to a harmonic map with non-orientable defects.

Magnetisation converges to a singular vector field with line defects.

Line defects connect non-orientable point defects via minimal connections.

Abstract

We study a variational model for ferronematics in two-dimensional domains, in the "super-dilute" regime. The free energy functional consists of a reduced Landau-de Gennes energy for the nematic order parameter, a Ginzburg-Landau type energy for the spontaneous magnetisation, and a coupling term that favours the co-alignment of the nematic director and the magnetisation. In a suitable asymptotic regime, we prove that the nematic order parameter converges to a canonical harmonic map with non-orientable point defects, while the magnetisation converges to a singular vector field, with line defects that connect the non-orientable point defects in pairs, along a minimal connection.