Lattice Solutions in a Ginzburg–Landau Model for a Chiral Magnet

Abstract

We examine micromagnetic pattern formation in chiral magnets, driven by the competition of Heisenberg exchange, Dzyaloshinskii–Moriya interaction, easy-plane anisotropy and thermodynamic Landau potentials. Based on equivariant bifurcation theory, we prove existence of lattice solutions branching off the zero magnetization state and investigate their stability. We observe in particular the stabilization of quadratic vortex–antivortex lattice configurations and instability of hexagonal skyrmion lattice configurations, and we illustrate our findings by numerical studies.

The mathematics of thin structures

“…Their stability, reduced size, and the small current densities sufficient to control them, make magnetic skyrmions extremely attractive for applications in modern spintronics [85]. An in-depth understanding of their rich structure (e.g., chirality, topological charge, stability) leads to challenging problems in a subject area where geometry and continuum mechanics meet topology and analysis, and this has raised interest in magnetic skyrmions also from a mathematical perspective [11,21,57,72,74,76,82,83,115,117,132,156,157,166,169,170,179].…”

Differential Equations of Quantum Mechanics

“…For recent work on vortices in the somewhat similar Landau-Lifshitz-type equations and the Weinberg-Salam model of electro-weak interactions (the U (2) Yang-Mills-Higgs system), see [174,219,151].…”