Goldstone mode of Skyrmion Crystal

TL;DR

This paper analyzes the Goldstone mode in a skyrmion crystal within a 2D ferromagnet model, deriving its dispersion relation and examining how magnetic fields influence its properties and displacement propagation.

Contribution

It introduces a stereographic projection method to construct skyrmion crystals and derives the Goldstone mode dispersion, highlighting the transition from isotropic to anisotropic displacement propagation.

Findings

Derived closed-form Goldstone mode dispersion relation.

Showed dependence of dispersion on magnetic field strength.

Identified transition from isotropic to anisotropic displacement propagation.

Abstract

We discuss the Goldstone mode of skyrmion crystal in a model of two-dimenssional ferromagnet with Dzyaloshinskii-Moriya interaction in magnetic field. We use stereographic projection approach to construct skyrmion crystal and consider skyrmion's displacement field. The small overlap of the individual skyrmion images restricts the potential energy to the interaction of nearest neighboring displacements. The closed form of the Goldstone mode dispersion is found and its dependence on the magnetic field is studied. We use semiclassical quantization to define the Green's function and show that the propagation of displacements through the crystal changes its tensorial form from isotropic to anisotropic one at large distances.