Magnon band structure of skyrmion crystals and stereographic projection approach

TL;DR

This paper develops a semiclassical and stereographic projection method to analyze magnon band structures in skyrmion crystals, revealing topologically trivial and non-trivial bands with distinct dispersion and Berry curvature properties.

Contribution

It introduces a novel stereographic projection approach combined with semiclassical analysis to study magnon dynamics in skyrmion crystals, providing new insights into their band topology.

Findings

Identified two sets of magnon bands with different topological properties.

Demonstrated robustness of certain bands to magnetic field variations.

Revealed Berry curvature sign-reversal in the magnon bands.

Abstract

Using semiclassical method combined with stereographic projection approach, we investigate the magnetic dynamics of the skyrmion crystal (SkX), formed in planar ferromagnet with both Dzyaloshinskii-Moriya interaction and uniform magnetic field. The topologically non-trivial ground state of SkX is described in stereographic projection by the complex valued function with simple poles at skyrmions' positions. We use the earlier proposed ansatz for this ground state function in the form of the sum of individual skyrmions. The dynamics follows from the second variation of the classical action. Numerical analysis yields the magnon band structure of tight-binding form in accordance with previously known results. There are two sets of bands, one set with a flat dispersion, topologically trivial and rapidly evolving with magnetic field. Another set is robust to magnetic field, characterized by pronounced dispersion and with the Berry curvature which may be sign-reversal in the Brillouin zone. The developed theory can be straightforwardly generalized for the analysis of magnetic dynamics in topological spin structures of other types.