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On the existence, structure, and stability of chiral magnetic skyrmions

University of British Columbia

This main focus of this thesis is on the existence and stability of topological solitons called chiral magenetic skyrmions, arising in planar ferromagnets including combinations of Zeeman, anisotropy, and chiral (Dzyoloshinskii-Moriya) interactions. The first part deals with the existence of skyrmions with co-rotational symmetry under small chiral interaction, which is a variational problem. By treating it as a perturbation problem, our results make precise the sense in which skyrmions are perturbations of bubbles (degree-one harmonic maps). The perturbation approach relies on delicate 2D resolvent expansion for a linearized operator with a zero-energy resonance. We show the uniqueness and (in a special case) the monotonicity of the skyrmion solution profile. By applying our precise bounds of the differences between skyrmions and harmonic maps, we obtain precise asymptotics of the skyrmion energy. In the second part, as an application of the first part, we study the spectrum of the second variation of the energy about the skyrmion, by perturbation theory in the presence of a threshold resonance. We first prove the linear (spectral) stability of chiral magnetic skyrmions against arbitrary perturbations; thus the skyrmion solution from the first part is a strict local minimizer (modulo translations) of the energy. We also prove the orbital stability of the skyrmions for the Landau-Lifshitz equation with dissipation, which is an alternate quantitative proof of the recent skyrmion stability result of Li-Melcher in Journal of Functional Analysis 2018 . These results are consistent with observations in the physics literature.

Text

2020-11-19

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/76552

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/