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Global perturbation of isolated equivariant chiral skyrmions from the harmonic maps

Slim Ibrahim, Ikkei Shimizu

TL;DR

This work analyzes isolated equivariant chiral skyrmions in a two-dimensional Landau-Lifshitz framework with Dzyaloshinskii-Moriya interaction, Zeeman field, and easy-plane anisotropy. By reducing to a scalar equivariant profile $f$ and employing a constrained variational setup, the authors construct solutions for all $β>0$ and $r>0$, prove exponential decay, and establish monotonicity under regime conditions on $(r,β)$. They develop a novel resolvent-absorption technique to handle nonlinear perturbations around the harmonic map, enabling precise control and a perturbative comparison to the harmonic map as $β o0$. The paper also delineates stability and instability regimes via a Fourier-decomposed Hessian analysis, identifying linear stability for small $r$ and instability for large $r$ as $β$ is taken small, with significant implications for skyrmion robustness in micromagnetic materials.

Abstract

Isolated skyrmion solutions to the 2D Landau-Lifshitz equation with the Dzyaloshinskii-Moriya interaction, Zeeman interaction, and easy-plane anisotropy are considered. In a wide range of parameters illustrating the various interaction strengths, we construct exact solutions and examine their monotonicity, exponential decay, and stability using a careful mathematical analysis. We also estimate the distance between the constructed solutions and the harmonic maps by exploiting the structure of the linearized equation and by proving a resolvent estimate for the linearized operator that is uniform in extra implicit potentials.

Global perturbation of isolated equivariant chiral skyrmions from the harmonic maps

TL;DR

This work analyzes isolated equivariant chiral skyrmions in a two-dimensional Landau-Lifshitz framework with Dzyaloshinskii-Moriya interaction, Zeeman field, and easy-plane anisotropy. By reducing to a scalar equivariant profile and employing a constrained variational setup, the authors construct solutions for all and , prove exponential decay, and establish monotonicity under regime conditions on . They develop a novel resolvent-absorption technique to handle nonlinear perturbations around the harmonic map, enabling precise control and a perturbative comparison to the harmonic map as . The paper also delineates stability and instability regimes via a Fourier-decomposed Hessian analysis, identifying linear stability for small and instability for large as is taken small, with significant implications for skyrmion robustness in micromagnetic materials.

Abstract

Isolated skyrmion solutions to the 2D Landau-Lifshitz equation with the Dzyaloshinskii-Moriya interaction, Zeeman interaction, and easy-plane anisotropy are considered. In a wide range of parameters illustrating the various interaction strengths, we construct exact solutions and examine their monotonicity, exponential decay, and stability using a careful mathematical analysis. We also estimate the distance between the constructed solutions and the harmonic maps by exploiting the structure of the linearized equation and by proving a resolvent estimate for the linearized operator that is uniform in extra implicit potentials.

Paper Structure

This paper contains 32 sections, 35 theorems, 359 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Introduction
  3. Main results
  4. Difficulty and Idea of the proof
  5. Notations and organization of the paper
  6. Existence: variational approach
  7. The regime of large values of $\beta$
  8. Direct method
  9. Topological change scenario
  10. Euler-Langrange equation
  11. The regime of the remaining values of $\beta$
  12. The regime of small values of $\beta$
  13. Verification of the Euler-Langrange equation
  14. Proof of Lemma \ref{['L2.3.1']}
  15. Improved pointwise bound
  16. ...and 17 more sections

Key Result

Theorem A

For $\beta>0$ and $r>0$, there exists a solution $f_{r,\beta}\in \mathcal{M}$ to E1.5 such that the following conditions hold.

Figures (2)

  • Figure 1: The $hk$ diagram summarizing the known mathematical results on $E_{1,h,k}$. Mel14 and LiMel18 investigate the bold line on $h$-axis. The gray region is covered by GusWan21. The Bogomol'nyi case corresponds to the sloped line lying in the region $k\le 0$, examined in BarSinRosSch20DorMel17IbrShi23.
  • Figure 2: The upper diagram represents the corresponding $(h,k)$ region where the existence of solutions to \ref{['E1.5']} is proved in Theorem \ref{['TD']}. The lower diagram indicates the $(h,k)$ region where monotonicity, stability, and instability are shown in Theorems \ref{['TD']} and \ref{['TE']}.

Theorems & Definitions (65)

  • Theorem A
  • Theorem B: Stability
  • Remark 1
  • Remark 2
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2: Topological change costs at least energy $2$
  • Lemma 3: Estimate of $E_{\beta,*}$
  • proof
  • ...and 55 more