Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Phase transition thresholds and chiral magnetic fields of general degree

Slim Ibrahim, Tatsuya Miura, Carlos Román, Ikkei Shimizu

TL;DR

This work provides a comprehensive variational analysis of the 2D Landau--Lifshitz energy with Dzyaloshinskii--Moriya interactions in the Bogomol'nyi regime, yielding exact minimal energies for all topological degrees and revealing two phase-transition thresholds at r = 1/√2 and r = 1. Using a Bogomol'nyi decomposition, the authors derive sharp energy formulas and prove rigidity: for 0 < r < 1, minimizers exist only for degrees k = 0 and k = −1 (up to translations), with higher degrees ruled out, while at r = 1 a richer family of minimizers appears. They also establish that the homogeneous state e3 is strictly stable in L^2 for all r > 0, and analyze stability under Zeeman perturbations, showing a critical transition at h = 0 between stability and instability. Collectively, the results rigorously connect energy minimization, topological degree, and magnetic-field-induced stability/failure of skyrmions and homogeneous states in 2D chiral magnets, providing a mathematically precise foundation for observed phase diagrams. $E_r[n] = D[n] + rH[n] + V[n]$, $Q[n] riangleq rac{1}{4 ext{π}} abla n ext{ dot } ( abla n imes abla n)$, and the Bogomol'nyi bound $E_r[n] - 4 ext{π} r^2 Q[n] = rac{r^2}{2} ext{∫} | abla^r_1 n + n imes abla^r_2 n|^2 dx + (1 - r^2) D[n]$ are central to the analysis.

Abstract

We study a variational problem for the Landau--Lifshitz energy with Dzyaloshinskii--Moriya interactions arising in 2D micromagnetics, focusing on the Bogomol'nyi regime. We first determine the minimal energy for arbitrary topological degree, thereby revealing two types of phase transitions consistent with physical observations. In addition, we prove the uniqueness of the energy minimizer in degrees $0$ and $-1$, and nonexistence of minimizers for all other degrees. Finally, we show that the homogeneous state remains stable even beyond the threshold at which the skyrmion loses stability, and we uncover a new stability transition driven by the Zeeman energy.

Phase transition thresholds and chiral magnetic fields of general degree

TL;DR

This work provides a comprehensive variational analysis of the 2D Landau--Lifshitz energy with Dzyaloshinskii--Moriya interactions in the Bogomol'nyi regime, yielding exact minimal energies for all topological degrees and revealing two phase-transition thresholds at r = 1/√2 and r = 1. Using a Bogomol'nyi decomposition, the authors derive sharp energy formulas and prove rigidity: for 0 < r < 1, minimizers exist only for degrees k = 0 and k = −1 (up to translations), with higher degrees ruled out, while at r = 1 a richer family of minimizers appears. They also establish that the homogeneous state e3 is strictly stable in L^2 for all r > 0, and analyze stability under Zeeman perturbations, showing a critical transition at h = 0 between stability and instability. Collectively, the results rigorously connect energy minimization, topological degree, and magnetic-field-induced stability/failure of skyrmions and homogeneous states in 2D chiral magnets, providing a mathematically precise foundation for observed phase diagrams. , , and the Bogomol'nyi bound are central to the analysis.

Abstract

We study a variational problem for the Landau--Lifshitz energy with Dzyaloshinskii--Moriya interactions arising in 2D micromagnetics, focusing on the Bogomol'nyi regime. We first determine the minimal energy for arbitrary topological degree, thereby revealing two types of phase transitions consistent with physical observations. In addition, we prove the uniqueness of the energy minimizer in degrees and , and nonexistence of minimizers for all other degrees. Finally, we show that the homogeneous state remains stable even beyond the threshold at which the skyrmion loses stability, and we uncover a new stability transition driven by the Zeeman energy.
Paper Structure (21 sections, 18 theorems, 126 equations, 10 figures)

This paper contains 21 sections, 18 theorems, 126 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properties of the functionals
  4. Equivariance
  5. Minimal energy
  6. Negative degree case
  7. Nonnegative degree case
  8. Unboundedness of the energy
  9. Rigidity of minimizers
  10. Nonnegative degree case
  11. Negative degree case
  12. Stability of the homogeneous state
  13. The critical case
  14. The supercritical case
  15. The subcritical case
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $k\in\mathbb{Z}$ and $r>0$. If $0<r\leq1$, then If $r>1$, then $\inf_{{\mathbf{n}}\in\mathcal{M}_k}E_r[{\mathbf{n}}]=-\infty$ holds for each $k$.

Figures (10)

  • Figure 1: Plots of the minimal energy for degrees $-3\leq k\leq3$.
  • Figure 2: Schematic illustration of the skyrmion (left) and anti-skyrmion (right) profiles.
  • Figure 3: The construction of ${\mathbf{n}}_R$ in \ref{['eq:defnR']}.
  • Figure 4: The construction of ${\mathbf{n}}_L$ in \ref{['eq:defnL']}.
  • Figure 5: Plots of $Z_0$ (blue) and $Z_1$ (red) for $v=-\frac{i}{2}\overline{z} + az^{2}$ with different values of $a$.
  • ...and 5 more figures

Theorems & Definitions (36)

  • Theorem 1.1
  • Remark 1.2: Two phase-transition thresholds
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Proposition \ref{['P2']}
  • Proposition 3.3
  • ...and 26 more