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On the formation of microstructure and the occurrence of vortices in a singularly perturbed energy related to helimagnetism: a scaling law result

Janusz Ginster

TL;DR

The paper analyzes singularly perturbed variational energies modeling helimagnetism, allowing curlful fields with vortices under incompatible boundary conditions. It establishes a sharp scaling law for the infimum across three parameters: boundary incompatibility θ, regularization strength σ, and interatomic distance ε, showing when uniform, highly oscillatory, or vortex-rich states minimize the energy. A novel adaptation of the ball-construction is used to balance bulk multi-well energy and the higher-order regularization, yielding matching upper and lower bounds. The results identify parameter regimes where vortices are energetically favorable and extend known gradient-only results to curl-bearing configurations. The work thus provides a rigorous continuum-to-discrete perspective on microstructure formation in helimagnetic systems.

Abstract

In this work, singularly perturbed energies arising from discrete $J_1$-$J_3$-models are studied. The energies under consideration consist of a non-convex bulk term and a higher-order regularizing term and are subject to incompatible boundary conditions. In contrast to existing results in the literature, in this work, admissible fields are not necessarily gradient fields, instead their curl is linked to topological singularities, so-called vortices, in the discrete $J_1$-$J_3$-model. The main result of this work is a scaling law for the minimal energy with respect to three parameters: one measuring the incompatibility of the boundary conditions, the second measuring the strength of the regularizing term, and the third being related to the interatomic distance in the discrete model. The shown result implies in particular that in certain parameter regimes, minimizers necessarily develop vortices. A key tool in the analysis is a careful modification of the celebrated ball-construction technique that, due to a lack of rigidity, considers simultaneously both the bulk energy and the regularizing term.

On the formation of microstructure and the occurrence of vortices in a singularly perturbed energy related to helimagnetism: a scaling law result

TL;DR

The paper analyzes singularly perturbed variational energies modeling helimagnetism, allowing curlful fields with vortices under incompatible boundary conditions. It establishes a sharp scaling law for the infimum across three parameters: boundary incompatibility θ, regularization strength σ, and interatomic distance ε, showing when uniform, highly oscillatory, or vortex-rich states minimize the energy. A novel adaptation of the ball-construction is used to balance bulk multi-well energy and the higher-order regularization, yielding matching upper and lower bounds. The results identify parameter regimes where vortices are energetically favorable and extend known gradient-only results to curl-bearing configurations. The work thus provides a rigorous continuum-to-discrete perspective on microstructure formation in helimagnetic systems.

Abstract

In this work, singularly perturbed energies arising from discrete --models are studied. The energies under consideration consist of a non-convex bulk term and a higher-order regularizing term and are subject to incompatible boundary conditions. In contrast to existing results in the literature, in this work, admissible fields are not necessarily gradient fields, instead their curl is linked to topological singularities, so-called vortices, in the discrete --model. The main result of this work is a scaling law for the minimal energy with respect to three parameters: one measuring the incompatibility of the boundary conditions, the second measuring the strength of the regularizing term, and the third being related to the interatomic distance in the discrete model. The shown result implies in particular that in certain parameter regimes, minimizers necessarily develop vortices. A key tool in the analysis is a careful modification of the celebrated ball-construction technique that, due to a lack of rigidity, considers simultaneously both the bulk energy and the regularizing term.

Paper Structure

This paper contains 11 sections, 19 theorems, 173 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Heuristic connection to a frustrated spin system and interpretation of the result
  3. Outline
  4. The mathematical setting and main result
  5. Notation
  6. Preliminaries
  7. Upper Bound
  8. Lower bound
  9. Energy estimates for vortices and the ball construction
  10. The lower bound for the energies E(1) sigma,theta,epsilon and E(2) sigma,theta,epsilon
  11. The lower bound for the anisotropic quadratic perturbation

Key Result

Theorem 2.1

There exist constants $C \geq c > 0$ such that it holds for all $\sigma > \sqrt{2} \pi \varepsilon >0$, $\theta \in (0,1/2)$ and $\epsilon \in \{1,2,a\}$ that where $s(\sigma, \varepsilon, \theta) = \min\left\{ \theta^2, \sigma \left( \frac{|\log \sigma|}{|\log \theta|} + 1 \right), \theta \frac{\sigma^3}{\varepsilon^2} + \theta \sigma \log\left( \frac{\sigma}{\varepsilon \theta} \right) \right\}

Figures (4)

  • Figure 1: Left: Sketch of neighboring spins such that $u(j) - \frac{\alpha}{2} u(j+e_1) + u(j+e_2) = 0$. The angle between these neighboring spins is given by $\arccos(\alpha/4)$. Right: Sketch of a ground state of the discrete energy in which the spin field (the sketched spin field is scaled to length of order $\varepsilon$ in order to fit the picture) rotates counter-clockwise in rows and columns.
  • Figure 2: Sketch of the branching construction from GinZwi:22. The different colors code the regions in which the values of the constructed gradient field $\nabla u$ are constant and in the set $K$. In the striped region the constructed function is interpolated towards the boundary so that $u$ satisfies $u(0,y) = (1-2\theta)y$. Left: sketch of the building block; one oscillation of $\partial_y u$ is refined into $\sim \theta^{-1}$ oscillations of $\partial_y u$. Right: branching construction using the isotropically rescaled building blocks so that $\partial_y u$ oscillates more and more towards the boundary. Close to the boundary it holds in a weak sense $\partial_y u \approx 1-2\theta$.
  • Figure 3: Sketch of the construction in the upper bound. Left: Sketch of $\nabla u$ which acts as a building block for $\tilde{\beta}$. The striped area is where $\nabla u$ is nonconstant and behaves roughly as $\frac{\sigma}{|(x,y) - (\frac{\sigma}{2\theta}, \frac{\sigma}{2\theta})}$. Right: Sketch of the function $\tilde{\beta}$. In the green and blue region $\tilde{\beta}$ is constantly $(1,1)^T$ and $(1,1-2\theta)^T$, respectively. The blue dots indicate the support of the measure $\operatorname{curl } \tilde{\beta} = \sigma \sum_{k=1}^{\lfloor \frac{\sigma}{2\theta} \rfloor}\delta_{(\frac{\sigma}{2\theta}, k \frac{\sigma}{2\theta})^T}$.
  • Figure 4: Sketch of $v(\frac{\sigma}{2\theta},\cdot)$ which only uses slope $1$ and approximates a function with slope $(1-2\theta)$ through jumps of distance $\frac{\sigma}{2\theta}$ with jump height $\sigma$. In the absolutely continuous part of $Dv$, denoted by $\tilde{\beta}$ in the proof of the upper bound, this induces the condition $\operatorname{curl } \tilde{\beta} = \sum_{k \in \mathbb{Z}} \sigma \, \delta_{(\frac{\sigma}{2\theta},k\frac{\sigma}{2\theta})}$.

Theorems & Definitions (38)

  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 4.1
  • Lemma 5.1
  • ...and 28 more