Energy concentration in a two-dimensional magnetic skyrmion model: variational analysis of lattice and continuum theories

TL;DR

The paper develops a rigorous variational framework for energy concentration in a two-dimensional baby Skyrme-type ferromagnet, showing that as the Zeeman term diverges, the topological charge density converges to an atomic measure with integer multiples of $4\pi$ and that the energy $\Gamma$-converges to the total variation of this measure in the continuum. It then constructs a discrete notion of topological charge on a triangular lattice, proves discrete compactness and $\Gamma$-convergence to a sum of cell energies $\psi(d_i)$ over the singularities, and establishes a precise continuum-discrete bridge, including extensions to full baby Skyrme energies with DMI and antisymmetric exchanges. The work also provides a careful treatment of the discrete topological charge by introducing admissible interpolation regions/surfaces and showing that the discrete charges converge to atomic measures with integer coefficients. Overall, the results unify continuum and lattice descriptions of skyrmion singularities and quantify energy costs for forming vortices of degree $d$ via a cell formula $\psi(d)$, offering a robust variational calculus framework for magnetic solitons. The findings have potential implications for the mathematical understanding of skyrmion energetics and for the accurate discretization of continuum micromagnetic models in simulations.

Abstract

We investigate the formation of singularities in a baby Skyrme type energy model, which describes magnetic solitons in two-dimensional ferromagnetic systems. In presence of a diverging anisotropy term, which enforces a preferred background state of the magnetization, we establish a weak compactness of its topological charge density, which converges to an atomic measure with quantized weights. We characterize the $Γ$-limit of the energies as the total variation of this measure. In the case of lattice type energies, we first need to carefully define a notion of discrete topological charge for $\mathbb{S}^2$-valued maps. We then prove a corresponding compactness and $Γ$-convergence result, thereby bridging the discrete and continuum theories.

Figures (5)

• Figure 1: Examples of the different possible interpolation regions $R$ according to Lemma \ref{['lm:mistero1']}. The dashed and the continuous lines denote edges in $\mathcal{N}_\varepsilon(u)$ and $\mathcal{C}_\varepsilon(u)$, respectively.
• Figure 2: Construction of the planar graph in the proof of Lemma \ref{['lm:mistero1']} (same notation as in Figure \ref{['fig:regions']}).
• Figure 3: The region $S_R(u)$ in the cases $N=2,3,4$.
• Figure 4: The edge $(i,j)_R$ precedes the edge $(j,k)_R$.
• Figure 5: Subdivision of an elementary cell of ${\mathcal{L}}_\varepsilon$ into cells of ${\mathcal{L}'_\varepsilon}$. The points of ${\mathcal{L}^m_\varepsilon}$ are indicated in white, while those in ${\mathcal{L}^c_\varepsilon}$ are indicated in gray.

Theorems & Definitions (40)

• Remark 2.1
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Lemma 4.1
• proof
• proof : Proof of Theorem \ref{['th:contcomp']}
• proof : Proof of Theorem \ref{['th:contmain']}
• Lemma 5.1