Radial Stabilization of Magnetic Skyrmions Under Strong External Magnetic Field

The skyrmion number density, $q\equiv\vec{n}\cdot\left(\partial_x\vec{n}\times\partial_y\vec{n}\right)/(4π)$, is one of the key quantities that characterizes the topological properties of a magnetic skyrmion. In this work, we propose a model for a two-dimensional magnetic system with Hamiltonian that contains an interaction term proportional to $q^2$ which preserves inversion symmetry. The proposed $q^2$ term is also known as the Skyrme term and is a two-dimensional version of the well-known quartic term in models of three-dimensional Hopfions. In contrast with the usual exchange interaction, the $q^2$ term persists at the strong external magnetic field limit. Using the Landau-Lifshitz-Gilbert equation for micromagnetic calculations, we show that the minimum energy configuration of this model exhibits skyrmion properties. Furthermore, this configuration remains stable under small linear radially symmetric perturbations, and we demonstrate that the total energy of the system is bounded from below, ensuring that it remains above the vacuum energy. This implies a topologically protected configuration. Our model provides a framework for describing skyrmions in materials without broken inversion symmetry, particularly in systems subjected to strong external magnetic fields, where conventional exchange interactions are significantly weaker than the Zeeman effect.

💡 Research Summary

The paper introduces a novel two‑dimensional magnetic model that incorporates a term proportional to the square of the skyrmion density, (q^{2}), into the Hamiltonian. This “Skyrme term” preserves inversion symmetry and remains operative even when a strong external magnetic field dominates the energetics, a regime where conventional exchange and Dzyaloshinskii‑Moriya (DM) interactions become negligible.

After reviewing the topological nature of magnetic skyrmions—specifically the skyrmion number density (q = \mathbf{n}\cdot(\partial_x\mathbf{n}\times\partial_y\mathbf{n})/4\pi) and its associated conserved charge—the authors point out that most existing models rely on broken inversion symmetry to generate the antisymmetric DM interaction. They propose instead to exploit the second invariant of Manton’s strain tensor, (\det D), which in two dimensions is exactly (|\varepsilon_{ij}\partial_i\mathbf{n}\times\partial_j\mathbf{n}|^{2}), i.e. (q^{2}). This term requires three spins forming a triangle, making it intrinsically two‑dimensional and distinct from the usual Heisenberg exchange that involves only nearest‑neighbor pairs.

The full Hamiltonian is written as
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