Hopfions in screw chiral magnets

TL;DR

The paper develops symmetry-transforming magnetic models to stabilize three-dimensional topological spin textures within arbitrary backgrounds. It constructs a screw chiral magnet with spatially modulated Dzyaloshinskii–Moriya interactions, enabling Hopfions and related textures to be metastable in a ferromagnetic background. Despiralization couples spatial translations with spin rotations, yielding distinct Goldstone modes and altering topological invariants such as the Hopf index via changes in emergent flux-tube linking. The findings suggest experimental pathways using rotating fields or structured light and position continuous symmetry transformations as a general design principle for magnetic solitons.

Abstract

Three-dimensional topological spin textures have attracted growing interest due to their rich geometry and potential for functional magnetic phenomena. In this work, we propose the concept of symmetry-transforming magnetic models as a novel route to generate and stabilize complex three-dimensional textures in an arbitrary magnetic background. Using this framework, we predict a screw chiral magnet model that stabilizes magnetic Hopfions and other three-dimensional magnetic textures within a ferromagnetic background. We show that the resulting solitons display distinctive physical properties, including unconventional Goldstone modes. Our results establish continuous symmetry transformations as a general strategy for uncovering new classes of magnetic solitons with unique dynamical signatures.

Figures (10)

• Figure 1: Spiralization and despiralization of magnetic textures and their corresponding models. Left: a Hopfion stabilized in the ferromagnetic ground state of the screw-symmetric model (Eq. \ref{['eq: despiralized energy functional']}). Right: a heliknoton embedded in the spiral ground state of a conventional chiral-magnet model. These two textures can be transformed into each other via (de-)spiralization, and the associated magnetic models indicated on the outer panels by highlighting their DMI interactions are related by the same screw transformation. Displayed are the isosurfaces of $m_y=0$ and the magnetization along edges of the cubic sample of size $2 L_D$. The colormap indicates the magnetization direction, as defined in the inset.
• Figure 2: Overview of the Goldstone modes of the Hopfion: screw (left), involving simultaneous changes of $z_0$ and $\phi$; gyration (middle), involving only a change of $\phi$; and swirl (right), involving only a change of $z_0$, which is a combination of screw and gyration motion. Shown are representative snapshots along each zero-mode trajectory. The background magnetization lies in the $xy$-plane, and its direction is indicated by the HSV colormap (see inset in the lower-left panel). In each image, either the $m_y = 0$ or $m_x = 0$ isosurface (white and black, respectively) is displayed, highlighting the characteristic toroidal structure. The $m_z = \pm 0.99$ isosurfaces are shown in all panels and are colored according to the in-plane magnetization component.
• Figure 3: Examples of stable magnetization configurations in the screw chiral magnet. Highlighted are isosurfaces of $m_z = \pm 0.75$ for the four three-dimensional spin textures: (a) a twisted skyrmion tube, (b) a twisted in-plane skyrmion tube, (c) a despiralized Twiston, and (d) a despiralized Hopfion configuration surrounding a skyrmion string. For configurations (a)--(c) a full period along the $z$-axis is shown and the subfigures depict $xy$-plane cross-sections at $z = 0$, $z = -L_D/4$, and $z = -L_D/2$. Configuration (d) displays five periods of the skyrmion string with a Hopfion ring located near the $z = 0$ plane. The subfigures for (d) show the $xy$-plane at $z = 0$, $z = -5L_D/4$, and $z = -5L_D/2$.
• Figure 4: Summary of Linkings and Hopf index calculation for the spin textures shown in Fig. \ref{['fig:other_textures']}. For (a)–(d), the Hopf index $H$ is calculated for one period along $z$. For (d) $H$ is evaluated for the region $-0.5<z<0.5$, see Fig. \ref{['fig:other_textures']}(d).
• Figure 5: Schematic illustration of the Hopfion ansatz described by Eqs. \ref{['eq:Hopfion_ansatz']} to \ref{['eq:Hopfion_ansatz2']}. The $\mathbf{m} = -\mathbf{e}_y$ isoline lies in the $xz$-plane on a circle of major radius $R$ (black circle). The angle $\psi$ locates the minor circle of radius $r$ (blue circle), which lies in a plane orthogonal to the tangent of the major circle. The local coordinates $(\rho,\phi)$ parametrize points within the minor circle.