Confinement-Induced Metastability and Structural Diversity of Hopfions in Chiral Magnetic Films

TL;DR

This work reveals how confinement and surface anchoring in thin chiral magnetic films stabilize four distinct hopfion textures by situating them near modulated finger phases CF-1 and CF-2. Using 3D micromagnetic simulations within a material-independent framework that also applies to chiral liquid crystals, the authors map a phase diagram in $k_u$ and $k_s$, showing that hopfions inflate toward finger phases or collapse into torons, while periodic hopfion lattices remain unstable and relax into finger textures. The study also uncovers repulsive inter-hopfion interactions, the possibility of composite and bag-like hopfion configurations, and the concept of hopfion lattices as inherently non-equilibrium, metastable states. These results establish general design rules for stabilizing and assembling 3D topological solitons in confined chiral systems and point toward practical texture engineering via external fields in both magnetic and liquid-crystal platforms.

Abstract

Topological particle-like excitations such as skyrmions and hopfions offer rich opportunities for spintronic and photonic applications. While skyrmions have been extensively studied, the stabilization mechanisms and phase behavior of three-dimensional hopfions remain largely unexplored. Here, we investigate the formation, stability, and interactions of hopfions in thin chiral magnetic films with surface anchoring, using three-dimensional micromagnetic simulations within a material-independent framework applicable to both magnetic and liquid crystalline systems. We identify four distinct types of isolated hopfions, generated by rotating bimeron and finger-like solitons around a central axis. The metastability regions of these precursor textures closely follow the boundaries of modulated finger phases, enabling their size to be continuously tuned through anisotropydriven inflation and collapse. Remarkably, we demonstrate that hopfions near their inflation threshold possess energies comparable with the homogeneous state, allowing them to enclose regions of modulated phases or other solitons, forming higher-order, bag-like domains. In contrast, periodic hopfion lattices remain intrinsically unstable under confinement, spontaneously relaxing into finger phases. These findings establish general principles for stabilizing, tuning, and assembling three-dimensional topological solitons in confined chiral systems, suggesting experimentally accessible routes for texture engineering in liquid crystals via electric-field control.

Figures (12)

• Figure 1: Creation of hopfions by rotating CF-1 and CF-2 fingers. (a) Schematics of two skyrmion filaments: one aligned with the surface normal and another oriented perpendicular to it. To accommodate the surrounding homogeneous state, the axisymmetric skyrmion structure must continuously deform into the configuration shown in (b). (b,c) Spin distributions within the two-dimensional CF-1 and CF-2 solitons. The green dash-dotted lines mark the rotation axes used to generate the corresponding three-dimensional hopfion textures. (d) A representative 3D hopfion obtained through this rotational construction.
• Figure 2: Procedure for constructing initial hopfion configurations for relaxation in mumax3. (a) Elementary cylinders are positioned along prescribed circular trajectories and assigned magnetization vectors of specified orientation. (b) Positions of circular trajectories in the middle plane of the film. (c) Five circular trajectories visualized in $xz$-plane cross-sections.
• Figure 3: Four types of isolated hopfions. (a, b) Trivial hopfions with Hopf index $Q_H = 0$. (c, d) Hopfions with $Q_H = +1$ and $Q_H = -1$, respectively. Upper panels show color maps of the $m_y$ component in $xz$ cross-sections. Lower panels display preimages plotted for $\theta = \pi/2$ and several values of the azimuthal angle $\psi$.
• Figure 4: Topological charge density $\rho_Q$ and local energy density $w$ for isolated hopfions. Color maps show the distributions of $\rho_Q$ and $w$ in the $xz$-plane for all hopfion varieties at $k_u = 0.22$, $k_s = 50$.
• Figure 5: Energy barriers between different hopfion states. The blue curve corresponds to $Q_H = 0$ hopfions, while the red curve represents $Q_H = \pm 1$ hopfions. Color plots depict three characteristic spin configurations for trivial $Q_H = 0$ hopfions along the transition pathway.