Magnetic Hopfions: Topological Solitons in Spintronics

• Magnetic hopfions are three-dimensional topological solitons defined by a Hopf index representing linked magnetization preimages.
• Their stability arises from a balance of exchange, Dzyaloshinskii–Moriya interactions, anisotropy, and magnetostatic effects.
• Dynamic modes and unique transport signatures make them promising for next-generation spintronic and neuromorphic applications.

Magnetic hopfions are three-dimensional topological solitons characterized by a nontrivial Hopf index, representing the integer-valued linking number of preimages under a continuous mapping from the compactified real space to the unit sphere of magnetization directions. These solitonic excitations exhibit complex internal topology, rich dynamical modes, and robustness rooted in their knot-like configuration, making them a versatile platform for fundamental studies and applications in next-generation spintronic devices.

1. Topological Structure and Mathematical Formulation

A magnetic hopfion is a smooth, localized magnetization configuration, $\mathbf{m}(\mathbf{r}) : \mathbb{R}^3 \to S^2$, where $|\mathbf{m}|=1$ and $\mathbf{m}(\infty) = \hat{\mathbf{z}}$. This compactification enables the identification of $\mathbb{R}^3 \cup \{\infty\} \cong S^3$, and the texture corresponds to a homotopy class in $\pi_3(S^2) \cong \mathbb{Z}$. The topological invariant—the Hopf index $H$ (or $Q_H$)—counts how many times the preimages of two distinct points on $S^2$ are linked in real space: $$H = \frac{1}{(4\pi)^2}\int d^3 r\, \mathbf{A} \cdot (\nabla \times \mathbf{A}),$$ where $\mathbf{A}$ is an emergent vector potential satisfying $$\nabla \times \mathbf{A} = \mathbf{B}_{\mathrm{em}}$$, with

$$\mathbf{B}_{\mathrm{em},i} = \frac{1}{2} \varepsilon_{ijk} \mathbf{m}\cdot(\partial_j \mathbf{m} \times \partial_k \mathbf{m}).$$

For $H=1$, the hopfion texture is typically realized as a closed, twisted loop of skyrmion string—any two distinct preimage loops are linked once. Configurations with higher $|H|$ correspond to multiply-linked structures with increased topological complexity (Tejo et al., 30 Aug 2025, Kasai et al., 28 Nov 2025, Saji et al., 2023).

2. Energetics and Micromagnetic Modeling

Hopfion stability requires a delicate balance of competing interactions encoded in the micromagnetic energy functional. In chiral magnets, the standard functional includes exchange ($A$), Dzyaloshinskii–Moriya interaction (DMI, $D$), uniaxial anisotropy ($K_u$), Zeeman energy, and magnetostatics: $$E[\mathbf{m}] = \int d^3 r \left\{ A |\nabla \mathbf{m}|^2 + D\,\mathbf{m} \cdot (\nabla \times \mathbf{m}) + K_u (1 - m_z^2) - \mu_0 M_s \mathbf{H}_{\mathrm{ext}} \cdot \mathbf{m} - \frac{\mu_0}{2} M_s \mathbf{m} \cdot \mathbf{H}_{d} \right\}.$$ Key stabilization mechanisms involve:

• Exchange–DMI competition: Establishes a twisted structure with finite core width.
• Perpendicular anisotropy and boundary pinning: Confine the texture and enforce robust top boundary conditions.
• Magnetostatic self-closure: Minimizes stray fields via flux closure in 3D.
• Topological protection: The nontrivial linking prohibits decay without singularities or crossing boundaries (Tejo et al., 30 Aug 2025, Metlov, 11 Apr 2024, Rybakov et al., 2019).

The introduction of higher-order exchange or frustration allows stabilization in the absence of DMI (purely Heisenberg models), with real materials requiring $A, B, C > 0$ and a scale-selection criterion $l_0 \sim \sqrt{\max(C,6B)/A}$ for the characteristic hopfion size (Rybakov et al., 2019, Sallermann et al., 2022).

3. Static Types, Phase Diagrams, and Stability

Recent variational and numerical studies have established the existence of multiple metastable hopfion types, distinguished by internal vortex–antivortex tube ordering, associated with the global phase $\chi$ in the Whitehead ansatz. For $H=1$:

• Type 1 ($\chi=\pi/2$): Tightly bound, smaller radius, lower energy—vortex filament resides inside, antivortex outside.
• Type 2 ($\chi=3\pi/2$): Larger, weaker exchange penalty, realized only in narrow regions of anisotropy and field.
• Both types are topologically equivalent ($H=1$) but differ in energetics and internal structure.

These phases are delineated in $(q, h)$ (anisotropy, external field) diagrams, with phase boundaries shifting under magnetostatic effects ($\mu^2$). As $\mu^2$ increases, hopfion stability is lost above material-dependent thresholds (Metlov, 11 Apr 2024, Metlov, 9 Jan 2025, Metlov, 2022). The presence of vortex–antivortex stacking also determines elliptical stability; only one sequence yields a shape-stable soliton in bulk (Metlov, 9 Jan 2025).

4. Dynamical Modes and Nonlinear Control

Magnetic hopfions exhibit a spectrum of intrinsic dynamical modes, including breathing, twisting, and gyrotropic resonances:

• Breathing mode: Radially symmetric oscillation of both core diameter and shell thickness, with frequency $f_0 \approx 0.54$–$0.56$ GHz in typical MnSi-like multilayers, well-captured by a rigid domain-wall interaction model. The breathing dynamics retain topological invariance and can be detected by FMR or BLS (Tejo et al., 30 Aug 2025).
• Spin-wave eigenmodes: Extend up to $\sim$15 GHz; localization on vortex lines or concentric tori can serve as real-space fingerprints and distinguish hopfions from torons or skyrmions. The topological transition to torons under field ($H_c \approx 32$ mT (Raftrey et al., 2021)) is marked by discontinuous spectral jumps.
• SOT-driven nonequilibrium control: Landau–Lifshitz–Gilbert dynamics with spin-orbit torque enable translation, precession, splitting, and recombination of hopfions. For $H>1$, controlled SOT protocols realize repeated splitting (e.g., $H=2\to1+1$) and fusion, governed by a universal tension hierarchy and thresholds for $\zeta$ (dimensionless SOT strength), paving the way for multilevel memory (Kasai et al., 28 Nov 2025, Kasai et al., 28 Nov 2025).

5. Electronic, Magnonic, and Transport Signatures

Hopfions possess several experimentally relevant transport and magnonic signatures, arising from their emergent electromagnetic field:

• Orbital Hall effect: The unique 3D orbital Hall response, with all three tensor elements $\sigma_{xy}^{L_z}$, $\sigma_{yz}^{L_x}$, $\sigma_{zx}^{L_y}$ nonzero, serves as an electrical hallmark distinguishing hopfions from skyrmions or skyrmioniums. This effect enables all-electrical detection and large orbital torques for spin-orbitronic manipulation (Göbel et al., 13 Jun 2025).
• Topological Hall effect: While the global topological Hall response vanishes (as $\langle \mathbf{B}_{\mathrm{em}} \rangle = 0$), local Hall measurements yield finite signals, switchable by field or current-induced reorientation, providing robust read-out for hopfion-based racetrack memories (Göbel et al., 2020).
• Magnonic Hall and magnonic lensing: Spin-wave packets experience effective Lorentz forces from $\mathbf{B}_{\mathrm{em}}$, resulting in direction-dependent focusing and skew scattering analogous to a magnonic Hall/Aharonov–Bohm effect. Magnonic holography resolves these via interference fringes, directly probing the Hopf index (Saji et al., 2023).

6. Materials Realization, Phase Diagrams, and Superstructures

Hopfions can be stabilized in both multilayer thin films and bulk chiral or frustrated magnets:

• Thin film nanodisks: Experimental realization in Ir/Co/Pt multilayers exploits interfacial DMI and engineered perpendicular anisotropy ratios, permitting direct 3D imaging via XMCD–PEEM and MTXM to confirm hopfion topology (Kent et al., 2020).
• Bulk chiral magnets and Heisenberg models: B20 materials (MnSi, FeGe), ferrimagnets, and alloys with tailored $A$, $D$, $K$, and $M_s$ are prime candidates. Theoretical bounds on $\mu^2 = \mu_0 M_s^2 C / D^2$ separate regions of stable and unstable hopfions (Metlov, 11 Apr 2024).
• Hierarchical topological superstructures: In frustrated easy-plane magnets, hopfions not only exist as isolated solitons but can self-organize into 3D crystals, with staggered chains ($H=+1$, $H=-1$) threaded by meron strings. These superstructures exhibit quantized skyrmion number per $xy$-plane and could support novel collective dynamics (Kasai et al., 1 Nov 2024).

System/class	Stabilizing interaction	Hopfion type(s)
Multilayer films	Interfacial DMI, PMA, boundary	$H=1$, rigid, 3D
Chiral bulk	Bulk DMI, exchange, anisotropy	Type 1 and 2, elliptic
Frustrated Heisenberg	Competing exchange	Arbitrary $|H|$

7. Detection Techniques and Applications

The experimental detection and device integration of hopfions utilize:

• 3D magnetic imaging: Off-axis electron holography, X-ray tomography, and spin-polarized STM resolve core geometry and linked preimage loops.
• SANS: Small-angle neutron scattering detects characteristic double-peak correlations and nonreciprocal spin-flip signals from toroidal magnetization and chirality (Metlov et al., 27 Apr 2024).
• Spectroscopic signatures: FMR, BLS, and magnonic transport provide mode-resolved access to hopfion dynamics (Tejo et al., 30 Aug 2025, Raftrey et al., 2021).
• Device paradigms: Racetrack and multilevel memory exploiting SOT-driven topology control, 3D topological logic exploiting robust, drift-free (vanishing gyrovector) motion (Wang et al., 2019).

The combination of topological robustness, dynamic tunability, and distinctive electromagnetic responses positions magnetic hopfions as a fundamental building block for topologically protected information carriers, reconfigurable spintronic elements, and spectrally encoded neuromorphic platforms (Tejo et al., 30 Aug 2025, Kasai et al., 28 Nov 2025, Göbel et al., 13 Jun 2025, Kent et al., 2020, Göbel et al., 2020).