We address the efficient computation of power-law-based interaction potentials of homogeneous d-dimensional bodies with an infinite n-dimensional array of copies, including their higher-order derivatives. This problem forms a serious challenge in micromagnetics with periodic boundary conditions and related fields. Nowadays, it is common practice to truncate the associated infinite lattice sum to a finite number of images, introducing uncontrolled errors. We show that, for general interacting geometries, the exact infinite sum for both dipolar interactions and generalized Riesz power-law potentials can be obtained by complementing a small direct sum by a correction term that involves efficiently computable derivatives of generalized zeta functions. We show that the resulting representation converges exponentially in the derivative order, reaching machine precision at a computational cost no greater than that of truncated summation schemes. In order to compute the generalized zeta functions efficiently, we provide a superexponentially convergent algorithm for their evaluation, as well as for all required special functions, such as incomplete Bessel functions. Magnetic fields and related quantities can thus be evaluated to machine precision in arbitrary cuboidal domains periodically extended along one or two dimensions. We benchmark our method against known formulas for magnetic interactions and against direct summation for Riesz potentials with sufficiently large exponents, consistently achieving full precision. In addition, we identify new corrections to the asymptotic limit of the demagnetization field and tabulate high-precision benchmark values that can be used as a reliable reference for micromagnetic solvers. The techniques developed are broadly applicable, with direct impact in other areas such as molecular dynamics.

Controlled generation of topological spin textures, such as merons and their bound state, the bimerons, is essential for advancing spintronic technologies and elucidating soliton physics in condensed matter. Using in situ Lorentz transmission electron microscopy coupled with femtosecond laser pulse, we demonstrate the creation of two distinct Bloch-type bimeron states in chiral magnet Co8Zn8Mn4 thin plates at room temperature. Magnetic imaging and micromagnetic simulations reveal that bimeron density varies with applied magnetic field strength, enabling dynamic topological control. We further establish that the topological classification of laser-generated bimerons is invariant with specimen thickness. Field-driven reversible transformations between elongated and circular bimeron morphologies are observed, governed by the competition of Zeeman energy and magnetic shape anisotropy. Micromagnetic simulations quantitatively reproduce these metastable states, validating a unified meron-skyrmion topological framework. This work establishes a single-pulse protocol for optical manipulation of topological spin textures.

Hopfions are three-dimensional topological solitons formed by closed loops of vortex strings, often taking the shape of rings. In magnetic crystals, hopfions have so far been observed only in unusual configurations in which hopfion rings are linked to skyrmion strings. Although theory predicts the existence of stable, isolated hopfions, their experimental realization has remained challenging. Here we demonstrate the laser-induced nucleation and direct observation of isolated magnetic hopfions in a cubic chiral magnet, FeGe, using transmission electron microscopy. The nucleation conditions are determined as a function of laser fluence and external magnetic field. Quantitative agreement between experimental data and micromagnetic simulations provides evidence for the emergence of isolated hopfions. We derive the topological invariant for hopfions under realistic rather than idealized boundary conditions and calculate its integer values for the observed objects. We also reveal that hopfions can coexist and interact with other topological spin textures over a broad field range. These findings demonstrate a contact-free approach for nucleating complex three-dimensional magnetic textures, thus providing a basis for further fundamental and applied research on magnetic hopfions.

Hopfions are three-dimensional (3D) topological solitons predicted to exist in diverse magnetic systems, yet their practical utility has been largely restricted to cryogenic environments. Here, we overcome this temperature constraint by demonstrating stable magnetic hopfions in the chiral magnet Co8Zn8Mn4 at and above room temperature. Using a transmission electron microscope equipped for in situ optical excitation, we generate magnetic hopfions with femtosecond laser pulses. Long-term observations further reveal Brownian-like motion at room temperature and thermally activated collapse upon approaching the high-temperature regime. Together with micromagnetic simulations and homotopy group analysis, our experimental observations uncover the hopfion formation mechanism through the fusion of bimeron pairs. These findings establish room-temperature magnetic hopfions and provide a framework for their further studies under technologically relevant conditions.

Magnetic hopfions are 3D topological solitons - knotted, vortex-like spin configurations. In chiral magnets, hopfions can appear as isolated structures or they can be linked to skyrmion strings. Previous studies employed a sophisticated protocol and a special sample geometry to nucleate such hopfions linked to one or a few skyrmion strings. Here, we introduce an electric-current-assisted nucleation protocol that is simple and independent of the sample shape and size. The resulting hopfions exhibit extraordinary stability in the presence of both positive and negative magnetic fields, in perfect agreement with micromagnetic simulations. We also present a comprehensive framework for classifying hopfions, skyrmions, and merons by deriving the corresponding homotopy group.

Magnetic solitons have recently attracted significant attention due to their intricate physical properties and potential applications in information processing. The majority of the studies in this field, however, are focused on a particular type of solitons known as skyrmions, characterised by a unit topological charge. Here, we present a shifted magnetic bilayer that can demonstrate magnetic solitons with arbitrary large topological charges. These configurations, which we call linked skyrmions , consist of multiple skyrmions linked together with topological point defects. These topological point defects, termed as anti-aligned points, originate from the mutually orthogonal Dzyaloshinskii-Moriya interaction in two different layers. By tuning the interlayer exchange coupling and the external magnetic field, one can also achieve different ground states in this bilayer. Additionally, the system also demonstrates conventional skyrmion-bags and k π -skyrmions. Finally we propose a suitable material candidate where these magnetic configurations can be realised. Our findings, thus, can provide a way to achieve solitons with large topological charge and realise them in realistic systems.

We present a shifted magnetic bilayer that exhibits various magnetic phases and magnetic textures with arbitrarily large topological numbers. The proposed system is characterised by a mutually orthogonal Dzyaloshinskii-Moriya interaction (DMI) in two different layers which can be induced by suitably placing non-magnetic atom with spin-orbit coupling. At weak interlayer coupling, the ground state resembles a checker-board pattern containing regions with unfavourable magnetic alignment which we call anti-aligned points. At finite interlayer coupling and finite external magnetic field, the bilayer can demonstrate a new class of magnetic solitons where multiple magnetic solitons can be connected by topological point defects which we call linked skyrmion. In addition to that the model also demonstrates conventional skyrmion-bags and kπ-skyrmions. Finally, with rigorous first principle calculations, we propose a suitable material candidate where these magnetic configurations can be observed.

Magnetic vortices and skyrmions are typically characterized by distinct topological invariants. This paper presents a unified approach for the topological classification of these textures, encompassing isolated objects and configurations where skyrmions and vortices coexist. Using homotopy group analysis, we derive topological invariants that form the free Abelian group, Z x Z. We provide an explicit method for calculating the corresponding integer indices in continuous and discrete systems. This unified classification framework extends beyond magnetism and is applicable to physical systems in general.

We investigate the behavior of minimizers of perturbed Dirichlet energies supported on a wire generated by a regular simple curve gamma and defined in the space of S2-valued functions. The perturbation K is represented by a matrix-valued function defined on S2 with values in R3x3. Under natural regularity conditions on K, we show that the family of perturbed Dirichlet energies converges, in the sense of Gamma-convergence, to a simplified energy functional on gamma. The reduced energy unveils how part of the antisymmetric exchange interactions contribute to an anisotropic term whose specific shape depends on the curvature of gamma. We also discuss the significant implications of our results for studies of ferromagnetic nanowires when Dzyaloshinskii-Moriya interaction (DMI) is present.

Additive nanotechnology enable curvilinear and three-dimensional (3D) magnetic architectures with tunable topology and functionalities surpassing their planar counterparts. Here, we experimentally reveal that 3D soft magnetic wireframe structures resemble compact manifolds and accommodate magnetic textures of high order vorticity determined by the Euler characteristic, χ . We demonstrate that self-standing magnetic tetrapods (homeomorphic to a sphere; χ = + 2) support six surface topological solitons, namely four vortices and two antivortices, with a total vorticity of + 2 equal to its Euler characteristic. Alternatively, wireframe structures with one loop (homeomorphic to a torus; χ = 0) possess equal number of vortices and antivortices, which is relevant for spin-wave splitters and 3D magnonics. Subsequent introduction of n holes into the wireframe geometry (homeomorphic to an n -torus; χ < 0) enables the accommodation of a virtually unlimited number of antivortices, which suggests their usefulness for non-conventional (e.g., reservoir) computation. Furthermore, complex stray-field topologies around these objects are of interest for superconducting electronics, particle trapping and biomedical applications.

Magnetic skyrmions are topologically nontrivial spin configurations that possess particle-like properties. Earlier research was mainly focused on a specific type of skyrmion with topological charge Q = -1. However, theoretical analyses of two-dimensional chiral magnets have predicted the existence of skyrmion bags -- solitons with arbitrary positive or negative topological charge. Although such spin textures are metastable states, recent experimental observations have confirmed the stability of isolated skyrmion bags in a limited range of applied magnetic fields. Here, by utilizing Lorentz transmission electron microscopy, we show the extraordinary stability of skyrmion bags in thin plates of B20-type FeGe. In particular, we show that skyrmion bags embedded within a skyrmion lattice remain stable even in zero or inverted external magnetic fields. A robust protocol for nucleating such embedded skyrmion bags is provided. Our results agree perfectly with micromagnetic simulations and establish thin plates of cubic chiral magnets as a powerful platform for exploring a broad spectrum of topological magnetic solitons.

We report tailed skyrmions—a new class of stable soliton solutions of the 2D chiral magnet model. Tailed skyrmions have elongated shapes and emerge in a narrow range of fields near the transition between the spin spirals and the saturated state. We analyze the stability range of these solutions in terms of external magnetic field and magnetocrystalline anisotropy. We calculate minimum energy paths and homotopies (continuous transitions) between tailed skyrmions of the same topological charge. The discovery of tailed skyrmions extends the diversity of already known solutions. This is illustrated by solitons with complex morphology, such as tailed skyrmion bags with and without chiral kinks.

We consider a new class of topological defects in chiral magnetic crystals such as FeGe and MnSi. These are composite topological defects that arise when skyrmions in the magnetic order intersect with twin boundaries in the underlying crystalline lattice. We show that the resulting stable configurations are a new type of defect that can be viewed as half-hopfions.

We present a metaheuristic conditional neural-network-based method aimed at identifying physically interesting metastable states in a potential energy surface of high rugosity. To demonstrate how this method works, we identify and analyze spin textures with topological charge $Q$ ranging from 1 to $-13$ (where antiskyrmions have $Q<0$) in the Pd/Fe/Ir(111) system, which we model using a classical atomistic spin Hamiltonian based on parameters computed from density functional theory. To facilitate the harvest of relevant spin textures, we make use of the newly developed Segment Anything Model (SAM). Spin textures with $Q$ ranging from $-3$ to $-6$ are further analyzed using finite-temperature spin-dynamics simulations. We observe that for temperatures up to around 20\,K, lifetimes longer than 200\,ps are predicted, and that when these textures decay, new topological spin textures are formed. We also find that the relative stability of the spin textures depend linearly on the topological charge, but only when comparing the most stable antiskyrmions for each topological charge. In general, the number of holes (i.e., non-self-intersecting curves that define closed domain walls in the structure) in the spin texture is an important predictor of stability -- the more holes, the less stable is the texture. Methods for systematic identification and characterization of complex metastable skyrmionic textures -- such as the one demonstrated here -- are highly relevant for advancements in the field of topological spintronics.

Magnetic skyrmions and hopfions are topological solitons1-well-localized field configurations that have gained considerable attention over the past decade owing to their unique particle-like properties, which make them promising objects for spintronic applications. Skyrmions2,3 are two-dimensional solitons resembling vortex-like string structures that can penetrate an entire sample. Hopfions4-9 are three-dimensional solitons confined within a magnetic sample volume and can be considered as closed twisted skyrmion strings that take the shape of a ring in the simplest case. Despite extensive research on magnetic skyrmions, the direct observation of magnetic hopfions is challenging10 and has only been reported in a synthetic material11. Here we present direct observations of hopfions in crystals. In our experiment, we use transmission electron microscopy to observe hopfions forming coupled states with skyrmion strings in B20-type FeGe plates. We provide a protocol for nucleating such hopfion rings, which we verify using Lorentz imaging and electron holography. Our results are highly reproducible and in full agreement with micromagnetic simulations. We provide a unified skyrmion-hopfion homotopy classification and offer insight into the diversity of topological solitons in three-dimensional chiral magnets.

In magnetic multilayers with perpendicular anisotropy, the competition of short-range and long-range interactions gives rise to the stability of cylindrical magnetic domains, also known as magnetic bubbles. The presence of Dzyaloshinsky-Moriya interaction induced by asymmetric interfaces between magnetic and nonmagnetic layers may lead to the formation of cylindrical bubble domains with Neel-type domain walls across the whole thickness of the multilayer. Such domain walls produce no contrast in Lorentz TEM under the normal incidence of the electron beam to the film. The latter is often used as an argument for the presence of Dzyaloshinskii-Moriya interaction in the system. Here we show that in magnetic multilayers, the absence of the Lorentz TEM contrast might also have another origin. In particular, in the absence of interfacial Dzyaloshinskii-Moriya interaction and weak interlayer exchange coupling, the magnetic bubbles might have Bloch-type domain walls of alternate chirality in adjacent layers. Such domain walls also do not produce magnetic contrast in Lorentz TEM at normal incidence of the electron beam. We show that, in the absence of interlayer exchange coupling, the magnetic bubble domains with the domain walls of fixed and alternate chirality have nearly identical energies and can coexist in the same range of magnetic fields. Using the geodesic nudged elastic band method, we prove that these states are separated by finite energy barriers. Furthermore, we demonstrate that magnetic multilayers with only dipolar coupling, besides the magnetic bubbles with nontrivial topology in all layers, can accommodate solutions with trivial topology within the internal layers.

Cubic chiral magnets exhibit a remarkable diversity of two-dimensional topological magnetic textures, including skyrmions. However, the experimental confirmation of topological states localized in all three spatial dimensions remains challenging. In this paper, we investigate a three-dimensional topological state called a heliknoton, which is a hopfion embedded into a helix or conic background. We explore the range of parameters at which the heliknoton can be stabilized under realistic conditions using micromagnetic modeling, harmonic transition state theory, and stochastic spin dynamics simulations. We present theoretical Lorentz TEM images of the heliknoton, which can be used for experimental comparison. Additionally, we discuss the stability of the heliknoton at finite temperatures and the mechanism of its collapse. Our study offers a pathway for future experimental investigations of three-dimensional topological solitons in magnetic crystals.

Raw data for Figures shown in the main text and Extended Data Figs 1-4 are provided here. Data is provided in *.dm4 format, which can be opened by Gatan Microscopy Suite (GMS) software 3.4.

Hopfions are an intriguing class of string-like solitons, named according to a classical topological concept classifying three-dimensional direction fields. The search for hopfions in real physical systems has been ongoing for nearly half a century, starting with the seminal work of Faddeev. However, so far, realizations in bulk solids are missing. Here, we show that hopfions appear as emergent particles of the classical Heisenberg model with competing exchange interactions. This requires going beyond the model approach used in prior work and deriving a general micromagnetic energy functional directly from a spin-lattice Hamiltonian. We present a definite parameter space in which the existence of hopfions is possible. This opens a concrete vista to combine computational approaches such as density functional theory with material informatics to find magnetic crystals that can host hopfions. As proof of principle, we show how zero-field hopfions can be visualized by the means of off-axis electron holography in a transmission electron microscope.

The diversity of three-dimensional magnetic states in a FeGe nanocylinder is studied using micromagnetic simulations and off-axis electron holography in the transmission electron microscope. In particular, we report the observation of a dipole string—a spin texture composed of two coupled Bloch points—which becomes stable under geometrical confinement. Quantitative agreement is obtained between experimental and theoretical phase shift images by taking into account the presence of a damaged layer on the surface of the nanocylinder. The theoretical model is based on the assumption that the damaged surface layer, which results from focused ion beam milling during sample preparation, has similar magnetic properties to those of an amorphous FeGe alloy. The results highlight the importance of considering the magnetic properties of the surface layers of such nanoscale samples, which influence their magnetic states.