Two hundred years ago, Ampère discovered that electric loops in which currents of electrons are generated by a penetrating magnetic field can mutually interact. Here we show that Ampères observation can be transferred to the quantum realm of interactions between triangular plaquettes of spins on a lattice, where the electrical currents at the atomic scale are associated with the orbital motion of electrons in response to the non-coplanarity of neighbouring spins playing the role of a magnetic field. The resulting topological orbital moment underlies the relation of the orbital dynamics with the topology of the spin structure. We demonstrate that the interactions of the topological orbital moments with each other and with the spins form a new class of magnetic interactions − topological-chiral interactions − which can dominate over the Dzyaloshinskii-Moriya interaction, thus opening a path for realizing new classes of chiral magnetic materials with three-dimensional magnetization textures such as hopfions. Exotic magnetic textures with particle-like properties 1-6 offer great potential for innovative spintronic applications 7 and brain-inspired computing 8,9 . Magnetic skyrmions, twodimensional (2D) localized solitons, are a prominent realization of chiral spin structures, first observed in the material class of non-centrosymmetric B20 bulk compounds 1 . The potential of spintronic applications would change fundamentally if the line of thought could be continued to the emergence of three-dimensional (3D) localized magnetic solitons, e.g. hopfions 10-12 . Recently, a 3D lattice of 3D magnetic textures on the nanometer scale was observed in the B20type cubic chiral magnets MnGe 13,14 . Despite the strong interest in this magnet, a complete theoretical model for the underlying magnetic interactions is remarkably elusive until now. While, for instance, the basic magnetic properties of the 2D skyrmions are determined by an intricate competition involving the Heisenberg exchange and the chiral relativistic Dzyaloshinskii-Moriya interaction 15,16 (DMI), such models fail to explain the 3D-magnetic texture observed in MnGe 17 .The 3D magnetization textures of 2D skyrmions gives rise to a scalar spin chirality, a driving force behind a plethora of macroscopic phenomena. Examples are the topological Hall effect 18,19 or a finite topological orbital moment (TOM) 20-25 , which can both serve as experimental fingerprints of skyrmions. Texture-induced contributions to these macroscopic phenomena were also predicted in frustrated magnets 26,27 , where they originate from the non-trivial spin topology associated with the real-space configuration of magnetic moments S i as reflected by the scalar spin chirality χ ijk = S i · (S j × S k ). Although the net spin magnetization might vanish, the symmetry of these chiral systems allows for lowering the energy by preferring orbital currents of specific rotational sense 26,28 . As a consequence, the motion of the electron in the complex magnetic background manifests itself in the finite T...

Thin-film sub-5 nm magnetic skyrmions constitute an ultimate scaling alternative for future digital data storage. Skyrmions are robust noncollinear spin textures that can be moved and manipulated by small electrical currents. Here we show here a technique to detect isolated nanoskyrmions with a current perpendicular-to-plane geometry, which has immediate implications for device concepts. We explore the physics behind such a mechanism by studying the atomistic electronic structure of the magnetic quasiparticles. We investigate from first principles how the isolated skyrmion local-density-of-states which tunnels into the vacuum, when compared with the ferromagnetic background, is modified by the site-dependent spin mixing of electronic states with different relative canting angles. Local transport properties are sensitive to this effect, as we report an atomistic conductance anisotropy of up to ∼20% for magnetic skyrmions in Pd/Fe/Ir(111) thin films. In single skyrmions, engineering this spin-mixing magnetoresistance could possibly be incorporated in future magnetic storage technologies.

Magnetic skyrmions are prime candidates for future spintronic devices. However, incorporating them as information carriers hinges on their interaction with defects ubiquitous in any device. Here we map from first-principles, the energy profile of single skyrmions interacting with single-atom impurities, establishing a generic shape as function of the defect’s electron filling. Depending on their chemical nature, foreign 3d and 4d transition metal adatoms or surface implanted defects can either repel or pin skyrmions in PdFe/Ir(111) thin films, which we relate to the degree of filling of bonding and anti-bonding electronic states inherent to the proximity of the non-collinear magnetic structure. Similarities with key concepts of bond theories in catalysis and surface sciences imbue the universality of the shape of the interaction profile and the potential of predicting its interaction. The resulting fundamental understanding may give guidance for the design of devices with surface implanted defects to generate and control skyrmions.

Magnetic atoms on heavy-element superconducting substrates are potential building blocks for realizing topological superconductivity in one-and two-dimensional atomic arrays. Their localized magnetic moments induce so-called Yu-Shiba-Rusinov (YSR) states inside the energy gap of the substrate. In the dilute limit, where the electronic states of the array atoms are only weakly coupled, proximity of the YSR states to the Fermi energy is essential for the formation of topological superconductivity in the band of YSR states. Here, we reveal via scanning tunnel spectroscopy and ab initio calculations of a series of 3d transition metal atoms (Mn, Fe, Co) adsorbed on the heavy-element superconductor Re that the increase of the Kondo coupling and sign change in magnetic anisotropy with d-state filling is accompanied by a shift of the YSR states through the energy gap of the substrate and a crossing of the Fermi level. The uncovered systematic trends enable the identification of the most promising candidates for the realization of topological superconductivity in arrays of similar systems.

When electrons are driven through unconventional magnetic structures, such as skyrmions, they experience emergent electromagnetic fields that originate several Hall effects. Independently, ground-state emergent magnetic fields can also lead to orbital magnetism, even without the spin–orbit interaction. The close parallel between the geometric theories of the Hall effects and of the orbital magnetization raises the question: does a skyrmion display topological orbital magnetism? Here we first address the smallest systems with nonvanishing emergent magnetic field, trimers, characterizing the orbital magnetic properties from first-principles. Armed with this understanding, we study the orbital magnetism of skyrmions and demonstrate that the contribution driven by the emergent magnetic field is topological. This means that the topological contribution to the orbital moment does not change under continuous deformations of the magnetic structure. Furthermore, we use it to propose a new experimental protocol for the identification of topological magnetic structures, by soft X-ray spectroscopy.