Topological–chiral magnetic interactions driven by emergent orbital magnetism

Abstract

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...

Prospecting chiral multisite interactions in prototypical magnetic systems

“…), either isotropic or anisotropic. The isotropic interactions include the biquadratic interaction (four-spin two-site) [12][13][14], the four-spin three-site interaction [15,16], and the ring exchange (four-spin four-site) [17][18][19], with a recent proposal for a six-spin three-site isotropic interaction [20], which we will show can be related to the six-spin six-site interaction derived in Ref. [19].…”

“…Phenomenological considerations can also be used to identify allowed forms for the interactions consistent with the symmetry of a target material. This led to the discovery of chiral four-spin three-site interactions in MnGe [20] (for which a derivation based on multiple scattering theory was also provided), and motivated their existence in an Fe chain on Re(0001) [52], while the magnetism of Ca 3 Ru 2 O 7 was rationalized by invoking higher order Lifshitz invariants in connection to a Ginzburg-Landau theory [53]. Lastly, the energy can also be expanded in a Taylor series in small deviations from a reference magnetic structure.…”

“…Lastly, the energy can also be expanded in a Taylor series in small deviations from a reference magnetic structure. This was developed into the very successful infinitesimal rotation method for two-spin interactions [54][55][56], with an extension to four-spin interactions recently proposed [20,57], and a growing body of work addressing its application for noncollinear magnetic structures [22,23,58,59].…”

Biquadratic exchange interactions in two-dimensional magnets

“…The best match between regression and DFT energies is for regressions beyond linear. This suggested that even more complex magnetic interactions may take place in 2D magnetic materials such as 3-, 4spin interactions 16,17 , and chiral biquadratic 18,19 which are not studied here. Materials with alike chemical environment (e.g., bond lengths, electron affinity, binding energy) such as CrI 3 , CrBr 3 and CrCl 3 present close variation of the energy and consequently similar magnitudes of the BQ exchange (Table 1).…”

Theory of current-induced angular momentum transfer dynamics in spin-orbit coupled systems

“…An exception to this picture is a noncollinear magnet, where orbital angular momentum is associated with the spin chirality [49][50][51] or density of topological charge [51,52]. Here, spin and orbital momenta may interact even without relativistic spin-orbit coupling [53]. Although such chiral or topological orbital angular momentum exhibits exotic dynamic phenomena associated with complex spin structures [54,55], we leave this case to future work.…”