Abstract
Atomistic spin models have found enormous success in addressing the properties of magnetic materials, grounded on the identification of the relevant underlying magnetic interactions. The huge development in the field of magnetic skyrmions and other noncollinear magnetic structures is largely due to our understanding of the chiral Dzyaloshinskii-Moriya interaction. Recently, various works have proposed new types of chiral interactions, with seemingly different forms, but the big picture is still missing. Here, we present a systematic construction of a generalized spin model containing isotropic and chiral multi-site interactions. These are motivated by a microscopic model that incorporates local spin moments and the spin-orbit interaction, and their symmetry properties are established. We show that the chiral interactions arise solely from the spin-orbit interaction and that the multi-site interactions do not have to follow Moriya's rules, unlike the Dzyaloshinskii-Moriya and chiral biquadratic interactions. The chiral multi-site interactions do not vanish due to inversion symmetry, and comply with a generalized Moriya rule: If all sites connected by the interaction lie in the same mirror plane, the chiral interaction vector must be perpendicular to this plane. We then illustrate our theoretical considerations with density functional theory calculations for prototypical magnetic systems. These are triangular trimers built out of Cr, Mn, Fe and Co adatoms on the Re(0001), Pt(111) and Au(111) surfaces, for which $C_\mathrm{3v}$ symmetry applies, and Cr and Fe square tetramers on Pt(001) with $C_\mathrm{4v}$ symmetry. The multi-site interactions are substantial in magnitude and cannot be neglected when comparing the energy of different magnetic structures. Finally, we discuss the recent literature in light of our findings, and clarify several unclear or confusing points.
Highlights
Atomistic spin models provide the foundation to understand the properties of magnetic materials: Complex magnetic ground-state structures, elementary excitations, solitons whether topologically trivial or nontrivial, thermal effects, and real-time dynamics [1]
The chiral multisite interactions do not vanish as a result of inversion symmetry and comply with a generalized Moriya rule: If all sites connected by the interaction lie in the same mirror plane, the chiral interaction vector must be perpendicular to this plane
In comparison to the full quantum-mechanical description, this type of model aims at a coarse-grained, low-energy description of a given material, by assuming that magnetism is well described by assigning rigid magnetic moments to specific spatial positions and specifying how these spins interact among each other
Summary
Atomistic spin models provide the foundation to understand the properties of magnetic materials: Complex magnetic ground-state structures, elementary excitations (spin waves), solitons whether topologically trivial or nontrivial (domain walls and magnetic skyrmions, respectively), thermal effects, and real-time dynamics [1]. Isotropic fourspin interactions explain why some bulk materials host short-period magnetic skyrmion lattices and other multiple-Q states [34,35,36,37,38] Another class of materials where the isotropic four-spin interactions play various roles are high-temperature superconductors, likely stabilizing the bicollinear antiferromagnetic ground state of FeTe [39,40] and modifying the spin wave spectrum of the parent compound La2CuO4 [41]. Phenomenological considerations can 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 provided), and motivated their existence in an Fe chain on Re(0001) [52], while the magnetism of Ca3Ru2O7 was rationalized by invoking higher order Lifshitz invariants in connection to a Ginzburg-Landau theory [53]. We discuss several recent works in light of our findings, and present our conclusions
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