We investigate long-range chiral magnetic interactions among adatoms mediated by surface states spin-splitted by spin-orbit coupling. Using the Rashba model, the tensor of exchange interactions is extracted wherein a thepseudo-dipolar interaction is found, in addition tothe usual isotropic exchange interaction and the Dzyaloshinskii-Moriya interaction. We find that, despite the latter interaction, collinear magnetic states can still be stabilized by the pseudo-dipolar interaction. The interadatom distance controls the strength of these terms, which we exploit to design chiral magnetism in Fe nanostructures deposited on aAu(111) surface. We demonstrate that these magnetic interactions are related to superpositions of the out-of-plane and in-plane components of the skyrmionic magnetic waves induced by the adatoms in the surrounding electron gas. We show that, even if the interatomic distance is large, the size and shape of the nanostructures dramatically impacts on the strength of the magnetic interactions, thereby affecting the magnetic ground state. We also derive an appealing connection between the isotropic exchange interaction and the Dzyaloshinskii-Moriya interaction, which relates the latter to the first-order change of the former with respect tospin-orbit coupling. This implies that the chirality defined by the direction of the Dzyaloshinskii-Moriya vector is driven by the variation of the isotropic exchange interaction due to the spin-orbit interaction.distance, which was recently confirmed experimentally usingscanning tunneling microscopy (STM) and theoretically usingabinitio simulations based on density functional theory [8]. We note that today, besides theory, state-of-the-art STM experiments can be used to learn about the magnitude, oscillatory behavior and decay ofRKKY interactions, as demonstrated in [28][29][30].Our goal is to address the DM interaction in an analytically tractable model and investigate its magnitude, sign and direction following a bottom-up approach, assembling nanostructures of different sizes and shapes, atom-by-atom. We are particularly interested in the long-range magnetic interactions that have been already investigated several times theoretically. For example, Imamura et al [31] considered pairs of localized spins interacting via the so-called two-dimensional Rashba gas of electrons [32,33] while Zhu et al [34] replaced the Rashba gas withthe surface of a topological insulator. We revisit the case of Rashba electrons and consider particularly the surface state of Au (111), where the Rashba spin splitting was observed experimentally [35]. 4 We report on selected nanostructures: dimers, wires, trimers, and two hexagonal structures deposited on the Au (111), where the interactions are mediated solely by the surface state. For the dimer case, we extract the analytical form of the magnetic exchange interactions tensor using the approximation of Imamura et al [31], labeled in the following RKKY approximation, without renormalizing the electronic structure of the Rashba el...
