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A spin Hamiltonian, which characterizes interatomic interactions between spinmoments, is highly valuable in predicting and comprehending the magneticproperties of materials. A deeper understanding of the microscopic origin ofmagnetic interactions can open new pathways toward realizing nanometer-scalesystems for future spintronic devices. Here, we explore a method for explicitlycalculating interatomic exchange interactions in non-collinear configurationsof magnetic materials considering only a bilinear spin Hamiltonian in a localscenario. Based on density-functional theory (DFT) calculations of dimersadsorbed on metallic surfaces, and with a focus on the Dzyaloshinskii-Moriyainteraction (DMI) which is essential for stabilizing chiral non-collinearmagnetic states, we discuss the interpretation of the DMI when decomposed intomicroscopic electron and spin densities and currents. We clarify the distinctorigins of spin currents induced in the system and their connection to the DMI.In addition, we reveal how non-collinearity affects the usual DMI, which issolely induced by spin-orbit coupling, and DMI-like interactions brought aboutby non-collinearity. We explain how the dependence of the DMI on the magneticconfiguration establishes a connection between high-order magneticinteractions, enabling the transition from a local to a global spinHamiltonian.

Unraveling the connection between high-order magnetic interactions and local-to-global spin Hamiltonian in non-collinear magnetic dimers