  A spin Hamiltonian, which characterizes interatomic interactions between spin
moments, is highly valuable in predicting and comprehending the magnetic
properties of materials. A deeper understanding of the microscopic origin of
magnetic interactions can open new pathways toward realizing nanometer-scale
systems for future spintronic devices. Here, we explore a method for explicitly
calculating interatomic exchange interactions in non-collinear configurations
of magnetic materials considering only a bilinear spin Hamiltonian in a local
scenario. Based on density-functional theory (DFT) calculations of dimers
adsorbed on metallic surfaces, and with a focus on the Dzyaloshinskii-Moriya
interaction (DMI) which is essential for stabilizing chiral non-collinear
magnetic states, we discuss the interpretation of the DMI when decomposed into
microscopic electron and spin densities and currents. We clarify the distinct
origins 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 is
solely induced by spin-orbit coupling, and DMI-like interactions brought about
by non-collinearity. We explain how the dependence of the DMI on the magnetic
configuration establishes a connection between high-order magnetic
interactions, enabling the transition from a local to a global spin
Hamiltonian.
