Dzyaloshinskii-Moriya interaction in strongly spin-orbit-coupled systems: General formula and application to topological and Rashba materials

We theoretically study the Dzyaloshinskii-Moriya interaction (DMI) mediated by band electrons with strong spin-orbit coupling (SOC). We first derive a general formula for the coefficient ${\bm D}_i$ of the DMI in free energy in terms of Green's functions, and examine its variations in relation to physical quantities. In general, the DMI coefficient can vary depending on physical quantities, i.e., whether one is looking at equilibrium spin structure (${\bm D}_i$) or spin-wave dispersion (${\bm D}_i^{(2)}$), and the obtained formula helps to elucidate their relations. By explicit evaluations for a magnetic topological insulator and a Rashba ferromagnet with perpendicular magnetization, we observe ${\bm D}_i^{(2)} \ne {\bm D}_i$ in general. In the latter model, or more generally, when the magnetization and the spin-orbit field are mutually orthogonal, ${\bm D}_i$ is exactly related to the equilibrium spin current for arbitrary strength of SOC, generalizing the similar relation for systems with weak SOC. Among various systems with strong SOC, magnetic Weyl semimetals are special in that ${\bm D}_i^{(2)} = {\bm D}_i$, and in fact, the DMI in this system arises as the chiral anomaly.