Dzyaloshinskii-Moriya interaction in absence of spin-orbit coupling

In contrast to conventional assumptions, we show that the Dzyaloshinskii-Moriya interaction can be of non-relativistic origin, in particular in materials with a non-collinear magnetic configuration, where non-relativistic contributions can dominate over spin-orbit effects. The weak antiferromagnetic phase of Mn$_{3}$Sn is used to illustrate these findings. Using electronic structure theory as a conceptual platform, all relevant exchange interactions are derived for a general, non-collinear magnetic state. It is demonstrated that non-collinearity influences all three types of exchange interaction and that physically distinct mechanisms, which connect to electron- and spin-density and currents, may be used as a general way to analyze and understand magnetic interactions of the solid state.