  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.
