(a) The transverse spiral magnetic order found in many multiferroic perovskite manganites. Here the spins rotate around an axis defined by the cross product , while the magnetic propagation is parallel to the vector joining these two spins . The direction of the ferroelectric polarization is given by the cross product . In the same figure the atoms (red circles) are coherently displaced from their paraelectric position by a distance (depicted as white circles) and are related to the Dzyaloshinskii vector by , where is the spin-orbit coupling constant. (Adapted from .) (b) A depiction of a weak ferromagnetism generated from a DM interaction from a small canting of antiferromagnetic spins that are stacked along the axis. For clarity only one column of spins along the axis is shown. In this arrangement the Dzyaloshinskii vector changes sign between pairs of spins.