Figure 1:
(a) The transverse spiral Mn magnetic order found in many multiferroic perovskite manganites. Here the Mn spins rotate around an axis defined by the cross product S_{i}×S_{i+1}, while the magnetic propagation Q is parallel to the vector joining these two spins r_{i,i+1}. The direction of the ferroelectric polarization is given by the cross product P=(S_{i}×S_{i+1})×Q. In the same figure the O atoms (red circles) are coherently displaced from their paraelectric position by a distance d (depicted as white circles) and are related to the Dzyaloshinskii vector by D_{i,i+1}∼λd×r_{i,i+1} , where λ is the spin-orbit coupling constant. (Adapted from [2].) (b) A depiction of a weak ferromagnetism generated from a DM interaction from a small canting of antiferromagnetic spins that are stacked along the c axis. For clarity only one column of spins along the c axis is shown. In this arrangement the Dzyaloshinskii vector changes sign between pairs of spins.