Background:symmetry argument Mostovoy,PRL 96,067601(06) If spin order is spatially inhomogeneous, symmetry allows for the 3rd-order coupling POM and then P may appear. Φem(P,M ）= yP.[M(V·M)-(M·7)M+··], D=Φem+P2/2mE →P∝[(M·a)M-M(a.M)] Background: symmetry argument If spin order is spatially inhomogeneous, symmetry allows for the 3rd-order coupling P∂M and then P may appear. =em+P2/2 Mostovoy, PRL 96, 067601 (06)