Electric field effect on spin waves and magnetization dynamics: role of magnetic moment current

We show that a static electric field $E_x$ gives rise to a shift of the spin wave dispersion relation $\omega(q_y-q_E)$ in the direction of the wavenumber $q_y$ of the quantity $q_E=-\gamma_LE_x/c^2$. This effect is caused by the magnetic moment current carried by the spin wave itself that generates an additional phase proportional to the electric field, as in the Aharonov-Casher effect. This effect is independent from the possibly present magneto-electric effects of insulating ferromagnets and superimposes to them. By extending this picture to arbitrary magnetization dynamics, we find that the electric field gives rise to a dynamic interaction term which has the same chiral from of the Dzyaloshinskii-Moriya interaction but is fully tunable with the applied electric field.