  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.
