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We derive a sum rule to demonstrate that the static magnetoelectric (ME)effect is governed by optical transitions that are simultaneously excited viathe electric and magnetic components of light. By a systematic analysis ofmagnetic point groups, we show that the ME sum rule is applicable to a broadvariety of non-centrosymmetric magnets including ME multiferroic compounds. Dueto the dynamical ME effect, the optical excitations in these materials canexhibit directional dichroism, i.e. the absorption coefficient can be differentfor counter-propagating light beams. According to the ME sum rule, themagnitude of the linear ME effect of a material is mainly determined by thedirectional dichroism of its low-energy optical excitations. Application of thesum rule to the multiferroic Ba$_2$CoGe$_2$O$_7$, Sr$_2$CoSi$_2$O$_7$ andCa$_2$CoSi$_2$O$_7$ shows that in these compounds the static ME effect ismostly governed by the directional dichroism of the spin-wave excitations inthe GHz-THz spectral range. On this basis, we argue that the studies ofdirectional dichroism and the application of ME sum rule can promote thesynthesis of new materials with large static ME effect.

Simultaneously Magnetic- and Electric-dipole Active Spin Excitations Govern the Static Magnetoelectric Effect in Multiferroic Materials