Symmetry of Multiferroicity in a Frustrated Magnet TbMn$_2$O$_5$

Based on measurements of soft x-ray magnetic scattering and symmetry considerations, we demonstrate that the magnetoelectric effect in TbMn$_2$O$_5$ arises from an internal field determined by $\vec{S}_{\vec{q}} \times {\vec{S}_{-\vec{q}}}$ with $\vec{S}_{\vec{q}}$ being the magnetization at modulation vector $\vec{q}$, whereas the magneto-elastic effect in the exchange energy governs the response to external electric fields. Our results set fundamental symmetry constraints on the microscopic mechanism of multiferroicity in frustrated magnets.