We theoretically study origins of the ferroelectricity in the multiferroic phases of the rare-earth ($R$) Mn perovskites, $R{\mathrm{MnO}}_3$, by constructing a realistic spin model including the spin-phonon coupling, which reproduces the entire experimental phase diagram in the plane of temperature and Mn-O-Mn bond angle for the first time. Surprisingly we reveal a significant contribution of the symmetric ($\mathit{S}·\mathit{S}$)-type magnetostriction to the ferroelectricity even in a spin-spiral-based multiferroic phase, which can be larger than the usually expected antisymmetric ($\mathit{S}\times \mathit{S}$)-type contribution. This explains well the nontrivial behavior of the electric polarization. We also predict the noncollinear deformation of the $E$-type spin structure and a wide coexisting regime of the $E$ and spiral states, which resolve several experimental puzzles.

• Received 21 February 2010

DOI:https://doi.org/10.1103/PhysRevLett.105.037205