Huge spin-driven polarizations at room temperature in bulk BiFeO$_{3}$*

Although BiFeO$_{3}$ is one of the most investigated multiferroics, its magnetoelectricity and spin-driven polarizations are barely understood on an atomistic level. By combining a first-principles approach with a spin-cycloid model, we report hidden but huge spin-driven polarizations at room temperature in bulk BiFeO$_{3}$. One of the polarizations reaches $\sim$ 0.03 C/m$^{2}$, which is larger than any other spin-driven polarization in a bulk material by one order of magnitude. By comparing our results with intrinsic measurements such as neutron scattering, Raman spectroscopy, IR directional dichroism, and high magnetic-field measurements, we disentangle all the hidden spin-driven polarizations due to exchange-striction, spin-current, and single-ion-anisotropy. We find that the broken inversion symmetries of the $R$3c structure of BiFeO$_{3}$ induce the strong response of the magnetic couplings to an electric field and are responsible for the associated huge spin-driven polarizations.