A Microscopic Model of Multiferroics RMn$_2$O$_5$

A microscopic model is developed to explain the phase diagram and the mechanism of magnetoelectric coupling in RMn$_2$O$_5$. We show that frustrated magnetic structure drives the system to a commensurate-incommensurate phase trasition which can be understood as a competition between a collinear order, which stems from the `order by disorder' mechanism, and a chiral symmetry order. The magnetoelectric interaction couples the collinear order to the electric polarization. The low energy excitation and the effect of external magnetic field are also analyzed.