Role of the Dzyaloshinskii-Moriya interaction in multiferroic perovskites

With the perovskite multiferroic $R\mathrm{Mn}{\mathrm{O}}_{3}$ $(R=\mathrm{Gd},\mathrm{Tb},\mathrm{Dy})$ as guidance, we argue that the Dzyaloshinskii-Moriya interaction (DMI) provides the microscopic mechanism for the coexistence and strong coupling between ferroelectricity and incommensurate magnetism. We use Monte Carlo simulations and zero-temperature exact calculations to study a model incorporating the double-exchange, superexchange, Jahn-Teller, and DMI terms. The phase diagram contains a multiferroic phase between $A$ and $E$ antiferromagnetic phases, in excellent agreement with experiments.