  Orthorhombically distorted perovskite manganites, RMnO3 with R being a
trivalent rare-earth ion, exhibit a variety of magnetic and electric phases
including multiferroic (i.e. concurrently magnetic and ferroelectric) phases
and fascinating magnetoelectric phenomena. We theoretically study the phase
diagram of RMnO3 by constructing a microscopic spin model, which includes not
only the superexchange interaction but also the single-ion anisotropy (SIA) and
the Dzyaloshinsky-Moriya interaction (DMI). Analysis of this model using the
Monte-Carlo method reproduces the experimental phase diagrams as functions of
the R-ion radius, which contain two different multiferroic states, i.e. the
ab-plane spin cycloid with ferroelectric polarization P//a and the bc-plane
spin cycloid with P//c. The orthorhombic lattice distortion or the
second-neighbor spin exchanges enhanced by this distortion exquisitely controls
the keen competition between these two phases through tuning the SIA and DMI
energies. This leads to a lattice-distortion-induced reorientation of P from a
to c in agreement with the experiments. We also discuss spin structures in the
A-type antiferromagnetic state, those in the cycloidal spin states, origin and
nature of the sinusoidal collinear spin state, and many other issues.
