  The main properties of realistic models for manganites are studied using
analytic mean-field approximations and computational numerical methods,
focusing on the two-orbital model with electrons interacting through
Jahn-Teller (JT) phonons and/or Coulombic repulsions. Analyzing the model
including both interactions by the combination of the mean-field approximation
and the exact diagonalization method, it is argued that the spin-charge-orbital
structure in the insulating phase of the purely JT-phononic model is not
qualitatively changed by the inclusion of the Coulomb interactions. As an
important application of the present mean-field approximation, the CE-type AFM
state, the charge-stacked structure along the z-axis, and the
(3x^2-r^2)/(3y^2-r^2)-type orbital ordering are successfully reproduced based
on the JT-phononic model for the half-doped manganite. Topological arguments
and the relevance of the Heisenberg exchange among localized t_{\rm 2g} spins
explains why the nearest-neighbor Coulomb interaction does not destroy the
charge stacking structure. It is also verified that the phase-separation
tendency is observed both in purely JT-phononic and purely Coulombic models in
the vicinity of the hole undoped region, as long as realistic hopping matrices
are used. This highlights the qualitative similarities of both approaches, and
the relevance of mixed-phase tendencies in the context of manganites. In
addition, the rich and complex phase diagram of the two-orbital Coulombic model
in one dimension is presented. Our results provide robust evidence that
Coulombic and JT-phononic approaches to manganites are not qualitatively
different ways to carry out theoretical calculations, but they share a variety
of common features.
