  Excitonic insulating system is studied from the viewpoints of the orbital
physics in strongly correlated electron systems. An effective model Hamiltonian
for low-energy electronic states is derived from the two-orbital Hubbard model
with a finite energy difference corresponding to the crystalline field
splitting. The effective model is represented by the spin operators and the
pseudo-spin operators for the spin-state degrees of freedom. The ground state
phase diagram is analyzed by the mean-field approximation. In addition to the
low-spin state and high-spin state phases, two kinds of the excitonic
insulating phases emerge as a consequence of the competition between the
crystalline field effect and the Hund coupling. The excitonic transition is
classified to be an Ising-like transition reflecting a spontaneous breaking of
the $Z_2$ symmetry. Magnetic structures in the two excitonic insulating phases
are different from each other; an antiferromagnetic order and a spin nematic
order. Collective excitations in each phase are examined by using the
generalized spin-wave method. The Goldstone modes in the excitonic insulating
phases appear in the dynamical correlation functions for the spins and
pseudo-spin operators. Both the transverse and longitudinal spin excitation
modes are active in the two excitonic insulating phases in contrast to the
low-spin state and high-spin state phases. Connections of the present results
to the perovskite cobalt oxides are discussed.
