  Effects of the spin-orbit coupling (SOC) and magnetic field on excitonic
insulating (EI) states are investigated. We introduce the two-orbital Hubbard
model with the crystalline field splitting, which is a minimal model for
discussing the exciton condensation in strongly correlated electron systems,
and analyze its effective Hamiltonian in the strong correlation limit by using
the mean-field theory. In the absence of the SOC and magnetic field, the ground
state changes from the nonmagnetic band-insulating state to the EI state by
increasing the Hund coupling. In an applied magnetic field, the magnetic moment
appears in the EI state, which is continuously connected to the forced
ferromagnetic state. On the other hand, in the presence of the SOC, they are
separated by a phase boundary. We find that the magnetic susceptibility is
strongly enhanced in the EI phase near the boundary with a small SOC. This
peculiar behavior is attributed to the low-energy fluctuation of the spin
nematicity inherent in the high-spin local state stabilized by the Hund
coupling. The present study not only reveals the impact of the SOC for the EI
state but also sheds light on the role of quantum fluctuations of the spin
nematicity for the EI state.
