  Effects of spontaneous parity breaking by charge, spin, and orbital orders
are investigated in a two-band Hubbard model on a honeycomb lattice. This is a
minimal model in which the inter-orbital hopping, atomic spin-orbit coupling,
and strong electron correlation give rise to fascinating properties, such as
the magnetoelectric effects, quantum spin Hall effect, and spin or valley
splitting in the band structure. We perform the symmetry analysis of possible
broken-parity states and the mean-field analysis of their competition. We find
that the model at 1/4 filling exhibits a spin-orbital composite ordered state
and a charge ordered state, in addition to a paramagnetic quantum spin-Hall
insulator. We show that the composite ordered phase exhibits two types of
magnetoelectric responses. The charge ordered state shows spin splitting in the
band structure, while the topological nature varies depending on electron
correlations.
