  Orbital degree of freedom plays an important role for understanding quantum
many-body phenomena. In this work, we study an experimentally related setup
with ultracold bosons loaded into hybridized bands of two-dimensional hexagonal
optical lattices. We find that the system supports various quantum many-body
phases at zero temperature, including chiral superfluid and chiral Mott
insulator by breaking time-reversal symmetry, and time-reversal-even insulating
phase, based on dynamical mean-field theory. To explain the time-reversal-even
phase, a fourth-order orbital-exchange model is derived to explain the
underlying mechanics. To relate to experimental situations, we make
band-structure calculations to obtain the Hubbard parameters, and show that
these orbital ordering phases persist also in the presence of
next-nearest-neighbor hopping.
