  Complex lattices provide a versatile ground for fascinating quantum many-body
physics. Here, we propose an exotic mechanics for generating orbital
frustration in hexagonal lattices. We study two-component (pseudospin-$1/2$)
Bose gases in $p$-orbital bands of two-dimensional hexagonal lattices, and find
that the system exhibits previously untouched orbital frustration as a result
of the interplay of spin and orbital degrees of freedom, in contrast to normal
Ising-type orbital ordering of spinless $p$-orbital band bosons in
two-dimensional hexagonal lattices. Based on the classification by symmetry
analysis, we find the interplay of orbital frustration and strong interaction
leads to exotic Mott and superfluid phases with spin-orbital intertwined
orders, in spite of the complete absence of spin-orbital interaction in the
Hamiltonian. Our study implies many-body correlations in a multi-orbital
setting could induce rich spin-orbital intertwined physics in complex lattice
structures.
