  Recent advanced experimental implementations of optical lattices with highly
tunable geometry open up new regimes for quantum many-body states of matter
that previously had not been accessible. Here we introduce a symmetry-based
method of utilizing the geometry of optical lattice to systematically control
topologically non-trivial orbital hybridization. Such an orbital mixing leads
to an unexpected and yet robust topological semimetal at single-particle level
for a gas of fermionic atoms. When considering s-wave attractive interaction
between atoms as for instance tuned by Feshbach resonance, topological
superfluid state with high Chern number is unveiled in the presence of on-site
rotation. This state supports chiral edge excitations, manifesting its
topological nature. An experimental realization scheme is designed, which
introduces a systematic way of achieving a new universality class (such as
Chern number of 2) of orbital-hybridized topological phases beyond
geometrically standard optical lattices.
