  Topological phases like topological insulators or superconductors are
fascinating quantum states of matter, featuring novel properties such as
emergent chiral edge states or Majorana fermions with non-Abelian braiding
statistics. The recent experimental implementation of optical lattices with
highly tunable geometry in cold gases opens up a new thrust on exploring these
novel quantum states. Here we report that the topological non-trivial Bloch
bands can arise naturally in a noncentrosymmetric lattice. It induces a
controllable orbital hybridization, producing the topological band structure.
In such bands, when considering attractive fermionic atoms, we find a
topological Fulde-Ferrell superfluid state with finite center-of-mass momentum
in the presence of onsite rotation. This topological superfluid supports
Majorana fermions on its edges. Experimental signatures are predicted for cold
gases in radio-frequency spectroscopy.
