  We study the topology and geometry of a fermionic model on the honeycomb
lattice with spin-dependent hopping which breaks the time-reversal and
charge-conjugation symmetries but preserves their composition. We show that in
such a case the Zak phases are topological invariants that characterize the
semi-metallic state at half-filling and determine the edge state structure. As
the strength of the spin-dependent hopping varies, the model shows several
Lifshitz transitions corresponding to creation and merging of multiple Dirac
points. We discuss the possible realization of this model in cold atom systems
and propose experimental signals of detecting the Dirac points and
Pancharathnam-Berry curvature of the bands.
