  We analyze a tight-binding model of ultracold fermions loaded in an optical
square lattice and subjected to a synthetic non-Abelian gauge potential
featuring both a magnetic field and a translationally invariant SU(2) term. We
consider in particular the effect of broken time-reversal symmetry and its role
in driving non-trivial topological phase transitions. By varying the spin-orbit
coupling parameters, we find both a semimetal/insulator phase transition and a
topological phase transition between insulating phases with different numbers
of edge states. The spin is not a conserved quantity of the system and the
topological phase transitions can be detected by analyzing its polarization in
time of flight images, providing a clear diagnostic for the characterization of
the topological phases through the partial entanglement between spin and
lattice degrees of freedom.
