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We consider a two-component Fermi gas with attractive interactions on asquare optical lattice, and study the interplay of Zeeman field, spin-orbitcoupling and next-nearest-neighbor hopping on the ground-state phase diagramsin the entire BCS-BEC evolution. In particular, we first classify anddistinguish all possible superfluid phases by the momentum-space topology oftheir zero-energy quasiparticle/quasihole excitations, and then numericallyestablish a plethora of quantum phase transitions in between. These transitionsare further signalled and evidenced by the changes in the correspondingtopological invariant of the system, \textit{i.e.}, its Chern number. Lastly,we find that the superfluid phase exhibits a reentrant structure, separated bya fingering normal phase, the origin of which is traced back to the changes inthe single-particle density of states.

Topological superfluids on a square optical lattice with non-Abelian gauge fields: Effects of next-nearest-neighbor hopping in the BCS-BEC evolution