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We explore the evolution of a ultracold quantum gas of interacting fermionscrossing from a Bardeen-Cooper-Schrieffer (BCS) superfluidity to aBose-Einstein condensation (BEC) of molecular bosons in the presence of atunable-range interaction among the fermions and of an artificial magneticfield, which can be used to simulate a pseudo-spin-orbit coupling (SOC) and toproduce topological states. We find that the crossover is affected by acompetition between the finite range of the interaction and the SOC and thatthe threshold $\lambda_B$ for the topological transition is affected by theinteractions only in the small pair size, BEC-like, regime. Below $\lambda_B$,we find persistence of universal behavior in the critical temperature, chemicalpotential, and condensate fraction, provided that the pair correlation lengthis used as a driving parameter. Above threshold, universality is lost in theregime of large pair sizes. Here, the limiting ground state departs from aweakly-interacting BCS-like, so that a different description is required. Ourresults can be relevant in view of current experiments with cold atoms inoptical cavities, where tunable-range effective atomic interactions can beengineered.

Interaction-Range Effects and Universality in the BCS-BEC Crossover of Spin-Orbit Coupled Fermi Gases