We review the BCS (Bardeen–Cooper–Schrieffer)–BEC (Bose–Einstein condensation) crossover phenomenon discussed in an ultracold Fermi atomic gas and a neutron superfluid in the low-density crust regime of a neutron star. A purpose of this paper is to show that these two very different atomic and nuclear systems can be closely related to each other from the viewpoint of this quantum many-body phenomenon. We explain how the BCS–BEC crossover is realized in the former atomic system by using the novel pairing mechanism called Feshbach resonance. We present a simple explanation for this crossover phenomenon to grasp the essence, as well as detailed microscopic theories that can cover the entire BCS–BEC crossover region. In the latter, we point out that the ordinary BCS theory already has the ability to describe the BCS–BEC crossover at T=0. At finite temperatures T>0, however, we need to go beyond this mean-field theory. Besides general aspects of the BCS–BEC crossover phenomenon, we also pick up special topics peculiar to each atomic gas and neutron fluid. The first one is the pseudogap phenomenon in the normal state of a Fermi atomic gas. The second one is the problem of non-zero effective range in an s-wave neutron superfluid.
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