Generalized Crossover in Interacting Fermions within the Low-Energy Expansion

We generalize the Bardeen-Cooper-Schrieffer-Bose-Einstein-condensation (BCS-BEC) crossover of two-component fermions, which is realized by tuning the $s$-wave scattering length $a$ between the fermions, to the case of an arbitrary effective range $r_{\mathrm{e}}$. By using the Nozi`{e}res-Schmitt-Rink (NSR) approach, we show another crossover by changing $r_{\mathrm{e}}$ and present several similarities and differences between these two crossovers. Furthermore, the region ($r_{\mathrm{e}}>a\mathrm{/}2$) where the effective range expansion breaks down and the Hamiltonian becomes non-Hermitian is found, being consistent with the Wigner’s causality bound. Our results are universal for low-density interacting fermions with low-energy constants $a$ and $r_{\mathrm{e}}$ and are directly relevant to ultracold Fermi atomic gases as well as dilute neutron matter.