Casimir interaction among heavy fermions in the BCS-BEC crossover

Abstract

We investigate a two-species Fermi gas with a large mass ratio interacting by an interspecies short-range interaction. Using the Born-Oppenheimer approximation, we determine the interaction energy of two heavy fermions immersed in the Fermi sea of light fermions as a function of the $s$-wave scattering length. In the BCS limit, we recover the perturbative calculation of the effective interaction between heavy fermions. The $p$-wave projection of the effective interaction is attractive in the BCS limit while it turns out to be repulsive near the unitarity limit. We find that the $p$-wave attraction reaches its maximum between the BCS and unitarity limits, where the maximal $p$-wave pairing of heavy minority fermions is expected. We also investigate the case where the heavy fermions are confined in two dimensions and the $p$-wave attraction between them is found to be stronger than that in three dimensions.