Fermion pairing in Bose-Fermi mixtures

An effective interaction between fermions in a Bose-Fermi mixture is derived. It is induced by density fluctuations of the bosonic background. The contributions from states containing both one and two virtual phonons are taken into account self-consistently. The time dependence of the effective interaction has been removed by assuming that the velocity of the fermions at the Fermi surface is much larger than the sound velocity in the Bose gas. This assumption is more appropriate for the actual experimental situations than the usual approximation of neglecting retardation effects. The effective interaction turns out to be attractive and, as a consequence, can give rise to a superconducting phase in the Fermi component of the mixture. The fermions are considered in only one magnetic state, so that pairing can be effective only in odd-$l$ channels. It has been found that the onset of the superconducting phase can occur at temperatures $(>\mathrm{}100\mathrm{}\mathrm{nK})$ of the same order of magnitude as the Fermi temperature $\left(\sim 300\mathrm{nK}\right),$ and the energy gap in the excitation spectrum is a small fraction $(\sim 1\%)$ of the Fermi energy.