BEC-polaron gas in a boson-fermion mixture: A many-body extension of Lee-Low-Pines theory

We investigate the ground state properties of the gaseous mixture of a single species of bosons and fermions at zero temperature, where bosons are major in population over fermions, and form the Bose-Einstein condensate (BEC). The boson-boson and boson-fermion interactions are assumed to be weakly repulsive and attractive, respectively, while the fermion-fermion interaction is absent due to the Pauli exclusion for the low energy $s$-wave scattering. We treat fermions as a gas of polarons dressed with Bogoliubov phonons, which is an elementary excitation of the BEC, and evaluate the ground state properties with the method developed by Lemmens, Devreese, and Brosens (LDB) originally for the electron polaron gas, and also with a general extension of the Lee-Low-Pines theory for many-body systems (eLLP), which incorporates the phonon drag effects as in the original LLP theory. The formulation of eLLP is developed and discussed in the present paper. The binding (interaction) energy of the polaron gas is calculated in these methods and shown to be finite (negative) for the dilute gas of heavy fermions with attractive boson-fermion interactions, though the suppression by the many-body effects exists.