Quantum field theory of dilute homogeneous Bose-Fermi mixtures at zero temperature: General formalism and beyond mean-field corrections

We consider a dilute homogeneous mixture of bosons and spin-polarized fermions at zero temperature. We first construct the formal scheme for carrying out systematic perturbation theory in terms of single particle Green’s functions. We especially focus on the description of the boson-fermion interaction. To do so we need to introduce the renormalized boson-fermion $T$ matrix, which we determine to second order in the boson-fermion $s$-wave scattering length. We also discuss how to incorporate the usual boson-boson $T$ matrix in mean field approximation to obtain the total ground-state properties of the system. The next-order term beyond mean field stems from the boson-fermion interaction and is proportional to $a_{\mathrm{BF}}{k}_{\mathrm{F}}.$ The total ground-state energy density to this order is the sum of the kinetic energy of the free fermions, the boson-boson mean-field interaction, the usual mean-field contribution to the boson-fermion interaction energy, and the first boson-fermion correction beyond mean field. We also compute the bosonic and the fermionic chemical potentials, the compressibilities, and the modification to the induced fermion-fermion interaction. We discuss the behavior of the total ground-state energy and the importance of the correction beyond mean field for various parameter regimes, in particular considering mixtures of ${}^6\mathrm{Li}$ and ${}^7\mathrm{Li}$ and of ${}^3\mathrm{He}$ and ${}^4\mathrm{He}.$ Moreover, we determine the modification of the induced fermion-fermion interaction due to the effects beyond mean field. We show that there is no effect on the depletion of the Bose condensate to first order in the boson-fermion scattering length $a_{\mathrm{BF}}.$