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 a new relevant object, 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 aBFkF. The total ground-state energy-density reads E/V = ǫF + ǫB + (2πh̄ aBFnBnF/m)[1 + aBFkFf(δ)/π]. The first term is the kinetic energy of the free fermions, the second term is the boson-boson mean-field interaction, the pre-factor to the additional term is the usual mean-field contribution to the boson-fermion interaction energy, and the second term in the 1 square brackets is the second-order correction, where f(δ) is a known function of δ = (mB − mF)/(mB + mF). 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 beyond mean-field correction for various parameter regimes, in particular considering mixtures of 6Li and 7Li and of 3He and 4He. Moreover we determine the modification of the induced fermion-fermion interaction due to the beyond mean-field effects. We show that there is no effect on the depletion of the Bose condensate to first order in the boson-fermion scattering length aBF. PACS numbers: 03.75.Fi, 03.70.+k , 01.55.+b Typeset using REVTEX 2