Quantum field theory of dilute homogeneous Bose-Fermi-mixtures at zero   temperature: general formalismand 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 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 $a_{\scriptsize BF}k_{\scriptsize F}$. The total ground-state energy-density reads $E/V = \epsilon_{\scriptsize F} + \epsilon_{\scriptsize B} + (2\pi\hbar^{2}a_{\rm BF}n_{\scriptsize B}n_{\scriptsize F}/m) [1 + a_{\scriptsize BF}k_{\scriptsize F}f(\delta)/\pi]$. 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 square brackets is the second-order correction, where $f(\delta)$ is a known function of $\delta= (m_{\scriptsize B} - m_{\scriptsize F})/(m_{\scriptsize B} + m_{\scriptsize F})$. We discuss the relevance of this new term, how it can be incorporated into existing theories of boson-fermion mixtures, and its importance in various parameter regimes, in particular considering mixtures of $^{6}$Li and $^{7}$Li and of $^{3}$He and $^{4}$He. 
Alexander P. Albus, Simon A. Gardiner, Fabrizio Illuminati, Martin   Wilkens