Quantum field theory of dilute homogeneous Bose-Fermi-mixtures at zero temperature: general formalismand beyond mean-field corrections

Abstract

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 $a_{BF}{k}_F$. The total ground-state energy-density reads $E/V={\epsilon }_F+{\epsilon }_B+(2\pi {\hslash }^2{a}_{\mathrm{BF}}{n}_B{n}_F/m)[1+a_{BF}{k}_{F}f(\delta )/\pi ]$. 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 $\delta =(m_B-m_F)/(m_B+m_F)$. 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.

submitted to Phys. Rev. A

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