arXiv:cond-mat/0207260 · Physical Review A

Ground-state properties of a dilute Bose-Fermi mixture

Luciano Viverit, S. Giorgini

Published Dec 19, 2002• 92 citations

Abstract

We investigate the properties of a dilute boson-fermion mixture at zero temperature. The ground-state energy of the system is calculated to second order in the Bose-Fermi coupling constant. The Green's function formalism is applied to obtain results for the effect of Bose-Fermi interactions on the spectrum of phonon excitations, the momentum distribution, the condensate fraction, and the superfluid density of the bosonic component. A quantitative discussion of these effects is presented in the case of mixtures with attractive interspecies interaction.

DOI: 10.1103/physreva.66.063604

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arXiv:cond-mat/0207260 · Physical Review A

Ground-state properties of a dilute Bose-Fermi mixture

Luciano Viverit, S. Giorgini

Published Dec 19, 2002• 92 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreva.66.063604

Related papers

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

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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}}.

{m}_{b}∕{m}_{f}

and the number ratio

{N}_{b}∕{N}_{f}

. We find that the attractive contribution to the energy is greatly enhanced for small values of the mass ratio. We study the role of Bose-Fermi pair correlations in the mixture by calculating the pole of the boson-fermion scattering amplitude in the medium. The pole shows a standard quasiparticle dispersion for a Bose-Fermi pair. In addition, we also study the fermion dispersion relation. We find two dispersion branches with the possibility of avoided crossings. This strongly depends on the number ratio

{N}_{b}∕{N}_{f}

E(\mathrm{k})

of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap.The Green's function method is applied to the calculation of the properties of a low-density boson gas. The next order term beyond that calculated by Lee and Yang, and Beliaev for the ground-state energy is obtained and the general form of the series expansion is found to be $(\frac{{E}_{0}}{\ensuremath{\Omega}})=\frac{1}{2}{n}^{2}{f}_{0}[1+a{(n{{f}_{0}}^{3})}^{\frac{1}{2}}+b(n{{f}_{0}}^{3})\mathrm{ln}n{{f}_{0}}^{3}+c(n{{f}_{0}}^{3})+d{(n{{f}_{0}}^{3})}^{\frac{3}{2}}\mathrm{ln}(n{{f}_{0}}^{3})+\ensuremath{\cdots}],$ where

n

is the density and

{f}_{0}

is the scattering length for the assumed two-body interaction between the bosons. The coefficients

a

and

b

are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient

b

is in agreement with the hard-sphere gas calculations of Wu and of Sawada.A discussion is given of the intermediate-density calculation of Brueckner and Sawada, and certain possible improvements in the method of summing a selected set of higher-order terms are proposed.

l

channels. It has been found that the onset of the superconducting phase can occur at temperatures

(>100\mathrm{nK})

of the same order of magnitude as the Fermi temperature $(\ensuremath{\sim}300\mathrm{nK}),$ and the energy gap in the excitation spectrum is a small fraction $(\ensuremath{\sim}1%)$ of the Fermi energy.