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arXiv:cond-mat/0305609 · Physical Review A

Fermion pairing in Bose-Fermi mixtures

F. Matera

Published Oct 24, 2003• 37 citations
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Abstract

An effective interaction between fermions in a Bose-Fermi mixture is derived. It is induced by density fluctuations of the bosonic background. The contributions from states containing both one and two virtual phonons are taken into account self-consistently. The time dependence of the effective interaction has been removed by assuming that the velocity of the fermions at the Fermi surface is much larger than the sound velocity in the Bose gas. This assumption is more appropriate for the actual experimental situations than the usual approximation of neglecting retardation effects. The effective interaction turns out to be attractive and, as a consequence, can give rise to a superconducting phase in the Fermi component of the mixture. The fermions are considered in only one magnetic state, so that pairing can be effective only in odd-$l$ 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.

DOI
10.1103/physreva.68.043624

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An effective interaction between fermions in a Bose-Fermi mixture is derived. It is induced by density fluctuations of the bosonic background. The contributions from states containing both one and two virtual phonons are taken into account self-consistently. The time dependence of the effective interaction has been removed by assuming that the velocity of the fermions at the Fermi surface is much larger than the sound velocity in the Bose gas. This assumption is more appropriate for the actual experimental situations than the usual approximation of neglecting retardation effects. The effective interaction turns out to be attractive and, as a consequence, can give rise to a superconducting phase in the Fermi component of the mixture. The fermions are considered in only one magnetic state, so that pairing can be effective only in odd-lll channels. It has been found that the onset of the superconducting phase can occur at temperatures (>100nK)(>100\mathrm{nK})(>100nK) of the same order of magnitude as the Fermi temperature (\ensuremath∼300nK),(\ensuremath{\sim}300\mathrm{nK}),(\ensuremath∼300nK), and the energy gap in the excitation spectrum is a small fraction (\ensuremath∼1(\ensuremath{\sim}1%)(\ensuremath∼1 of the Fermi energy.
We develop a strong-coupling theory for the superfluidity of fermion pairing phase in a Bose-Fermi mixture. Dynamical screening, self-energy renormalization, and a pairing gap function are included self-consistently within the adiabatic limit (i.e., the phonon velocity is much smaller than the Fermi velocity). An analytical solution for the transition temperature (T(c)) is derived within reasonable approximations. Using typical parameters of a 40K-87Rb mixture, we find that the calculated T(c) is several times larger than that obtained in the weak coupling theory, and can be up to several percent of the Fermi temperature.
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.
The spectral properties of a mixture of bosons hybridized with fermion pairs are studied in the superconducting state. Such a model exhibits two limiting behaviors. One is reminiscent of a BCS system, provided that the total concentration n of particles is below a certain critical concentration nc{\mathit{n}}_{\mathit{c}}nc​, in which case bosons exist only in thermally excited states. The other limit for n\ensuremath{\geqslant}nc{\mathit{n}}_{\mathit{c}}nc​ describes a situation largely controlled by the condensation of the bosons. The variation of the single-boson as well as collective excitations as a function of n and for the entire temperature regime from T=0 K to Tc{\mathit{T}}_{\mathit{c}}Tc​ will be discussed, both with and without long-range Coulomb interaction between the charge carriers. The method for calculation applied here is based on the dielectric formalism and the use of Ward identities, currently used in the theory of the interacting Bose gas, which allows us to derive the density and current correlation functions consistent with local charge-current conservation laws. In the absence of a Coulomb interaction the behavior of the system is characterized by the existence of a sound-wave-like branch in the Bose single-particle and collective excitation spectrum. In the BCS regime the sound velocity is proportional to the Fermi velocity while in the Bose regime it is proportional to the strength of the boson-fermion hybridization and decreases with the superfluid density as we approach Tc{\mathit{T}}_{\mathit{c}}Tc​. In the presence of a long-range Coulomb interaction and in the Bose regime one observes, besides the usual optical plasmon mode at high frequency and corresponding to in-phase density oscillation of the two subsytems, an out-of-phase collective mode in the two particle continuum just above the gap. Moreover, upon approaching Tc{\mathit{T}}_{\mathit{c}}Tc​ a relatively well-defined quasiparticle feature starts to emerge within the gap and is seen in both the Bose single-particle excitation and the density oscillation spectrum. In the BCS regime, due to the very small number of occupied bosonic states, the low-energy quasiparticle features are absent. \textcopyright{} 1996 The American Physical Society.
We investigate the quantum phases of mixed-dimensional cold atom mixtures. In particular, we consider a mixture of a Fermi gas in a two-dimensional lattice, interacting with a bulk Fermi gas or a Bose-Einstein condensate in a three-dimensional lattice. The effective interaction of the two-dimensional system mediated by the bulk system is determined. We perform a functional renormalization group analysis, and demonstrate that by tuning the properties of the bulk system, a subtle competition of several superconducting orders can be controlled among $s\ensuremath{-}\mathrm{wave}, p\ensuremath{-}\mathrm{wave}, {d}_{{x}^{2}\ensuremath{-}{y}^{2}}\ensuremath{-}\mathrm{wave}$, and ${g}_{xy({x}^{2}\ensuremath{-}{y}^{2})}\ensuremath{-}\mathrm{wave}$ pairing symmetries. Other instabilities such as a charge-density-wave order are also demonstrated to occur. In particular, we find that the critical temperature of the $d\ensuremath{-}\mathrm{wave}$ pairing induced by the next-nearest-neighbor interactions can be an order of magnitude larger than that of the same pairing induced by doping in the simple Hubbard model. We expect that by combining the nearest-neighbor interaction with the next-nearest-neighbor hopping (known to enhance $d\ensuremath{-}\mathrm{wave}$ pairing), an even higher critical temperature may be achieved.
We show that in dilute boson-fermion mixtures with fermions in two internal states, even when the bare fermion-fermion interaction is repulsive, the exchange of density fluctuations of the Bose condensate may lead to an effective fermion-fermion attraction, and thus to a Cooper instability in the s-wave channel. We give a simple condition to know when this is going to happen, and an analytic expression for the associated Tc{T}_{c}Tc​ in the most important limit where the phonon branch of the Bogoliubov excitation spectrum of the bosons is important. We find a Tc{T}_{c}Tc​ of the same order as for a pure Fermi gas with bare attraction.
We consider a mixture of a single-component Bose gas and a two-component Fermi gas at temperatures where the Bose gas is almost fully condensed. In such a mixture, two fermionic atoms can interact with each other by exchanging a phonon that propagates through the Bose condensate. We calculate the interaction potential due to this mechanism, and determine the effective s-wave scattering length for two fermions that interact, both directly by the interatomic potentials as well as by the above-mentioned exchange mechanism. We find that the effective scattering length is quite sensitive to changes in the condensate density, and becomes strongly energy dependent. In addition, we consider the mechanical stability of these mixtures, and also calculate the dispersion and the damping of the various collisionless collective modes of the gas.