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arXiv:cond-mat/0512628 · Physical Review A

Effective field theory of boson-fermion mixtures and bound fermion states on a vortex of boson superfluid

Yusuke Nishida, Dam Thanh Son

Published Jul 18, 2006• 9 citations
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Abstract

We construct a Galilean invariant low-energy effective field theory of boson-fermion mixtures and study bound fermion states on a vortex of boson superfluid. We derive a simple criterion to determine for which values of the fermion angular momentum $l$ there exist an infinite number of bound energy levels. We apply our formalism to two boson-fermion mixed systems: the dilute solution of $^{3}\mathrm{He}$ in $^{4}\mathrm{He}$ superfluid and the cold polarized Fermi gas on the BEC side of the ``splitting point.'' For the $^{3}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ mixture, we determine parameters of the effective theory from experimental data as functions of pressure. We predict that infinitely many bound $^{3}\mathrm{He}$ states on a superfluid vortex with $l=\ensuremath{-}2,\ensuremath{-}1,0$ are realized in a whole range of pressure $0--20\phantom{\rule{0.3em}{0ex}}\mathrm{atm}$, where experimental data are available. As for the cold polarized Fermi gas, while only $S$-wave $(l=0)$ and $P$-wave $(l=\ifmmode\pm\else\textpm\fi{}1)$ bound fermion states are possible in the BEC limit, those with higher negative angular momentum become available as one moves away from the BEC limit.

DOI
10.1103/physreva.74.013615

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We construct a Galilean invariant low-energy effective field theory of boson-fermion mixtures and study bound fermion states on a vortex of boson superfluid. We derive a simple criterion to determine for which values of the fermion angular momentum lll there exist an infinite number of bound energy levels. We apply our formalism to two boson-fermion mixed systems: the dilute solution of 3He^{3}\mathrm{He}3He in 4He^{4}\mathrm{He}4He superfluid and the cold polarized Fermi gas on the BEC side of the ``splitting point.'' For the 3He\ensuremath-4He^{3}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}3He\ensuremath-4He mixture, we determine parameters of the effective theory from experimental data as functions of pressure. We predict that infinitely many bound 3He^{3}\mathrm{He}3He states on a superfluid vortex with l=\ensuremath−2,\ensuremath−1,0l=\ensuremath{-}2,\ensuremath{-}1,0l=\ensuremath−2,\ensuremath−1,0 are realized in a whole range of pressure 0−−20atm0--20\phantom{\rule{0.3em}{0ex}}\mathrm{atm}0−−20atm, where experimental data are available. As for the cold polarized Fermi gas, while only SSS-wave (l=0)(l=0)(l=0) and PPP-wave (l=\ifmmode±\else\textpm\fi1)(l=\ifmmode\pm\else\textpm\fi{}1)(l=\ifmmode±\else\textpm\fi1) bound fermion states are possible in the BEC limit, those with higher negative angular momentum become available as one moves away from the BEC limit.
We consider a system of repulsively interacting Bose-Fermi mixtures of spin-polarized uniform atomic gases at zero temperature. We examine possible realization of ppp-wave superfluidity of fermions due to an effective attractive interaction via density fluctuations of Bose-Einstein condensate within mean-field approximation. We find the ground state of the system by direct energy comparison of the ppp-wave superfluid state in a single uniform mixed phase and normal Fermi gas state in a phase-separated phase, and suggest an occurrence of the ppp-wave superfluidity for a strong boson-fermion interaction regime. We study some signatures of the ppp-wave superfluid, such as anisotropic energy gap and quasiparticle energy in the axial state, that have not been observed in spin-unpolarized superfluid of atomic fermions. We also show that a Cooper pair is a tightly bound state, such as a diatomic molecule in the strong boson-fermion coupling regime, and suggest an observable indication of the ppp-wave superfluid in the real experiment.
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 the renormalized boson-fermion TTT matrix, which we determine to second order in the boson-fermion sss-wave scattering length. We also discuss how to incorporate the usual boson-boson TTT 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.{a}_{\mathrm{BF}}{k}_{\mathrm{F}}.aBF​kF​. 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 6Li{}^{6}\mathrm{Li}6Li and 7Li{}^{7}\mathrm{Li}7Li and of 3He{}^{3}\mathrm{He}3He and 4He.{}^{4}\mathrm{He}.4He. 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 aBF.{a}_{\mathrm{BF}}.aBF​.
Using boson-vortex duality, we formulate a low-energy effective theory of a two-dimensional vortex lattice in a bosonic Galilean-invariant compressible superfluid. The excitation spectrum contains a gapped Kohn mode and an elliptically polarized Tkachenko mode that has quadratic dispersion relation at low momenta. External rotation breaks parity and time-reversal symmetries and gives rise to Hall responses. We extract the particle number current and stress tensor linear responses and investigate the relations between them that follow from Galilean symmetry. We argue that elementary particles and vortices do not couple to the spin connection which suggests that the Hall viscosity at zero frequency and momentum vanishes in a vortex lattice.
We study the low-energy properties of the spinful Bose-Fermi Hubbard model on a simple cubic lattice for attractive fermion-fermion and repulsive boson-boson interactions at half-filling for the fermions and unit-filling for the bosons. Recent DMFT results predict a variety of phases, including superfluids, charge density waves, mixed states and supersolids. We develop a self-consistent mean-field scheme which allows for all orders to appear and determine the ground state of the system by comparing the corresponding energies. In the double superfluid phase, we consider possibilities for exotic extended s-, p- and d-wave boson-assisted superfluidity.
After an introduction into 100 years of research on superfluidity and the concept of the BCS-BEC crossover, we describe recent experimental studies of a spin-polarized Fermi gas with strong interactions. Tomographically resolving the spatial structure of an inhomogeneous trapped sample, we have mapped out the superfluid phases in the parameter space of temperature, spin polarization, and interaction strength. Phase separation between the superfluid and the normal component occurs at low temperatures, showing spatial discontinuities in the spin polarization. The critical polarization of the normal gas increases with stronger coupling. Beyond a critical interaction strength all minority atoms pair with majority atoms, and the system can be effectively described as a boson-fermion mixture. Pairing correlations have been studied by rf spectroscopy, determining the fermion pair size and the pairing gap energy in a resonantly interacting superfluid.
We have analyzed a single vortex at T=0 in a 3D superfluid atomic Fermi gas across a Feshbach resonance. On the BCS side, the order parameter varies on two scales: k(F)(-1)and the coherence length xi, while only variation on the scale of xi is seen away from the BCS limit. The circulating current has a peak value jmax which is a nonmonotonic function of 1/k(F)a(s) implying a maximum critical velocity approximately v(F) at unitarity. The number of fermionic bound states in the core decreases as we move from the BCS to the BEC regime. Remarkably, a bound state branch persists even on the BEC side reflecting the composite nature of bosonic molecules.