The Hartree energy shift is calculated for a unitary Fermi gas. By including the momentum dependence of the scattering amplitude explicitly, the Hartree energy shift remains finite even at unitarity. Extending the theory also for spin-imbalanced systems allows calculation of polaron properties. The results are in good agreement with more involved theories and experiments.
The Hartree energy shift is calculated for a unitary Fermi gas. By including the momentum dependence of the scattering amplitude explicitly, the Hartree energy shift remains finite even at unitarity. Extending the theory also for spin-imbalanced systems allows calculation of polaron properties. The results are in good agreement with more involved theories and experiments.
In this paper we consider a two-component gas of fermions on the BCS side of the BCS-BEC crossover at zero temperature. We use a momentum dependent interaction that reproduces the s-wave scattering phase shifts of a contact interaction up to a momentum cutoff that is scaled with the Fermi momentum. Using a diagrammatic formulation of Bogoliubov many-body perturbation theory, suitably augmented by self-consistency conditions, we obtain the Hartree shift and the pairing gap to third order. In the weak-coupling regime, our results are not only well-converged but also agree with the well-established Gor'kov-Melik-Barkhudarov corrections for the gap and the Galitskii result for the Hartree shift. Near the unitary regime, our results for the Nambu-Gor'kov self-energy are less converged, but there is still reasonable agreement with experiments as well as with quantum Monte-Carlo results. Perspectives for improvements and applications of this approach to neutron matter are discussed.
We consider an imbalanced mixture of two different ultracold Fermi gases, which are strongly interacting. Calling spin-down the minority component and spin-up the majority component, the limit of small relative density $x={n}_{\ensuremath{\downarrow}}/{n}_{\ensuremath{\uparrow}}$ is usually considered as a gas of noninteracting polarons. This allows us to calculate, in the expansion of the total energy of the system in powers of x, the terms proportional to x (corresponding to the binding energy of the polaron) and to x5/3 (corresponding to the kinetic energy of the polaron Fermi sea). We investigate in this paper terms physically due to an interaction between polarons and which are proportional to x2 and x7/3. We find three such terms. The first one corresponds to the overlap between the clouds dressing two polarons. The two other ones are due to the modification of the single polaron binding energy caused by the nonzero density of polarons. The second term is due to the restriction of the polaron momentum by the Fermi sea formed by the other polarons. The last one results from the modification of the spin-up Fermi sea brought by the other polarons. The calculation of all these terms is made at the simplest level of a single particle-hole excitation. It is performed for all the possible interaction strengths within the stability range of the polaron. At unitarity the last two terms give a fairly weak contribution while the first one is strong and leads to a marked disagreement with Monte Carlo results. The possible origins of this discrepancy are discussed.
I study the properties of a spin-down neutron impurity immersed in a low-density free Fermi gas of spin-up neutrons. In particular, I analyze its energy (${E}_{\ensuremath{\downarrow}}$), effective mass (${m}_{\ensuremath{\downarrow}}^{*}$), and quasiparticle residue (${Z}_{\ensuremath{\downarrow}}$). Results are compared with those of state-of-the-art quantum Monte Carlo calculations of the attractive Fermi polaron realized in ultracold atomic gases experiments, and with those of previous studies of the neutron polaron. Calculations are performed within the Brueckner-Hartree-Fock approach using the chiral two-body nucleon-nucleon interaction of Entem and Machleidt at N3LO with a 500 MeV cut-off and the Argonne V18 phenomenological potential. Only contributions from the 1S0 partial wave, which is the dominant one in the low-density region considered, are included. Contributions from three-nucleon forces are expected to be irrelevant at these densities and, therefore, are neglected in the calculation. It is shown that for Fermi momenta between $\ensuremath{\approx}0.25$ and $\ensuremath{\approx}0.45\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}1}$ the energy, effective mass, and quasiparticle residue of the impurity vary only slightly, respectively, in the ranges $\ensuremath{-}0.604\phantom{\rule{0.16em}{0ex}}{E}_{F}<{E}_{\ensuremath{\downarrow}}<\ensuremath{-}0.635\phantom{\rule{0.16em}{0ex}}{E}_{F}$, $1.300\phantom{\rule{0.16em}{0ex}}m<{m}_{\ensuremath{\downarrow}}^{*}<1.085\phantom{\rule{0.16em}{0ex}}m$, and $0.741<{Z}_{\ensuremath{\downarrow}}<0.836$ in the case of the chiral interaction, and $\ensuremath{-}0.621\phantom{\rule{0.16em}{0ex}}{E}_{F}<{E}_{\ensuremath{\downarrow}}<\ensuremath{-}0.643\phantom{\rule{0.16em}{0ex}}{E}_{F}$, $1.310\phantom{\rule{0.16em}{0ex}}m<{m}_{\ensuremath{\downarrow}}^{*}<1.089\phantom{\rule{0.16em}{0ex}}m$, and $0.739<{Z}_{\ensuremath{\downarrow}}<0.832$ when using the Argonne V18 potential. These results are compatible with those derived from ultracold atoms and show that a spin-down neutron impurity in a free Fermi gas of spin-up neutrons with a Fermi momentum in the range $0.25\ensuremath{\lesssim}{k}_{F}\ensuremath{\lesssim}0.45\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}1}$ exhibits properties very similar to those of an attractive Fermi polaron in the unitary limit.
We analyze strongly interacting Fermi gases in the unitary regime by considering the generalization to an arbitrary number N of spin-1∕2 fermion flavors with Sp(2N) symmetry. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$ this problem is exactly solved by the Bardeen-Cooper-Schrieffer--Bose-Einstein condensate mean-field theory, with corrections small in the parameter 1∕N. The large-N expansion provides a systematic way to determine corrections to mean-field predictions, allowing the calculation of a variety of thermodynamic quantities at (and in proximity to) unitarity, including the energy, the pairing gap, and the upper-critical polarization (in the case of a polarized gas) for the normal to superfluid instability. For the physical case of N=1, among other quantities, we predict in the unitarity regime, the energy of the gas to be $\ensuremath{\xi}=0.28$ times that for the noninteracting gas and the pairing gap to be 0.52 times the Fermi energy.
In scattering theory, the unitary limit is defined by an infinite scattering-length and a zero effective range, corresponding to a phase-shift π/2, independent of energy. This condition is satisfied by a rank-1 separable potential V(k,k')=-v(k)v(k') with v^{2}(k)=(4π)^{2}(Λ^{2}-k^{2})^{-1/2}, Λbeing the cut-off in momentum space.Previous calculations using a Pauli-corrected ladder summation to calculate the energy of a zero temperature many body system of spin 1/2 fermions with this interaction gave ξ=0.24 (in units of kinetic energy) independent of density and with Λ---->infinity. This value of ξis appreciably smaller than the experimental and that obtained from other calculations, most notably from Monte Carlo, which in principle would be the most reliable. Our previous work did however also show a strong dependence on effective range r_0 (with r_0=0 at unitarity). With an increase to r_0=1.0 the energy varied from ξ~0.38 at k_f=0.6 1/fm to ~0.45 at k_f=1.8 1/fm which is somewhat closer to the Monte-Carlo results. These previous calculations are here extended by including the effect of the previously neglected mean-field propagation, the dispersion correction. This is repulsive and found to increase drastically with decreasing effective range. It is large enough to suggest a revised value of ξ~0.4 ~0.5 independent of r_0. Off-shell effects are also investigated by introducing a rank-2 (phase-shift equivalent) separable potential. Effects of 10% or more in energy could be demonstrated for r_0>0. It is pointed out that a computational cut-off in momentum-space brings in another scale in the in otherwise scale-less unitary problem.
These lecture notes give a brief introduction to the so-called Fermi-polaron problem, which explores the behaviour of a mobile impurity introduced into an ideal Fermi gas. While this problem has been considered now for more than a decade in ultracold atomic gases, it continues to generate surprises and insights as new quantum mixtures emerge, both in atomic gases and in the solid state. Here we summarise the basic theory for the Fermi polaron with a focus on the three-dimensional case, although the results can be straightforwardly generalised to two dimensions. Our aim is to provide a pedagogical treatment of the subject and we thus cover fundamental topics such as scattering theory and renormalisation. We discuss the ground state of the Fermi polaron and how it is connected to the phase diagram of the spin-imbalanced Fermi gas, and we also give a brief overview of the energy spectrum and non-equilibrium dynamics. Throughout, we highlight how the static and dynamic behaviour of the Fermi polaron is well described using intuitive variational approaches.