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arXiv:cond-mat/0602191 · Physical Review A

Stability of trapped fermionic gases with attractive interactions

Benjamin M. Fregoso, Gordon Baym

Published Apr 27, 2006• 17 citations
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Abstract

We present a unified overview, from the mean-field to the unitarity regime, of the stability of a trapped Fermi gas with short range attractive interactions. Unlike in a system of bosons, a Fermi gas is always stable in these regimes, no matter how large the particle number. However, when the interparticle spacing becomes comparable to the range of the interatomic interactions, instability is not precluded.

DOI
10.1103/physreva.73.043616

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We present a unified overview, from the mean-field to the unitarity regime, of the stability of a trapped Fermi gas with short range attractive interactions. Unlike in a system of bosons, a Fermi gas is always stable in these regimes, no matter how large the particle number. However, when the interparticle spacing becomes comparable to the range of the interatomic interactions, instability is not precluded.
Two-component equal-mass Fermi gases, in which unlike atoms interact through a short-range two-body potential and like atoms do not interact, are stable even when the interspecies sss-wave scattering length becomes infinitely large. Solving the many-body Schr\"odinger equation within a hyperspherical framework and by Monte Carlo techniques, this paper investigates how the properties of trapped two-component gases change if a third or fourth component are added. If all interspecies scattering lengths are equal and negative, our calculations suggest that both three- and four-component Fermi gases become unstable for a certain critical set of parameters. The relevant length scale associated with the collapse is set by the interspecies scattering length and we argue that the collapse is similar to the collapse of an attractive trapped Bose gas, a many-body phenomenon. Furthermore, we consider a three-component Fermi gas in which two interspecies scattering lengths are negative while the other interspecies scattering length is zero. In this case, the stability of the Fermi system is predicted to depend appreciably on the range of the underlying two-body potential. We find parameter combinations for which the system appears to become unstable for a finite negative scattering length and parameter combinations for which the system appears to be made up of weakly bound trimers that consist of one fermion of each species.
This work is inspired by recent experimental observations in ultracold atomic Bose-Fermi mixtures [DeSalvo et al., Nature 568 (2019)]. These experiments reveal the emergence of an attractive fermion-mediated interaction between bosons, as well as a stability-instability transition. We give the first mathematical demonstration of this transition by studying the low-energy spectrum of a many-body interspecies Hamiltonian. More precisely, we show the convergence of its eigenvalues towards those of an effective Bose Hamiltonian, which includes fermion-mediated effects. Applying this result to a model with short-range potentials, we derive a stability-instability transition in the bosonic subsystem, driven by the Bose-Fermi coupling strength ggg. For small ∣g∣|g|∣g∣, the bosons form a stable Bose-Einstein condensate with the energy per particle uniformly bounded from below. For large ∣g∣|g|∣g∣, the energy per particle is no longer uniformly bounded from below, signaling the collapse of the condensate.
Abstract. Trapped degenerate dipolar Bose and Fermi gases of cylindrical symmetry with the polarization vector along the symmetry axis are only stable for the strength of dipolar interaction below a critical value. In the case of bosons, the stability of such a dipolar Bose-Einstein condensate (BEC) is investigated for different strengths of contact and dipolar interactions using variational approximation and numerical solution of a mean-field model. In the disk shape, with the polarization vector perpendicular to the plane of the disk, the atoms experience an overall dipolar repulsion and this fact should contribute to the stability. However, a complete numerical solution of the dynamics leads to the collapse of a strongly disk-shaped dipolar BEC due to the long-range anisotropic dipolar interaction. In the case of fermions, the stability of a trapped single-component degenerate dipolar Fermi gas is studied including the Hartree-Fock exchange and Brueckner-Goldstone correlation energies in the local-density approximation valid for a large number of atoms. Estimates for the maximum allowed number of polar Bose and Fermi molecules in BEC and degenerate Fermi gas are given. PACS numbers: 03.75.Hh,03.75.Ss,03.75.Kk,05.30.Fk ar
The structure and stability of dilute degenerate Fermi gases trapped in an external potential is discussed with special emphasis on the influence of s- and p-wave interactions. In a first step an effective contact interaction for all partial waves is derived, which reproduces the energy spectrum of the full potential within a mean-field model space. Using the s- and p-wave part the energy density of the multicomponent Fermi gas is calculated in Thomas-Fermi approximation. On this basis the stability of the one- and two-component Fermi gas against mean-field induced collapse is investigated. Explicit stability conditions in terms of density and total particle number are given. For the single-component system attractive p-wave interactions limit the density of the gas. In the two-component case a subtle competition of s- and p-wave interactions occurs and gives rise to a rich variety of phenomena. A repulsive p-wave part, for example, can stabilize a two-component system that would otherwise collapse due to an attractive s-wave interaction. It is concluded that the p-wave interaction may have important influence on the structure of degenerate Fermi gases and should not be discarded from the outset.
Trapped degenerate dipolar Bose and Fermi gases of the cylindrical symmetry with the polarization vector along the symmetry axis are only stable for the strength of dipolar interaction below a critical value. In the case of bosons, the stability of such a dipolar Bose–Einstein condensate (BEC) is investigated for different strengths of contact and dipolar interactions using a variational approximation and a numerical solution of a mean-field model. In the disc shape, with the polarization vector perpendicular to the plane of the disc, the atoms experience an overall dipolar repulsion and this fact should contribute to the stability. However, a complete numerical solution of the dynamics leads to the collapse of a strongly disc-shaped dipolar BEC due to the long-range anisotropic dipolar interaction. In the case of fermions, the stability of a trapped single-component degenerate dipolar Fermi gas is studied including the Hartree–Fock exchange and Brueckner–Goldstone correlation energies in the local-density approximation valid for a large number of atoms. Estimates for the maximum allowed number of polar Bose and Fermi molecules in the BEC and degenerate Fermi gas are given.
The probability amplitude for NNN particles in a quantum gas with negligible range of interparticle interaction potentials to come to a small region of size rrr scales like rγr^γrγ. It is shown that γγγ is quantitatively related to the ground state energy of these NNN fermions in the unitarity limit, confined by an isotropic harmonic potential. For large NNN, the short range density distribution of these NNN particles is predominantly the same as the Thomas-Fermi profile of the gas in the unitarity limit confined by such a harmonic potential. These results may shed light on strongly interacting ultracold atomic Fermi gases, in a trap or on an optical lattice.