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Physical review. B, Condensed matter

Collective modes of the one-dimensional Fermi gas within the quasiparticle random-phase approximation

T. Alm, Peter Schuck

Published Jul 15, 1996• 13 citations
Publisher

Abstract

The one-dimensional Fermi gas with attractive \ensuremath{\delta} interaction is treated in the quasiparticle random-phase approximation at zero temperature. The collective modes are evaluated numerically in the high-density (weak-coupling) and in the low-density (strong-coupling) case. Whereas in the weak-coupling limit the ordinary (particle-hole) random-phase approximation is approached for low momenta, the collective mode in the strong-coupling limit reproduces the Bogoliubov mode for the weakly interacting gas of particle-particle pairs (bosons). We find a smooth evolution of the collective modes from weak to strong coupling. Analytical approximations for the dispersion relation in the long-wavelength limit are derived for both weak and strong coupling. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.2471

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The one-dimensional Fermi gas with attractive \ensuremath{\delta} interaction is treated in the quasiparticle random-phase approximation at zero temperature. The collective modes are evaluated numerically in the high-density (weak-coupling) and in the low-density (strong-coupling) case. Whereas in the weak-coupling limit the ordinary (particle-hole) random-phase approximation is approached for low momenta, the collective mode in the strong-coupling limit reproduces the Bogoliubov mode for the weakly interacting gas of particle-particle pairs (bosons). We find a smooth evolution of the collective modes from weak to strong coupling. Analytical approximations for the dispersion relation in the long-wavelength limit are derived for both weak and strong coupling. \textcopyright{} 1996 The American Physical Society.
In this paper, we have studied the one-dimensional Fermi gas with attractive δ interaction potential in the quasi-particle random phase approximation at zero temperature. Using the theoretical formalism of Alm and Schuck, we have studied the behavior of collective modes over the whole coupling regime from weak coupling (high density) to strong coupling (low density). We observe that in the weak- coupling limit particle-hole RPA is approached to low momenta and the collective mode in the strong coupling limit reproduces the Bogoliubov mode for the weakly interacting bosons.
Following the idea of the density-functional approach, we develop a generalized Bogoliubov theory of an interacting Bose gas confined in a one-dimensional harmonic trap by using a local chemical potential, calculated with the Lieb-Liniger exact solution, as the exchange energy. At zero temperature, we use the theory to describe collective modes of a finite-particle system in all interaction regimes from the ideal gas limit to the mean-field Thomas-Fermi regime and to the strongly interacting Tonks-Girardeau regime. At finite temperature, we investigate the temperature dependence of collective modes in the weak-coupling regime by means of a Hartree-Fock-Bogoliubov theory with Popov approximation. By emphasizing the effects of finite particle number and nonzero temperature on collective-mode frequencies, we compare our results with a recent experimental measurement [E. Haller et al., Science 325, 1224 (2009)] and some previous theoretical predictions. We show that the experimental data are still not fully explained within current theoretical framework.
We study the collective excitation and stability of superfluid Fermi gases flowing with a constant velocity in three-dimensional free space. In particular, we investigate a possible gapless superfluid state induced by the superflow using the mean-field theory and the generalized random-phase approximation (GRPA). For weak attractive interactions, we find that the mean-field superfluid order parameter can take a nonzero value even after the superflow velocity exceeds the threshold for the onset of Bogoliubov quasiparticle excitations. Since the Cooper pairs are only partially broken by the quasiparticle excitations, a gapless superfluid state can be formed over a certain range of superflow velocity above the pair-breaking onset. In addition to the usual quasiparticle-pair continuum and the Anderson-Bogoliubov collective mode, the GRPA excitation spectrum of the gapless superfluid state has a quasiparticle-quasihole continuum and a second collective mode. We find that the long-wavelength excitations of the second collective mode eventually cause dynamical instability of the system when the superflow velocity increases. However, the gapless superfluid state still remains stable in a narrow but finite range of superflow velocity.
We examine collective modes, stability, and BCS pairing in a quasi-two-dimensional gas of dipolar fermions aligned by an external field. By using the (conserving) Hartree-Fock approximation, which treats direct and exchange interactions on an equal footing, we obtain the spectrum of single-particle excitations and long-wavelength collective modes (zero sound) in the normal phase. It appears that exchange interactions result in strong damping of zero sound when the tilting angle between the dipoles and the normal to the plane of confinement is below some critical value. In particular, zero sound cannot propagate if the dipoles are perpendicular to the plane of confinement. At intermediate coupling, we find unstable modes that can lead either to collapse of the system or to the formation of a density wave. The BCS transition to a superfluid phase, on the other hand, occurs at arbitrarily weak strengths of the dipole-dipole interaction, provided the tilting angle exceeds a critical value. We determine the critical temperature of the transition, taking into account many-body effects as well as virtual transitions to higher excited states in the confining potential, and discuss prospects of experimental observations.
We demonstrate the existence of a collective excitation branch in the pair-breaking continuum of superfluid Fermi gases and BCS superconductors. At zero temperature, we analytically continue the equation on the collective mode energy in Anderson's Random Phase Approximation or Gaussian fluctuations through its branch cut associated with the continuum, and obtain the full complex dispersion relation, including in the strong coupling regime. The branch exists as long as the chemical potential μ is positive and the wave number below sqrt[2mμ]/ℏ (with m the fermion mass). In the long wavelength limit, the branch varies quadratically with the wave number, with a complex effective mass that we compute analytically for an arbitrary interaction strength.
We consider the collective mode spectrum of a normal Fermi gas in a spherical harmonic trap. Using a self-consistent random-phase approximation, we systematically examine the effects of the two-body interactions on modes of various symmetries. For weak coupling, where the spectrum is shifted very little away from the ideal gas case, a sum-rule approach is shown to work well. For stronger coupling, the interplay between the single particle and collective excitations causes effects such as mode splitting and Landau damping. A finite low-frequency response present at stronger coupling is predicted for the quadrupole mode. Finally, we briefly discuss the effect of a finite temperature on the spectrum, and the excitation of higher collective modes.