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arXiv:cond-mat/0607340 · Physical Review A

Hyperspherical description of the degenerate Fermi gas: s -wave interactions

Seth T. Rittenhouse, M. J. Cavagnero, Javier von Stecher, Chris H. Greene

Published Nov 28, 2006• 20 citations
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Abstract

We present a unique theoretical description of the physics of the spherically trapped $N$-atom degenerate Fermi gas at zero temperature based on an ordinary Schr\"odinger equation with a microscopic, two-body interaction potential. With a careful choice of coordinates and a variational wave function, the many-body Schr\"odinger equation can be accurately described by a linear, one-dimensional effective Schr\"odinger equation in a single collective coordinate, the rms radius of the gas. Comparisons of the energy, rms radius, and peak density of the ground-state energy are made to those predicted by Hartree-Fock (HF) theory. Also the lowest radial excitation frequency (the breathing mode frequency) agrees with a sum rule calculation, but deviates from a HF prediction.

DOI
10.1103/physreva.74.053624

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We present a unique theoretical description of the physics of the spherically trapped NNN-atom degenerate Fermi gas at zero temperature based on an ordinary Schr"odinger equation with a microscopic, two-body interaction potential. With a careful choice of coordinates and a variational wave function, the many-body Schr"odinger equation can be accurately described by a linear, one-dimensional effective Schr"odinger equation in a single collective coordinate, the rms radius of the gas. Comparisons of the energy, rms radius, and peak density of the ground-state energy are made to those predicted by Hartree-Fock (HF) theory. Also the lowest radial excitation frequency (the breathing mode frequency) agrees with a sum rule calculation, but deviates from a HF prediction.
We propose an unconventional description for the ground state and collective oscillations of the two-component normal Fermi gas with two-body zero-range interactions. The many-body problem can be accurately reduced to a linear, one-dimensional Schroedinger equation in a single collective coordinate, the hyperradius R of the N-atom system which is the root mean square radius. The calculated properties of the Fermi gas ground state are shown to agree accurately with results from the Hartree-Fock (HF) approximation over a wide range of interspecies scattering lengths. The breathing mode excitation frequency deviates qualitatively from HF predictions, but we show that this reflects a failure of the HF approach for this observable.
We present a simple implementation of a density-dependent, zero-range interactions in a degenerate Fermi gas described in hyperspherical coordinates. The method produces a 1D effective potential which accurately describes the ground state energy as a function of the hyperradius, the rms radius of the two spin component gas throughout the unitarity regime. In the unitarity regime the breathing mode frequency is found to limit to the non-interacting value. A dynamical instability, similar to the Bosenova, is predicted to be possible in gases containing more than three spin components, for large, negative, two-body scattering lengths.
We present a simple implementation of a density-dependent, zero-range interaction in a degenerate Fermi gas using hyperspherical coordinates in the large particle number limit. The method produces a 1D effective potential that accurately describes the ground-state energy as a function of the hyperradius of the two spin component gas throughout the vicinity of the unitarity regime. In the unitarity regime the breathing mode frequency is found to limit to the non-interacting value. A dynamical instability, similar to the Bosenova, is predicted to be possible for a gas containing more than three spin components, for large, negative, two-body scattering lengths. While the situation is less clear for the gas with three components, collapse might also be possible.
We investigate the ``twist'' mode (rotation of the upper against the lower hemisphere) of a dilute atomic Fermi gas in a spherical trap. The normal and superfluid phases are considered. The linear response to this external perturbation is calculated within the microscopic Hartree-Fock-Bogoliubov approach. In the normal phase the excitation spectrum is concentrated in a rather narrow peak very close to the trapping frequency. In the superfluid phase the strength starts to be damped and fragmented and the collectivity of the mode is progressively lost when the temperature decreases. In the weak-pairing regime some reminiscence of the collective motion still exists, whereas in the strong-pairing regime the twist mode is completely washed out. The disappearance of the twist mode in the strong-pairing regime with decreasing temperature is interpreted in the framework of the two-fluid model.
We treat the trapped two-component Fermi system, in which unlike fermions interact through a two-body short-range potential having no bound state but an infinite scattering length. By accurately solving the Schrödinger equation for up to N=6 fermions, we show that no many-body bound states exist other than those bound by the trapping potential, and we demonstrate unique universal properties of the system: Certain excitation frequencies are separated by 2variant Planck's over 2piomega, the wave functions agree with analytical predictions and a virial theorem is fulfilled. Further calculations up to N=30 determine the excitation gap, an experimentally accessible universal quantity, and it agrees with recent predictions based on a density functional approach.