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On the kinetic energy of unitary Fermi gas in a harmonic trap

Alexander L. Zubarev, M. Zoubarev

Published Nov 27, 2012• 3 citations
OpenAlex

Abstract

We have considered the orbital-free approximation of the kinetic energy functional to investigate the zero temperature properties of dilute harmonically trapped two component Fermi gas at unitarity. It is shown that our approach provides a realible and inexpensive method to study superfluid strongly interacting dilute Fermi gases. PACS numbers: 31.15.Ew, 71.15.Mb, 03.75.SsThe modern density functional theory (DFT) is based on the Kohn-Sham (KS) approach [1], where the noninteracting kinetic energy, T, is calculated in terms of the KS orbitals, although Ref.[2] proved the basic existence of functional T(ρ), where ρ is the density, ρ(⃗r)d

Related papers

On the kinetic energy of unitary Fermi gas in a harmonic trap

A. L. Zubarev, M. Zoubarev

We have considered the orbital-free approximation of the kinetic energy functional to investigate the zero-temperature properties of dilute harmonically trapped two-component Fermi gas at unitarity. It is shown that our approach provides a realible and inexpensive method to study superfluid strongly interacting dilute Fermi gases.

arXiv:0906.1239•Europhysics Letters (EPL)•Aug 1, 2009•5 citations

Weizsacker energy of unitary Fermi gas in a harmonic trap

Alexander L. Zubarev

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Harmonically trapped fermion gases: exact and asymptotic results in arbitrary dimensions

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We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive differential equations connecting these densities, which so far have been proved only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas–Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is uniquely identified. We derive an analytical expansion for the remaining oscillating parts and obtain very simple closed forms for the leading-order oscillating densities. Finally, we show that the simple TF functional relation τ TF [ρ] between kinetic and particle densities is also fulfilled for the asymptotic quantum densities τ( r ) and ρ( r ) including their leading-order oscillating terms.

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We present an ab initio calculation of small numbers of trapped, strongly interacting fermions using the Green's function Monte Carlo method. The ground-state energy, density profile, and pairing gap are calculated for particle numbers $N=2--22$ using the parameter-free ``unitary'' interaction. Trial wave functions are taken in the form of correlated pairs in a harmonic oscillator basis. We find that the lowest energies are obtained with a minimum explicit pair correlation beyond that needed to exploit the degeneracy of oscillator states. We find that the energies can be well fitted by the expression ${a}_{\mathit{TF}}{E}_{\mathit{TF}}+\ensuremath{\Delta}\phantom{\rule{0.2em}{0ex}}\mathrm{mod}(N,2)$ where ${E}_{\mathit{TF}}$ is the Thomas-Fermi energy of a noninteracting gas in the trap and $\ensuremath{\Delta}$ is the pairing gap. There is no evidence of a shell correction energy in the systematics, but the density distributions show pronounced shell effects. We find the value $\ensuremath{\Delta}=0.7\ifmmode\pm\else\textpm\fi{}0.2\ensuremath{\omega}$ for the pairing gap. This is smaller than the value found for the uniform gas at a density corresponding to the central density of the trapped gas.

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Harmonically trapped fermion gases: exact and asymptotic results in arbitrary dimensions

M. Brack, M. V. N. Murthy

We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive differential equations connecting these densities, which so far have been proved only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas-Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is uniquely identified. We derive an analytical expansion for the remaining oscillating parts and obtain very simple closed forms for the leading-order oscillating densities. Finally, we show that the simple TF functional relation tau(TF)[rho] between kinetic and particle densities is also fulfilled for the asymptotic quantum densities tau(r) and rho(r) including their leading-order oscillating terms.

•Jan 1, 2003•23 citations
We have considered the orbital-free approximation of the kinetic energy functional to investigate the zero temperature properties of dilute harmonically trapped two component Fermi gas at unitarity. It is shown that our approach provides a realible and inexpensive method to study superfluid strongly interacting dilute Fermi gases. PACS numbers: 31.15.Ew, 71.15.Mb, 03.75.SsThe modern density functional theory (DFT) is based on the Kohn-Sham (KS) approach [1], where the noninteracting kinetic energy, T, is calculated in terms of the KS orbitals, although Ref.[2] proved the basic existence of functional T(ρ), where ρ is the density, ρ(⃗r)d
We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive differential equations connecting these densities, which so far have been proved only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas–Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is uniquely identified. We derive an analytical expansion for the remaining oscillating parts and obtain very simple closed forms for the leading-order oscillating densities. Finally, we show that the simple TF functional relation τ TF [ρ] between kinetic and particle densities is also fulfilled for the asymptotic quantum densities τ( r ) and ρ( r ) including their leading-order oscillating terms.
In the unitary regime, fermions interact strongly via two-body potentials that exhibit a zero range and a (negative) infinite scattering length. The energy density is proportional to the energy density of the free Fermi gas with a proportionality constant $\ensuremath{\xi}$. The author uses a simple density functional parametrized by an effective mass and the universal constant $\ensuremath{\xi}$, and employs Kohn-Sham density-functional theory to obtain the parameters from fit to one exactly solvable two-body problem. This yields $\ensuremath{\xi}=0.42$ and a rather large effective mass. The form of the density functional is checked by similar Kohn-Sham calculations for the exactly solvable Calogero model.
We present an ab initio calculation of small numbers of trapped, strongly interacting fermions using the Green's function Monte Carlo method. The ground-state energy, density profile, and pairing gap are calculated for particle numbers N=2−−22N=2--22N=2−−22 using the parameter-free ``unitary'' interaction. Trial wave functions are taken in the form of correlated pairs in a harmonic oscillator basis. We find that the lowest energies are obtained with a minimum explicit pair correlation beyond that needed to exploit the degeneracy of oscillator states. We find that the energies can be well fitted by the expression ${a}_{\mathit{TF}}{E}_{\mathit{TF}}+\ensuremath{\Delta}\phantom{\rule{0.2em}{0ex}}\mathrm{mod}(N,2)$ where ETF{E}_{\mathit{TF}}ETF​ is the Thomas-Fermi energy of a noninteracting gas in the trap and $\ensuremath{\Delta}$ is the pairing gap. There is no evidence of a shell correction energy in the systematics, but the density distributions show pronounced shell effects. We find the value $\ensuremath{\Delta}=0.7\ifmmode\pm\else\textpm\fi{}0.2\ensuremath{\omega}$ for the pairing gap. This is smaller than the value found for the uniform gas at a density corresponding to the central density of the trapped gas.
We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive differential equations connecting these densities, which so far have been proved only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas-Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is uniquely identified. We derive an analytical expansion for the remaining oscillating parts and obtain very simple closed forms for the leading-order oscillating densities. Finally, we show that the simple TF functional relation tau(TF)[rho] between kinetic and particle densities is also fulfilled for the asymptotic quantum densities tau(r) and rho(r) including their leading-order oscillating terms.