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Laser Physics Letters

Density functional theory of the trapped Fermi gas in the unitary regime

Brandon P. van Zyl, David A. W. Hutchinson

Published Sep 26, 2007• 8 citations
Publisher

Abstract

We investigate a density-functional theory (DFT) approach for an unpolarized trapped dilute Fermi gas in the unitary limit. A reformulation of the recent work of T. Papenbrock [1] in the language of fractional exclusion statistics allows us to obtain an estimate of the universal factor, ξ3D, in three dimensions (3D), in addition to providing a systematic treatment of finite-N corrections. We show that in 3D, finite-N corrections lead to unphysical values for ξ3D, thereby suggesting that a simple DFT applied to a small number of particles may not be suitable in 3D. We then perform an analogous calculation for the two-dimensional (2D) system in the infinite-scattering length regime, and obtain a value of ξ2D=1. Owing to the unique properties of the Thomas-Fermi energy density-functional in 2D our result, in contrast to 3D, is exact and therefore requires no finite-N corrections.

DOI
10.1002/lapl.200710091

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•Nov 27, 2012•3 citations
We investigate a density-functional theory (DFT) approach for an unpolarized trapped dilute Fermi gas in the unitary limit. A reformulation of the recent work of T. Papenbrock [1] in the language of fractional exclusion statistics allows us to obtain an estimate of the universal factor, ξ3D, in three dimensions (3D), in addition to providing a systematic treatment of finite-N corrections. We show that in 3D, finite-N corrections lead to unphysical values for ξ3D, thereby suggesting that a simple DFT applied to a small number of particles may not be suitable in 3D. We then perform an analogous calculation for the two-dimensional (2D) system in the infinite-scattering length regime, and obtain a value of ξ2D=1. Owing to the unique properties of the Thomas-Fermi energy density-functional in 2D our result, in contrast to 3D, is exact and therefore requires no finite-N corrections.
A recent paper by T. Papenbrock [Phys. Rev. A 72, 041602(R) (2005)] suggests that a determination of the universal factor ${\ensuremath{\xi}}_{3\mathrm{D}}$ for a harmonically trapped, unpolarized dilute Fermi gas in the unitary regime may be obtained within the framework of a simple density-functional theory (DFT). One of the key results of that work is an estimate for the universal factor ${\ensuremath{\xi}}_{3\mathrm{D}}\ensuremath{\approx}0.54$, which is within 20% of the generally accepted quantum Monte Carlo simulation value of ${\ensuremath{\xi}}_{3\mathrm{D}}=0.44\ifmmode\pm\else\textpm\fi{}0.01$. Motivated by this result, this report investigates the determination of the universal factor ${\ensuremath{\xi}}_{2\mathrm{D}}$ for a two-dimensional trapped Fermi gas, and suggests that the simple DFT approach of Papenbrock provides an exact result of ${\ensuremath{\xi}}_{2\mathrm{D}}=1$ in two dimensions.
We determine the energy density $\ensuremath{\xi}(3∕5)n{\ensuremath{\epsilon}}_{F}$ and the gradient correction $\ensuremath{\lambda}{\ensuremath{\hbar}}^{2}{(\mathbf{\ensuremath{\nabla}}n)}^{2}∕(8mn)$ of the extended Thomas-Fermi (ETF) density functional, where nnn is the number density and ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy, for a trapped two-component Fermi gas with infinite scattering length (unitary Fermi gas) on the basis of recent diffusion Monte Carlo (DMC) calculations [Phys. Rev. Lett. 99, 233201 (2007)]. In particular we find that $\ensuremath{\xi}=0.455$ and $\ensuremath{\lambda}=0.13$ give the best fit of the DMC data with an even number NNN of particles. We also study the odd-even splitting $\ensuremath{\gamma}{N}^{1∕9}\ensuremath{\hbar}\ensuremath{\omega}$ of the ground-state energy for the unitary gas in a harmonic trap of frequency $\ensuremath{\omega}$ determining the constant $\ensuremath{\gamma}$. Finally we investigate the effect of the gradient term in the time-dependent ETF model by introducing generalized Galilei-invariant hydrodynamics equations.
We consider interacting Fermi systems close to the unitary regime and compute the corrections to the energy density that are due to a large scattering length and a small effective range. Our approach exploits the universality of the density functional and determines the corrections from the analyical results for the harmonically trapped two-body system. The corrections due to the finite scattering length compare well with the result of Monte Carlo simulations. We also apply our results to symmetric neutron matter.
We use two fundamental theoretical frameworks to study the finite-size (shell) properties of the unitary gas in a periodic box: (1) an ab initio quantum Monte Carlo (QMC) calculation for boxes containing 4 to 130 particles provides a precise and complete characterization of the finite-size behavior, and (2) a new density functional theory (DFT) fully encapsulates these effects. The DFT predicts vanishing shell structure for systems comprising more than 50 particles, and allows us to extrapolate the QMC results to the thermodynamic limit, providing the tightest bound to date on the ground-state energy of the unitary gas: ξ(S)≤0.383(1). We also apply the new functional to few-particle harmonically trapped systems, comparing with previous calculations.
Abstract We revisit the properties of the two-component Fermi gas with short-range interactions in three dimensions, in the limit where the s -wave scattering length diverges. Such a unitary Fermi gas possesses universal thermodynamic and dynamical observables that are independent of any interaction length scale. Focusing on trapped systems of N fermions, where N ≤ 10 , we investigate how well we can determine the zero-temperature behavior of the many-body system from published few-body data on the ground-state energy and the contact. For the unpolarized case, we find that the Bertsch parameters extracted from trapped few-body systems all lie within 15% of the established value. Furthermore, the few-body values for the contact are well within the range of values determined in the literature for the many-body system. In the limit of large spin polarization, we obtain a similar accuracy for the polaron energy, and we estimate the polaron’s effective mass from the dependence of its energy on N . We also compute an upper bound for the squared wave-function overlap between the unitary Fermi system and the non-interacting ground state, both for the trapped and uniform cases. This allows us to prove that the trapped unpolarized ground state at unitarity has zero overlap with its non-interacting counterpart in the many-body limit N → ∞ .
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