Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0809.1820 · Physical Review A

Extended Thomas-Fermi density functional for the unitary Fermi gas

Luca Salasnich, Flavio Toigo

Published Nov 18, 2008• 85 citations
arXivHTMLPDF

Abstract

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 $n$ 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 $N$ 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.

DOI
10.1103/physreva.78.053626

Related papers

Study of a density functional for a unitary Fermi gas

Nicola Manini, Luca Salasnich

In this thesis, following L. Salasnich and F. Toigo [Phys. Rev. A 78, 053626 (2008)] we use an extended Thomas-Fermi (TF) density functional to describe a gas of interacting fermions composed of an even number N of particle in the unitary regime confined by a spherically symmetric harmonic trap. The great advantage in taking such a functional, relative to functionals based on single-fermion wave functions, is to depend only on the density number n and on its derivatives, so that the numerical load is independent of the number of particles. By minimizing the ETF functional with respect to the density we obtain a time-independent non-linear Schrodinger equation which we solve using the imaginary time propagation method, by integrating numerically the corresponding time-dependent Schrodinger equation. We determine the two characteristic parameters of the functionals ξ and λ by fitting the Monte Carlo total energy data obtained by D. Blume et al. [Phys. Rev. Lett. 99, 233201 (2007)]. We obtain the values ξ = 0.469 and λ = 0.082 leading to a χ2 = 42. These values are to be compared with ξ = 0.455 and λ = 0.13, leading to a χ2 = 282 as proposed by Salasnich and Toigo. We also study the dependence of the ground-state density profile on the values of the two parameters, finding that ξ enhances the width of the profile, while increasing λ smooths out the outer boundary of the fermionic cloud. Advisor: Dr. Nicola Manini Co-Advisor: Dr. Luca Salasnich

•Jan 1, 2010•0 citations

Surface effects in the unitary Fermi gas

Luca Salasnich, Francesco Ancilotto, Flavio Toigo

We study the extended Thomas-Fermi (ETF) density functional of the superfluid unitary Fermi gas. This functional includes a gradient term which is essential to describe accurately the surface effects of the system, in particular with a small number of atoms, where the Thomas-Fermi (local density) approximation fails. We find that our ETF functional gives density profiles which are in good agreement with recent Monte Carlo results and also with a more sophisticated superfluid density functional based on Bogoliubov-de Gennes equations. In addition, by using extended hydrodynamics equations of superfluids, we calculate the frequencies of collective surface oscillations of the unitary Fermi gas, showing that quadrupole and octupole modes strongly depend on the number of trapped atoms.

arXiv:0909.2344•Laser Physics Letters•Sep 24, 2009•19 citations

Unitary Fermi gas in a harmonic trap

S. Y. Chang, G. F. Bertsch

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.

arXiv:physics/0703190•Physical Review A•Aug 2, 2007•74 citations

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

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

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.

Laser Physics Letters•Sep 26, 2007•8 citations

Density-functional theory of the trapped two-dimensional Fermi gas in the unitary regime

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

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.

arXiv:cond-mat/0610256•Physical Review A•Aug 14, 2007•5 citations

Time-dependent density-functional theory for trapped strongly interacting fermionic atoms

Yeong E. Kim, Alexander L. Zubarev

The dynamics of strongly interacting trapped dilute two-component Fermi gases (dilute in the sense that the range of interatomic potential is small compared with interparticle spacing) is investigated in a single-equation approach to the time-dependent density-functional theory. For the ground-state energy per particle of the system in the homogeneous phase, we have constructed an Pad\'e parametrization based on Monte Carlo data and asymptotic behavior. Our numerical results for collective frequencies in the BCS-BEC crossover regime are in good agreement with recent experimental data obtained by the Duke University group. In addition, we show that the calculated corrections to the hydrodynamic approximation may be important, even for systems with a rather large number of atoms.

arXiv:cond-mat/0404513•Physical Review A•Sep 27, 2004•141 citations
We determine the energy density \ensuremathξ(3∕5)n\ensuremathϵF\ensuremath{\xi}(3∕5)n{\ensuremath{\epsilon}}_{F}\ensuremathξ(3∕5)n\ensuremathϵF​ and the gradient correction \ensuremathλ\ensuremathℏ2(\ensuremath∇n)2∕(8mn)\ensuremath{\lambda}{\ensuremath{\hbar}}^{2}{(\mathbf{\ensuremath{\nabla}}n)}^{2}∕(8mn)\ensuremathλ\ensuremathℏ2(\ensuremath∇n)2∕(8mn) of the extended Thomas-Fermi (ETF) density functional, where nnn is the number density and \ensuremathϵF{\ensuremath{\epsilon}}_{F}\ensuremathϵF​ 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 \ensuremathξ=0.455\ensuremath{\xi}=0.455\ensuremathξ=0.455 and \ensuremathλ=0.13\ensuremath{\lambda}=0.13\ensuremathλ=0.13 give the best fit of the DMC data with an even number NNN of particles. We also study the odd-even splitting \ensuremathγN1∕9\ensuremathℏ\ensuremathω\ensuremath{\gamma}{N}^{1∕9}\ensuremath{\hbar}\ensuremath{\omega}\ensuremathγN1∕9\ensuremathℏ\ensuremathω of the ground-state energy for the unitary gas in a harmonic trap of frequency \ensuremathω\ensuremath{\omega}\ensuremathω determining the constant \ensuremathγ\ensuremath{\gamma}\ensuremathγ. Finally we investigate the effect of the gradient term in the time-dependent ETF model by introducing generalized Galilei-invariant hydrodynamics equations.
In this thesis, following L. Salasnich and F. Toigo [Phys. Rev. A 78, 053626 (2008)] we use an extended Thomas-Fermi (TF) density functional to describe a gas of interacting fermions composed of an even number N of particle in the unitary regime confined by a spherically symmetric harmonic trap. The great advantage in taking such a functional, relative to functionals based on single-fermion wave functions, is to depend only on the density number n and on its derivatives, so that the numerical load is independent of the number of particles. By minimizing the ETF functional with respect to the density we obtain a time-independent non-linear Schrodinger equation which we solve using the imaginary time propagation method, by integrating numerically the corresponding time-dependent Schrodinger equation. We determine the two characteristic parameters of the functionals ξ and λ by fitting the Monte Carlo total energy data obtained by D. Blume et al. [Phys. Rev. Lett. 99, 233201 (2007)]. We obtain the values ξ = 0.469 and λ = 0.082 leading to a χ2 = 42. These values are to be compared with ξ = 0.455 and λ = 0.13, leading to a χ2 = 282 as proposed by Salasnich and Toigo. We also study the dependence of the ground-state density profile on the values of the two parameters, finding that ξ enhances the width of the profile, while increasing λ smooths out the outer boundary of the fermionic cloud. Advisor: Dr. Nicola Manini Co-Advisor: Dr. Luca Salasnich
We study the extended Thomas-Fermi (ETF) density functional of the superfluid unitary Fermi gas. This functional includes a gradient term which is essential to describe accurately the surface effects of the system, in particular with a small number of atoms, where the Thomas-Fermi (local density) approximation fails. We find that our ETF functional gives density profiles which are in good agreement with recent Monte Carlo results and also with a more sophisticated superfluid density functional based on Bogoliubov-de Gennes equations. In addition, by using extended hydrodynamics equations of superfluids, we calculate the frequencies of collective surface oscillations of the unitary Fermi gas, showing that quadrupole and octupole modes strongly depend on the number of trapped atoms.
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 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.
The dynamics of strongly interacting trapped dilute two-component Fermi gases (dilute in the sense that the range of interatomic potential is small compared with interparticle spacing) is investigated in a single-equation approach to the time-dependent density-functional theory. For the ground-state energy per particle of the system in the homogeneous phase, we have constructed an Pad\'e parametrization based on Monte Carlo data and asymptotic behavior. Our numerical results for collective frequencies in the BCS-BEC crossover regime are in good agreement with recent experimental data obtained by the Duke University group. In addition, we show that the calculated corrections to the hydrodynamic approximation may be important, even for systems with a rather large number of atoms.