arXiv:cond-mat/0610256 · Physical Review A

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

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

Published Aug 14, 2007• 5 citations

Abstract

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.

DOI: 10.1103/physreva.76.025601

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arXiv:cond-mat/0610256 · Physical Review A

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

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

Published Aug 14, 2007• 5 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physreva.76.025601

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Jan 1, 20100 citations

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}}

{\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.

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.

ν

and, by extrapolating to

ν\to 0

, determine the universal critical temperature of the continuum unitary Fermi gas in units of Fermi energy:

T_c/ε_F=0.152(7)

. We also determine the thermodynamic functions and show how the Monte Carlo results can be used for accurate thermometry of a trapped unitary gas.