Extended Thomas-Fermi Density Functional for the Unitary Fermi Gas

TL;DR

This paper refines the extended Thomas-Fermi density functional for the unitary Fermi gas by fitting parameters to diffusion Monte Carlo data, and explores its implications for ground-state energies and hydrodynamics.

Contribution

It introduces specific parameter values for the ETF functional based on DMC data and analyzes the impact of gradient corrections on the gas's properties.

Findings

Optimal parameters: ξ=0.455, λ=0.13

Determined odd-even energy splitting constant γ

Studied gradient term effects in hydrodynamics

Abstract

We determine the energy density $\xi (3/5) n \epsilon_F$ and the gradient correction $\lambda \hbar^2(\nabla n)^2/(8m n)$ of the extended Thomas-Fermi (ETF) density functional, where $n$ is number density and $\epsilon_F$ is Fermi energy, for a trapped two-components Fermi gas with infinite scattering length (unitary Fermi gas) on the basis of recent diffusion Monte Carlo (DMC) calculations [Phys. Rev. Lett. {\bf 99}, 233201 (2007)]. In particular we find that $\xi=0.455$ and $\lambda=0.13$ give the best fit of the DMC data with an even number $N$ of particles. We also study the odd-even splitting $\gamma N^{1/9} \hbar \omega$ of the ground-state energy for the unitary gas in a harmonic trap of frequency $\omega$ determining the constant $\gamma$. Finally we investigate the effect of the gradient term in the time-dependent ETF model by introducing generalized Galilei-invariant hydrodynamics equations.