Predicting energies of small clusters from the inhomogeneous unitary Fermi gas

We investigate the inhomogeneous unitary Fermi gas and use the long-wavelength properties to predict the energies of small clusters of unitary fermions trapped in harmonic potentials. The large pairing gap and scale invariance place severe restrictions on the form of the density functional. We determine the relevant universal constants needed to constrain the functional from calculations of the bulk in oscillating external potentials. Comparing with exact quantum Monte Carlo calculations, we find that the same functional correctly predicts the lack of shell closures for small clusters of fermions trapped in harmonic wells as well as their absolute energies. A rapid convergence to the bulk limit in three dimensions, where the surface-to-volume ratio is quite large, is demonstrated. The resulting functional can be tested experimentally, and is a key ingredient in predicting possible polarized superfluid phases and the properties of the unitary Fermi gas in optical lattices.

Figure 1

Static structure function for the unitary Fermi gas. The open squares are the BCS result, and the open circles are for the BCS-J variational trial function. The solid circles are the results of the DMC calculation (see text).

Figure 2

Energy of the unitary Fermi gas in a periodic potential vs strength of the interaction for $q=k_F/2$ (lower curves) and $q=k_F$ (upper curves). Quantum Monte Carlo calculations are shown as symbols. The bands are density functional results for ${\mathcal{E}}_2$ using $c_2=0.30\left(5\right)$ and for ${\mathcal{E}}_4$ with $c_2$ and $c_4$ extracted from fits to all the bulk QMC data. See the text for details. The error ellipse obtained for $c_2$ and $c_4$ from the fit is shown in the inset.

Figure 3

Densities of the unitary Fermi gas in external potentials of frequency $k_F/2$ (upper row) and $k_F$ (middle row) for potential strengths $V_0=0$, 0.25, 0.4, and 0.8 from left to right. The lower row shows the predicted density distributions (in the $z=0$ plane) for systems of 8, 14, 30, and 50 fermions (left to right) in a harmonic trap. Scale invariance requires the energies depend only upon the shape of the density distribution, except for an overall scale of ${\rho }^{2/3}$.

Figure 4

Ground-state energy ${[E\left(N\right)/E_{\text{TF}}]}^2$ of trapped fermions at unitarity vs particle number N. Present DMC calculations are shown as open circles, and AFMC calculations as diamonds. Density functional results ${\mathcal{E}}_2$ and ${\mathcal{E}}_4$ are shown (see text). The inset shows the extrapolation of the same data to the bulk (large N) limit.