Resonantly Interacting Fermions in a Box

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: ${\xi }_S\le 0.383(1)$. We also apply the new functional to few-particle harmonically trapped systems, comparing with previous calculations.

Figure 1

Ground-state energy density $\xi =\mathcal{E}/{\mathcal{E}}_{\mathrm{FG}}$ of $N_{+}$ fermions in a periodic cubic box at the unitary limit. The circles with error bars are the result of using a quadratic least-squares extrapolation to zero effective range of our new QMC results. The solid curve is the best fit SLDA DFT. The light dotted curve is the functional considered in 19 with $\alpha =0.69$. For comparison, we have plotted the previous best estimate ${\xi }_S=0.40(1)$ (red square) and the current estimate ${\xi }_S=0.383(1)$ below it to the far right of the figure. Inset: We show the typical effective-range dependence $\xi (k_F{r}_e)$ with the best fit $1\sigma$ error bounds for all-point cubic (solid dark green region) and five-point quadratic (hatched light yellow region) polynomial fits. Note that (a) the five-point quadratic model is consistent with the full cubic model and has a comparable extrapolation error, and (b) the inflection point near $k_F{r}_e\approx 0.16$ necessitates a higher-order fit for larger ranges (cubic is sufficient for the ranges shown here). Results for $N_{+}=40$ show the same qualitative behavior; hence, for the other points we use the five-point quadratic extrapolation. (Values in supplemental material [35].)

Figure 2

Ground-state energy of the harmonically trapped unitary Fermi gas (in units where $\hslash \omega =1$) scaled to demonstrate the asymptotic form $16E^2/(3N_{+}{)}^{8/3}={\xi }_S[1+cx+O(x^{7/6}){]}^2$ predicted in Ref. 32. The best fit SLDA (solid blue line) is compared with zero-range results for $N_{+}\in \{4,6\}$ from Ref. 33 and finite-range QMC results from Ref. 23 (upper red dots) and Ref. 34 (green pluses). The latter have significantly lower energy, despite having a slightly larger effective range, suggesting that the wave functions in Ref. 23 were not fully optimized. We expect careful optimization and zero-range extrapolation to bring the QMC results for large $N_{+}$ in line with the DFT as discussed in the text.