Effective-range dependence of resonant Fermi gases

A Fermi gas of cold atoms allows precise control over the dimensionless effective range, $k_{\mathrm{F}}{R}_{\mathrm{eff}}$, of the Feshbach resonance. Our pseudopotential formalism allows us to create smooth potentials with effective range, $-2\le k_{\mathrm{F}}{R}_{\mathrm{eff}}\le 2$, which we use for a variational and diffusion Monte Carlo study of the ground state of a unitary Fermi gas. We report values for the universal constants of $\xi =0.388\left(1\right)$ and $\zeta =0.087\left(1\right)$, and compute the condensate fraction, momentum distribution, and pair correlations functions. Finally, we show that a gas with $k_{\mathrm{F}}{R}_{\mathrm{eff}}\gtrsim 1.9$ is thermodynamically unstable.

Figure 1

Scattering phase shift $\delta \left(k\right)=arccot(k{R}_{\mathrm{eff}}/2)$ for the three different cases of $R_{\mathrm{eff}}$. For $R_{\mathrm{eff}}<0$ the phase shift of a realistic potential with the same low energy scattering properties is indicated by the red dotted line. The noninteracting phase shift is shown by the gray dashed line.

Figure 2

Plots of different potentials normalized by the reciprocal Fermi energy $E_{\mathrm{F}}$ as a function of radius for effective ranges $k_{\mathrm{F}}{R}_{\mathrm{eff}}=\{-1,0.1,1\}$. For the case $k_{\mathrm{F}}{R}_{\mathrm{eff}}=0.1$, the potentials are shown using solid lines, whereas the red and blue dashed lines show the Pöschl-Teller and potential well interactions with $k_{\mathrm{F}}{R}_{\mathrm{eff}}=0.2$, and the shaded purple region shows the variation of the UTP for effective range $0\le k_{\mathrm{F}}{R}_{\mathrm{eff}}\le 0.2$. The parameters of the well-barrier potential potential were chosen so that its depth and height correspond to those of the UTP.

Figure 3

Root-mean-square (rms) phase-shift error for the potential well, well barrier, and UTP for $k$ ranging from 0 to $k_{\mathrm{F}}$. On average, the rms phase shift error for the UTP is about two orders of magnitude less than that for the potential well and well-barrier potentials.

Figure 4

Ground-state energy per particle of the unitary Fermi gas as a fraction of that of a noninteracting gas with effective range $k_{\mathrm{F}}{R}_{\mathrm{eff}}$. Results obtained using the UTP are shown in purple, and results obtained using the Pöschl-Teller interaction for positive effective range in red. For large negative effective range the mean-field (MF) theory of Ref. [12] is indicated by the dotted green line.

Figure 5

Ground-state energy per particle of the unitary Fermi gas as a fraction of that of a noninteracting gas with effective range $k_{\mathrm{F}}{R}_{\mathrm{eff}}$. The UTP shown in purple and Pöschl-Teller in red are results from this work. Also shown are other numerical results from Refs. [11, 14, 27, 29, 30, 41], where for Ref. [30] we extrapolated their results to infinite system size as shown by the solid gray line, with the uncertainty in our extrapolation indicated by the dashed lines. Experimental results at $k_{\mathrm{F}}{R}_{\mathrm{eff}}=0$ from Refs. [42, 43, 44, 45] are shown in the box on the left, slightly offset from unitarity for improved readability.

Figure 6

Condensate fraction as a function of effective range calculated using the UTP in purple and Pöschl-Teller interaction in red for comparison. We also plot the results obtained near the zero-range limit by Refs. [14, 28, 29].

Figure 7

(Top) Momentum distribution $n\left(k\right)$ for $k_{\mathrm{F}}{R}_{\mathrm{eff}}=\{-1,0,1\}$. The noninteracting distribution is indicated by the gray dashed line. (Bottom) Tail of the momentum distribution on logarithmic axes. The lines indicate a weighted least-squares fit of $n\left(k\right)=C/k^{\alpha }$ to the momentum tail $k>2.2k_{\mathrm{F}}$.

Figure 8

Pair-correlation function for opposite (top) and equal (bottom) spins for $k_{\mathrm{F}}{R}_{\mathrm{eff}}=\{-1,0,1\}$. The noninteracting correlation function is indicated by the gray dashed line.

Figure 9

(Top) Energy density as a function of the Fermi gas density on the bottom axes and Fermi wave vector on the top axes. HF is the Hartree-Fock theory equation (4) shown by the green dashed line. We also show our DMC results obtained using the UTP in purple, fitted with cubic splines. (Bottom) Curvature for the Hartree-Fock theory and UTP. The spinodal point estimated from UTP data at $n{R}_{\mathrm{eff}}^3\approx 0.25$ ($k_{\mathrm{F}}{R}_{\mathrm{eff}}\approx 1.9$) is near the spinodal point calculated using the Hartree-Fock theory at $n{R}_{\mathrm{eff}}^3=0.235$ ($k_{\mathrm{F}}{R}_{\mathrm{eff}}=1.91$).

Figure 10

Ground-state energy for $k_{\mathrm{F}}{R}_{\mathrm{eff}}=\{-1,0,1\}$ as a function of time step (bottom axes) and walker population (top axes). The straight lines show a weighted least-squares fit to the data points in the linear regime.

Figure 11

Ground-state energy for $k_{\mathrm{F}}{R}_{\mathrm{eff}}=\{-1,0,1\}$ as a function of the number of particles. The straight lines show a weighted least-squares fit to the data points with $N\ge 162$.