Quantum Monte Carlo studies of superfluid Fermi gases

We report results of quantum Monte Carlo calculations of the ground state of dilute Fermi gases with attractive short-range two-body interactions. The strength of the interaction is varied to study different pairing regimes which are characterized by the product of the $s$-wave scattering length and the Fermi wave vector, $a{k}_F$. We report results for the ground-state energy, the pairing gap $\Delta$, and the quasiparticle spectrum. In the weak-coupling regime, $1∕a{k}_F<-1$, we obtain Bardeen-Cooper-Schrieffer (BCS) superfluid and the energy gap $\Delta$ is much smaller than the Fermi gas energy $E_{\mathrm{FG}}$. When $a>0$, the interaction is strong enough to form bound molecules with energy $E_{\mathrm{mol}}$. For $1∕a{k}_F\gtrsim 0.5$, we find that weakly interacting composite bosons are formed in the superfluid gas with $\Delta$ and gas energy per particle approaching $\mid E_{\mathrm{mol}}\mid ∕2$. In this region, we seem to have Bose-Einstein condensation (BEC) of molecules. The behavior of the energy and the gap in the BCS-to-BEC transition region, $-0.5<1∕a{k}_F<0.5$, is discussed.

Figure 1

Ground-state energy per particle of dilute Fermi gases as a function of $a{k}_F$. The full and dashed curves give the LOCV results for $\cosh \left(\mu r_0=12\right)$ and $\delta$-function potentials, and the circles show the essentially exact results for the $\cosh$ potential obtained with the GFMC method described in Sec. 3. The dotted and dash-dotted curves correspond to the conventional BCS results with $\cosh$ and $\delta$-function potentials, respectively.

Figure 2

Correlation function $f\left(r\right)$ for different values of $a{k}_F$ in the LOCV approximation using the $\cosh$ potential with $\mu r_0=12$.

Figure 3

The LOCV $E∕A$ in units of $E_{\mathrm{FG}}$ vs $a{k}_F$ for attractive $\delta$-function potential. The dashed line corresponds to $f\left(r\right)$ having one node, and the solid line shows the results with nodeless $f\left(r\right)$.

Figure 4

Mixed and growth estimates of the GFMC energy. The $\tau \to \infty$ asymptotic value is reached after $\sim 5000$ time steps. Each time step $\Delta \tau$ is $1.2{10}^{-3}\hslash ∕E_{\mathrm{FG}}$.

Figure 5

$E\left(A\right)$ when $a{k}_F=-\infty$ from Ref. 16.

Figure 6

$E\left(A\right)$ for $1∕a{k}_F⩽1∕3$.

Figure 7

$E\left(A\right)$ for $1∕a{k}_F>1∕3$. The results for $a{k}_F=0.5$ have been multiplied by 0.5 for graphing.

Figure 8

$E∕A$ and $E_{\mathrm{mol}}∕2$ for positive values of $1∕a{k}_F$ for the $\cosh \left(\mu r_0=12\right)$ potential and $E_{\mathrm{mol}}∕2$ for the $\delta$-function potential.

Figure 9

$\Delta \left(A\right)$ for $1∕a{k}_F⩽1∕3$.

Figure 10

$\Delta \left(A\right)$ for $1∕a{k}_F>1∕3$.

Figure 11

Calculated values of ${\Delta }_{\mathrm{GFMC}}\left(a{k}_F\right)$ ($\cosh$, $\mu r_0=12$ potential) are compared with various estimates of $\Delta \left(a{k}_F\right)$ and $-E_{\mathrm{mol}}∕2$. The BCS and Gorkov estimates do not depend on the shape of the potential, while $-E_{\mathrm{mol}}∕2$ is shown for both $\cosh$ (solid line) and $\delta$-function (dash double dot) potentials. ${\Delta }_{\mathrm{BCS}}$ and ${\Delta }_{\text{Gorkov}}$ assume the chemical potential $\approx T_F$ throughout the whole range of $a{k}_F$ [see Eq. (34)].

Figure 12

Pressure ($P$, dashed curve) and adiabatic index ($\Gamma$, continuous curve) in the BCS regime ($a{k}_F<0$). In the dilute limit $\left(a{k}_F\to 0\right)$ we have $P∕\left(\rho E_{\mathrm{FG}}\right)\to 2∕3$ and $\Gamma \to 5∕3$. In the dense limit $\left(a{k}_F\to -\infty \right)$ we have $P∕\left(\rho E_{\mathrm{FG}}\right)\to 0.44(2∕3)$ and $\Gamma \to 5∕3$. $\Gamma$ has a minimum at $a{k}_F\sim -1.3$.