Ground-state properties of strongly interacting Fermi gases in two dimensions

Exact calculations are performed on the two-dimensional strongly interacting unpolarized uniform Fermi gas with a zero-range attractive interaction. Two auxiliary-field approaches are employed which accelerate the sampling of imaginary-time paths using BCS trial wave functions and a force bias technique. Their combination enables calculations on large enough lattices to reliably compute ground-state properties in the thermodynamic limit. An equation of state is obtained with a parametrization provided, which can serve as a benchmark and allow accurate comparisons with experiments. The pressure, contact parameter, and condensate fraction are determined systematically vs $k_{F}a$. The momentum distribution, pairing correlation, and the structure of the pair wave function are computed. The use of force bias to accelerate the Metropolis sampling of auxiliary fields in determinantal approaches is discussed.

Figure 1

Calculated equation of state. The top panel shows the energy, relative to the final AFQMC results, for a finite number of particles $N$. Also shown are the DMC results of Ref. [18], which are variational. Note the small scale of the vertical axis. The bottom panel shows the AFQMC (and DMC) results at the TL, relative to the BCS result. A fit has been performed on the AFQMC results for the EOS. The result is given in Eqs. (4) and (5) and shown as the solid line. The inset in panel (b) compares the calculated pressure from the AFQMC (solid line) and DMC (dashed line, taken from Ref. [29]) with experiment [29] (points) in the crossover region.

Figure 2

The contact parameter $C$. The main figure shows the result of $C$ (relative to the BCS result) obtained from Eq. (6). The statistical uncertainty is smaller than the line thickness. DMC [18] and BCS results are also shown for comparison. The inset shows $n\left(\mathbf{k}\right)k^4$ vs $k\equiv \left|\mathbf{k}\right|$ at $x=0.5$. The horizontal lines give the $C$ values from DMC, AFQMC, and BCS (top to bottom), indicated by the arrows in the main figure. The $n\left(\mathbf{k}\right)$ data are from two systems with $L=45$ (circles) and 51 (squares), respectively, and $N=58$. Results are plotted for $\mathbf{k}$ along both the horizontal (solid symbols) and the diagonal (open) directions.

Figure 3

Momentum distribution and pair wave functions in three regimes of interaction strengths $x\equiv ln\left(a{k}_F\right)$. In each panel, the vertical tick labels on the left are for $n\left(\mathbf{k}\right)$, and those on the right are for ${\varphi }_{\uparrow \downarrow }\left(\mathbf{k}\right)$, both plotted vs $k$ (in units of $k_F$). Note the different scales between the three panels. The inset shows the real-space wave function ${\psi }_{\uparrow \downarrow }\left(\mathbf{r}\right)$ vs $\mathbf{r}$ in a 3D plot. The lattice has ${\mathcal{N}}_s=2025$ sites with density $n=0.0286$.

Figure 4

Condensate fraction and pairing correlation functions. In the main graph, the uncertainty in the QMC data (from extrapolation to the TL) is estimated by multiple runs with different sizes and is indicated by the thickness of the line. Also shown are BCS results and, in the BEC limit, Bogoliubov results for a Bose gas for reference. In the inset, the pairing correlation function $C\left(\mathbf{r}\right)$ is plotted vs $r$ for three interaction strengths [from top to bottom, the same parameters as in (a)–(c) of Fig. 3]. The dashed lines are from BCS, and the solid lines are QMC results (error bars smaller than symbol size).

Figure 5

Extrapolation of a finite-size lattice to the continuum limit in few-body problems. Panel (a) shows exact diagonalization results for the two-body problem at $ln\left(a{k}_F\right)=0.5$, whereas panel (b) shows QMC solutions for the four-body problem at $ln\left(a{k}_F\right)=0.0$. In each case, results are obtained for both the Hubbard and the quadratic dispersions. A fourth-order polynomial function in $1/L$ fits both dispersions well, and the extrapolated results in the continuum limit agree well with each other. The insets indicate that the coefficients on $1/L$ are negligible in both cases.