Diffusion Monte Carlo study of strongly interacting two-dimensional Fermi gases

Ultracold atomic Fermi gases have been a popular topic of research, with attention being paid recently to two-dimensional (2D) gases. In this work, we perform $T=0$ ab initio diffusion Monte Carlo calculations for a strongly interacting two-component Fermi gas confined to two dimensions. We first go over finite-size systems and the connection to the thermodynamic limit. After that, we illustrate pertinent 2D scattering physics and properties of the wave function. We then show energy results for the strong-coupling crossover, in between the Bose-Einstein condensation (BEC) and Bardeen-Cooper-Schrieffer (BCS) regimes. Our energy results for the BEC-BCS crossover are parametrized to produce an equation of state, which is used to determine Tan's contact. We carry out a detailed comparison with other microscopic results. Finally, we calculate the pairing gap for a range of interaction strengths in the strong coupling regime, following from variationally optimized many-body wave functions.

Figure 1

Error associated with finite-size periodic systems, defined in units of $E_{\mathrm{FG}}$ as the absolute value of the energy difference between $N$ particles in a finite-size box and the TL system where $N\to \infty$ and $A\to \infty$. The largest error is seen for small $N$ where dotted lines are drawn to guide the eye. The inset displays this range on a linear scale. For large values of $N$ the error approaches zero as marked by the dashed line.

Figure 2

Wave function solutions to Eq. (2) in the limit of zero scattering energy ($k^2\to 0$) for the modified Pöschl-Teller potential. Parameters $\mu$ and $v_0$ in $V\left(r\right)$ are varied to describe a range of interaction strengths $\eta =ln\left(k_F{a}_{2\mathrm{D}}\right)$, labeled by line thickness. The scattering length is defined as the point where the wave function crosses the $r$ axis, which is marked with a dashed line. The inset shows the effective range integrand in Eq. (4) for two interaction strengths, $\eta =-1$ (thick line) and $\eta =3$ (thin line).

Figure 3

The relationship between interaction strength $\eta =ln\left(k_F{a}_{2\mathrm{D}}\right)$ and $v_0$ of the modified Pöschl-Teller potential $V\left(r\right)$ is shown in the top panel. The effective range is maintained constant at $k_F{r}_e=0.006$ by tuning $\mu$ and $v_0$ in unison. The absolute value of the binding energy per particle ${\epsilon }_b/2$ is plotted in the bottom panel. A dramatic increase in $|{\epsilon }_b|$ is seen as paired particles form increasingly stronger bound states.

Figure 4

Variationally optimized pairing functions for various interaction strengths. These are labeled by line thickness, i.e., the thickest line is for $\eta =-1$ and the thinnest line is for $\eta =3$. The top panel plots $\varphi \left(\mathbf{r}\right)$ through a path parallel to the box sides, corresponding to the (1,0) or equivalently the (0,1) direction. A path along the (1,1) direction is taken for $\varphi \left(\mathbf{r}\right)$ in the bottom panel plot. The dashed lines are included to show node locations, where $\varphi \left(\mathbf{r}\right)=0$.

Figure 5

Total energy per particle of the 2D strongly interacting Fermi gas. Thermodynamic-limit-extrapolated QMC results are plotted with a solid line (blue). At this energy resolution, all QMC results are in good agreement. The mean-field result is shown as a dashed line (black) for comparison. This is expected to be reasonably accurate for $\left|\eta \right|\gg 1$. Half of the binding energy per particle is plotted with a dotted line (orange). There is logarithmically small binding energy in the BCS limit and $|{\epsilon }_b|$ becomes very large in the BEC limit.

Figure 6

Our DMC energy results for the strongly interacting Fermi gas in the BEC-BCS crossover. The binding energy per particle has been subtracted and the units are $E_{\mathrm{FG}}={\hslash }^2{k}_F^2/4m$. Our DMC calculations for the Jastrow-BCS and Jastrow-Slater wave functions are shown with circles (blue) and triangles (purple), respectively. With decreasing $\eta$, we see increasingly significant improvements over the Slater energy results by using the optimized BCS wave function.

Figure 7

The energy-per-particle results for various ab initio QMC methods are compared. Our optimized Jastrow-BCS wave function results are shown with circles (blue). These closely follow the AFQMC energy results of Shi et al [21], plotted as diamonds (green). Previous DMC energy results of Bertaina and Giorgini [20] are plotted as squares (red); since DMC is variational, the new results show a notable improvement. Note that the error bars, which are provided for all data, are comparable to the symbol size (or larger) only for previous DMC results.

Figure 8

The contact parameter as determined by the EOS derivative. The result from our DMC method is plotted as the solid line (blue) and we compare to other QMC results. We show previous DMC results [20] with a dotted line (red), and AFQMC results [21] with a dashed line (green). Qualitatively, the contact reaches a maximum on the BCS side of the strong coupling regime and decays more rapidly on the BEC side of the crossover.

Figure 9

The pairing gap in the strong coupling crossover. The binding energy per particle has been subtracted and the units are ${\epsilon }_F={\hslash }^2{k}_F^2/2m=2E_{\mathrm{FG}}$. Our results are shown with circular symbols (blue) and the mean-field prediction is shown with a solid line (black). Our calculated chemical potential $\mu$ becomes zero for the interaction strength $\eta \approx 0.65$. This location is marked with a dotted line (blue). Mean-field theory predicts that $\mu =0$ at $\eta \approx 0.12$.