Effective-range dependence of two-dimensional Fermi gases

The Feshbach resonance provides precise control over the scattering length and effective range of interactions between ultracold atoms. We propose the ultratransferable pseudopotential to model effective interaction ranges $-1.5\le k_{\mathrm{F}}^2{R}_{\mathrm{eff}}^2\le 0$, where $R_{\mathrm{eff}}$ is the effective range and $k_{\text{F}}$ is the Fermi wave vector, describing narrow to broad Feshbach resonances. We develop a mean-field treatment and exploit the pseudopotential to perform a variational and diffusion Monte Carlo study of the ground state of the two-dimensional Fermi gas, reporting on the ground-state energy, contact, condensate fraction, momentum distribution, and pair-correlation functions as a function of the effective interaction range across the BEC-BCS crossover. The limit $k_{\mathrm{F}}^2{R}_{\mathrm{eff}}^2\to -\infty$ is a gas of bosons with zero binding energy, whereas $ln\left(k_{\mathrm{F}}a\right)\to -\infty$ corresponds to noninteracting bosons with infinite binding energy.

Figure 1

Scattering phase shift $\delta \left(k\right)$ for $a=1$ and $R_{\mathrm{eff}}^2=\{-1,0,1\}$ (all lengths are in units of inverse momentum). For $R_{\mathrm{eff}}^2=-1$ 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

Plot of the potential well (blue), the UTP for the contact interaction ${\left(k_{\mathrm{F}}{R}_{\mathrm{eff}}\right)}^2=0$ (solid purple), and the UTP for finite effective range ${\left(k_{\mathrm{F}}{R}_{\mathrm{eff}}\right)}^2=-1$ (dashed purple), normalized by the reciprocal Fermi energy $E_{\mathrm{F}}$ as a function of the dimensionless radius. All potentials are calibrated for interaction strength $ln\left(k_{\mathrm{F}}a\right)=0$.

Figure 3

(Top) Difference in absolute value of the bound-state energy of the potential and the exact bound-state energy of the potential well (blue), the UTP for the contact interaction ${\left(k_{\mathrm{F}}{R}_{\mathrm{eff}}\right)}^2=0$ (solid purple), and the UTP for finite effective range ${\left(k_{\mathrm{F}}{R}_{\mathrm{eff}}\right)}^2=-1$ (dashed purple). (Bottom) Root-mean-square (rms) scattering phase-shift error of the same potentials, with $k$ averaged over the interval from zero to the Fermi momentum $k_{\mathrm{F}}$.

Figure 4

Ground-state energy per particle minus half the two-body binding energy, divided by the energy per particle in a noninteracting gas, as a function of the interaction parameter $ln\left(k_{\mathrm{F}}a\right)$ across the BEC-BCS crossover. Next to our result we show the results from Refs. [21, 22, 23].

Figure 5

Contact minus the contact contribution from the molecular bound state, normalized by the fourth power of the Fermi wave vector. For comparison we also show the results from Refs. [21, 22, 23].

Figure 6

Condensate fraction as a function of the interaction parameter $ln\left(k_{\mathrm{F}}a\right)$. We show the VMC, DMC and extrapolated estimates, along with the auxiliary-field quantum Monte Carlo result from Shi et al. [22]. Also shown are the perturbative Bogoliubov theory for the BEC limit [56] and the mean-field prediction [57].

Figure 7

DMC ground-state energy per particle in units of that of a noninteracting gas as a function of the dimensionless effective range squared $k_{\mathrm{F}}^2{R}_{\mathrm{eff}}^2$ shown by the solid purple line. The mean-field result is shown by the dashed green line.

Figure 8

Condensate fraction as a function of the dimensionless effective range squared. We show the extrapolated result, as well as the bare VMC and DMC estimates.

Figure 9

(Top) Extrapolated data for the momentum distribution $n\left(k\right)$ for $k_{\mathrm{F}}^2{R}_{\mathrm{eff}}^2\in \{0,-1\}$. The inset additionally shows DMC, VMC, and extrapolated results separately for $k_{\mathrm{F}}^2{R}_{\mathrm{eff}}^2=0$. (Bottom) Tail of the momentum distribution on logarithmic axes, with the lines indicating $n\left(k\right)\sim 1/k^4$.

Figure 10

Pair-correlation function for opposite (top) and equal (bottom) spins for $k_{\mathrm{F}}^2{R}_{\mathrm{eff}}^2\in \{-1,0\}$. The noninteracting correlation function for fermions is indicated by the gray dashed line. The insets show DMC, VMC, and extrapolated data separately.

Figure 11

Variation of the dimensionless ground-state energy per particle with DMC time step and walker population with interaction parameter $\eta =ln\left(k_{\mathrm{F}}a\right)$ in the zero-range limit $k_{\mathrm{F}}^2{R}_{\mathrm{eff}}^2=0$. The straight lines show a weighted least-squares fit to the data points.

Figure 12

Variation of the dimensionless ground-state energy per particle with the difference in the noninteracting kinetic energy for a finite and infinite system (top) and the number of particles (bottom) for various interaction parameters $\eta =ln\left(k_{\mathrm{F}}a\right)$ in the zero effective-range limit $k_{\mathrm{F}}^2{R}_{\mathrm{eff}}^2=0$. The straight lines show a weighted least-squares fit to the data points.