Quantum fluctuations in the BCS-BEC crossover of two-dimensional Fermi gases

We present a theoretical study of the ground state of the BCS-BEC crossover in dilute two-dimensional Fermi gases. While the mean-field theory provides a simple and analytical equation of state, the pressure is equal to that of a noninteracting Fermi gas in the entire BCS-BEC crossover, which is not consistent with the features of a weakly interacting Bose condensate in the BEC limit and a weakly interacting Fermi liquid in the BCS limit. The inadequacy of the two-dimensional mean-field theory indicates that the quantum fluctuations are much more pronounced than those in three dimensions. In this work, we show that the inclusion of the Gaussian quantum fluctuations naturally recovers the above features in both the BEC and the BCS limits. In the BEC limit, the missing logarithmic dependence on the boson chemical potential is recovered by the quantum fluctuations. Near the quantum phase transition from the vacuum to the BEC phase, we compare our equation of state with the known grand canonical equation of state of two-dimensional Bose gases and determine the ratio of the composite boson scattering length $a_B$ to the fermion scattering length $a_{2\mathrm{D}}$. We find $a_B\simeq 0.56a_{2\mathrm{D}}$, in good agreement with the exact four-body calculation. We compare our equation of state in the BCS-BEC crossover with recent results from the quantum Monte Carlo simulations and the experimental measurements and find good agreements.

Figure 1

Curves of the function $R\left(\nu \right)$ for various values of the gas parameter $\eta =ln\left(k_F{a}_{2\mathrm{D}}\right)$. For comparison, we show the mean-field prediction $R_{\mathrm{MF}}\left(\nu \right)=-{\nu }^2+2\nu$ by the dashed line.

Figure 2

Evolution of the chemical potential $\mu$ in the BCS-BEC crossover. We show the quantity $\nu =(\mu +{\varepsilon }_B/2)/{\varepsilon }_F$ as a function of the gas parameter $\eta =ln\left(k_F{a}_{2\mathrm{D}}\right)$. The mean-field prediction is represented by the dashed line.

Figure 3

The order parameter or pairing gap $\mathrm{\Delta }$ (divided by ${\varepsilon }_F$) as a function of the gas parameter $\eta =ln\left(k_F{a}_{2\mathrm{D}}\right)$. The dashed line is the mean-field prediction. The dashed-dotted line corresponds to the prediction with the GMB effect in the weak coupling regime $\left(\eta >2\right)$.

Figure 4

Evolution of the energy density $E$ in the BCS-BEC crossover. The quantity $R=(E+n{\varepsilon }_B/2)/E_{\mathrm{FG}}$ is shown as a function of the gas parameter $\eta =ln\left(k_F{a}_{2\mathrm{D}}\right)$. The dashed line represents the mean-field prediction. The (blue) circles and (red) squares represent the predictions from the diffusion Monte Carlo simulation [46] and the auxiliary-field Monte Carlo simulation [47], respectively. The bottom-left dashed (green) line represents the Bogoliubov EOS of a weakly interacting two-dimensional Bose gas with the boson scattering length $a_B=0.56a_{2\mathrm{D}}$ [see Eq. (74)]. The top-right dashed (purple) line shows the EOS of a weakly interacting two-dimensional Fermi gas [see Eq. (76)].

Figure 5

Evolution of the pressure $P$ in the BCS-BEC crossover. $P/P_{\mathrm{FG}}$ as a function of the gas parameter $\eta =ln\left(k_F{a}_{2\mathrm{D}}\right)$ is shown. The mean-field prediction is represented by the dashed line. The (blue) circles with error bars are the experimental data taken from [37].

Figure 6

Evolution of the contact $C$ in the BCS-BEC crossover. $(C-C_{\mathrm{MF}})/k_F^4$ as a function of the gas parameter $\eta =ln\left(k_F{a}_{2\mathrm{D}}\right)$, where $C_{\mathrm{MF}}=\alpha /2$ is the mean-field prediction. The dashed (blue) line and the dash-dotted (red) line represent the predictions from the diffusion Monte Carlo simulation [46] and the auxiliary-field Monte Carlo simulation [47], respectively.

Figure 7

The quantity $f\left(\zeta \right)+ln\left(\zeta \right)$ as a function of $\zeta$ in the range ${10}^{-6}<\zeta <{10}^{-2}$. In the calculation we use Eq. (A16) for $f\left(\zeta \right)$.