Precision Many-Body Study of the Berezinskii-Kosterlitz-Thouless Transition and Temperature-Dependent Properties in the Two-Dimensional Fermi Gas

We perform large-scale, numerically exact calculations on the two-dimensional interacting Fermi gas with a contact attraction. Reaching much larger lattice sizes and lower temperatures than previously possible, we determine systematically the finite-temperature phase diagram of the Berezinskii-Kosterlitz-Thouless (BKT) transitions for interaction strengths ranging from BCS to crossover to BEC regimes. The evolutions of the pairing wave functions and the fermion and Cooper pair momentum distributions with temperature are accurately characterized. In the crossover regime, we find that the contact has a nonmonotonic temperature dependence, first increasing as temperature is lowered, and then showing a slight decline below the BKT transition temperature to approach the ground-state value from above.

Figure 1

BKT transition temperatures and phase diagram of the 2D interacting Fermi gas. Empty red circles show our exact results for a finite system of $L=45$, $N_e=58$ with quadratic dispersion. Filled red circles show finite-size scaling results to the continuum and thermodynamic limits ($L=\infty$, $N_e=\infty$) for a subset of the interaction strengths. The solid red line connecting these are the results of interpolation, with the shaded band indicating statistical error bars based on both sets of results. For comparison, results are also shown from BCS mean-field theory and its improvement on the BCS side (Petrov et al. [36]), the weakly interacting Bose gas on the BEC side (Petrov et al. [36]), one-loop Gaussian fluctuation theory (Bighin et al. [36]), Luttinger-Ward theory (Bauer et al. [40]), Gaussian pair fluctuation theory (Mulkerin et al. [60]), and experimental measurements (Ries et al. [24] and Sobirey et al. [34]).

Figure 2

The singlet pairing wave function in reciprocal space versus $|\mathbf{k}|/k_F$ (main plots), and its Fourier transform, the real space pairing wave function plotted versus $k_F|\mathbf{r}|$ along the diagonal line ($x=y$) (insets). Three temperatures are displayed, from high to low, in the three panels from left to right: (a) $T/T_F=0.25$, (b) $T/T_F=0.125$, and (c) $T/T_F=0.0625$. In each panel five $\log (k_{F}a)$ values are shown, as indicated by the legends on top, to span the entire range of the interaction strength. The statistical uncertainties are smaller than the symbol size in all cases and are not shown. In the main plots, the Fermi surface is indicated by a vertical dot-dashed line. These calculations are performed with $L=45$ and $N_e=58$.

Figure 3

The momentum distributions for Cooper pairs $n_{\mathbf{Q}}$ versus $|\mathbf{Q}|/k_F$ (main plot) and fermions $n(\mathbf{k})$ versus $|\mathbf{k}|/k_F$ (insets), for three interaction strengths: (a) $\log (k_{F}a)=-1.0$ in BEC regime, (b) $\log (k_{F}a)=+0.5$ in crossover regime, and (c) $\log (k_{F}a)=+3.5$ in BCS regime, with temperatures $T/T_F=0.0625\sim 0.5$. These calculations are performed with $L=45$ and $N_e=58$, and the corresponding $T_{\mathrm{BKT}}/T_F$ values from Fig. 1 are indicated.

Figure 4

The contact density $C/k_F^4$ as a function of temperature in the crossover regime [$\log (k_{F}a)=+0.50$]. The red open circles represent the $T>0$ results from our AFQMC calculations. Results at ground state (green solid circle, from Ref. [45]) and from the BCS mean-field theory (dark yellow line for the continuum limit, magenta solid squares for $L=45$, $N_e=58$) are also included for comparisons. The inset is an enlargement of the contact density in $T/T_F\in [0,0.5]$ containing our AFQMC results from different systems up to $L=75$, and the previous lattice QMC results with $L=19$ [47].