Emergence of a Pseudogap in the BCS-BEC Crossover

Strongly correlated Fermi systems with pairing interactions become superfluid below a critical temperature $T_c$. The extent to which such pairing correlations alter the behavior of the liquid at temperatures $T>T_c$ is a subtle issue that remains an area of debate, in particular regarding the appearance of the so-called pseudogap in the BCS-BEC crossover of unpolarized spin-$1/2$ nonrelativistic matter. To shed light on this, we extract several quantities of crucial importance at and around the unitary limit, namely, the odd-even staggering of the total energy, the spin susceptibility, the pairing correlation function, the condensate fraction, and the critical temperature $T_c$, using a nonperturbative, constrained-ensemble quantum Monte Carlo algorithm.

Figure 1

Left: The condensate fraction $\alpha$ as a function of temperature at different scattering lengths; at a fixed temperature, $\alpha$ increases toward the BEC limit. At all scattering lengths, the condensate fraction tends to decrease with an increase in lattice size. At $1/(k_{F}a)=0.2$, Astrakharchik et al. [56] estimated the zero-temperature condensate fraction as $\alpha (T=0)\approx 0.65$. Right (top): characteristic temperatures in the BCS-BEC crossover; $T_c$ is the superfluid critical temperature; $T_s$ is a lower bound on the temperature at which the spin susceptibility peaks; and $T^{*}$ is the temperature at which the pairing gap disappears. Our estimate for $T_c$ agrees with the experimental value from Ku et al. [61]. Right (bottom): $\alpha$ at unitarity; error bars for our results are typically within the marker size. Also shown: the experimental results of Ku et al. [61], Sanner et al. [62], and the previous AFQMC studies of Bulgac et al. [57] (BDM) and Jensen et al. [63] (JGA). We also plot zero-temperature results by Astrakharchik et al. [56] (ABCG) and He et al. [64] (HLLL). The large condensate fraction measured by Sanner et al. is relevant to our comparison of the spin susceptibility in Fig. 2. The JGA estimates, derived from the maximum eigenvalue of the two-body density matrix, are closer to the experimental results especially at high temperature, whereas the finite-size scaling of our results, derived from the asymptotic values of $h(k_{F}r)$, yields more accurate estimates of the critical temperature $T_c$. The discrepancy between our results and BDM, which are also derived from the asymptotic behavior of $h(k_{F}r)$, support the argument of Jensen et al. [63] that the difference is due to the BDM spherical momentum cutoff. $T_c$ estimates are compatible with previous estimates by Burovski et al. [58] and Bulgac et al. [57]. Estimates for $T^{*}$ are compatible with previous results by Magierski et al. [65].

Figure 2

Top: AFQMC results for the spin susceptibility ${\chi }_S$ at four different scattering lengths, scaled by its zero-temperature noninteracting counterpart, ${\chi }_0=3N/(2V{ϵ}_F)$. Bottom: At unitarity, we compare our result to two previous AFQMC studies: Jensen et al. [60] (JGA) and Wlazłowski et al. [67] (WMDBR); the experimental result of Sanner et al. [62]; a self-consistent Luttinger-Ward result (EH) [68]; the normal Fermi liquid theory prediction; and a self-consistent NSR result (PDU) [69].

Figure 3

Top: AFQMC results for ${\mathrm{\Delta }}_E$ at four different scattering lengths, scaled by the Fermi energy ${ϵ}_F$. We incorporate results for all lattices with $N_x\ge 8$ using a regression technique described in the Supplemental Material [9]. Bottom: At unitarity, we compare our results to the AFQMC results of Jensen et al. [60] (JGA) and Magierski et al. [65] (MWB); the zero-temperature QMC prediction of Carlson and Reddy [78] (CR); and the experimental results of Hoinka et al. [76] and Schirotzek et al. [77].

Figure 4

AFQMC results for the temperature-dependent Bertsch parameter, $\xi [T/{ϵ}_F,1/(k_{F}a)]$ at four different scattering lengths. We incorporate results for all lattices with $N_x\ge 8$ using a regression technique described in the Supplemental Material [9]. At unitarity, we compare our results to the experimental measurements of Ku et al. [61] and the high-precision AFQMC results of Drut et al. [79] (DLWM). We also show the zero temperature predictions of Carlson et al. [81] (CGSZ) at unitarity and of Astrakharchik et al. [82] (ABCG) at all scattering lengths.