Critical Temperature Curve in BEC-BCS Crossover

The strongly correlated regime of the crossover from Bardeen-Cooper-Schrieffer pairing to Bose-Einstein condensation can be realized by diluting a system of two-component fermions with a short-range attractive interaction. We investigate this system via a novel continuous-space-time diagrammatic determinant Monte Carlo method and determine the universal curve $T_c/{\epsilon }_F$ for the transition temperature between the normal and the superfluid states as a function of the scattering length with the maximum on the Bose-Einstein condensation side. At unitarity, we confirm that $T_c/{\epsilon }_F=0.152(7)$.

Figure 1

The universal results for the critical temperature in the units of the Fermi energy plotted versus $\kappa =1/k_{F}a$ (circles). The solid lines for negative and positive $\kappa$ represent the limiting behavior of the BCS theory (with the Gorkov-Melik-Barkhudarov correction) and the ideal BEC, respectively. For reference, we also plot nonuniversal results for hard-sphere (triangles) and soft-sphere (squares) bosons (Ref. 23).

Figure 2

Finite-size analysis for $c=0.83$, $\mu =0.36$, corresponding to $U\approx -7.519$ and $a\approx 1.52$ (in the units of $m=1/2$, $l_0=1$) yielding ${\beta }_c=1.290(8)$. The error bars are 1 standard deviation and were calculated using the blocking method. Inset: The thermodynamic limit value of the number density is obtained via a linear fit of $n(L)$ vs $1/L$. In this case, $n{l}_0^3=0.119(2)$, which results in $\zeta =0.492(3)$, $T_c^{(\zeta )}/{\epsilon }_F^{(\zeta )}=0.335(6)$, and $\kappa =0.432(3)$.

Figure 3

The extrapolation of the simulation results to the universal limit $\zeta =n^{1/3}{l}_0\to 0$. The procedure yields $T_c/{\epsilon }_F=0.152(9)$, 0.202(9), and 0.252(15) for $\kappa =1/k_{F}a=0$ (squares), 0.217(2) (circles), and 0.474(8) (triangles), correspondingly. For comparison, we also plot our results for the Hubbard model (open squares) adapted from Ref. 14. The estimate of Ref. 16 at $\kappa =0$ (obtained for finite $\zeta \approx 0.93$) is shown by the diamond. Solid lines are linear fits.