Modified Fermi sphere, pairing gap, and critical temperature for the BCS-BEC crossover

We investigate the phase diagram of two-component fermions in the BCS-BEC (Bose-Einstein condensate) crossover. Using functional renormalization-group equations we calculate the effect of quantum fluctuations on the fermionic self-energy parametrized by a wave-function renormalization, an effective Fermi radius, and the gap. This allows us to follow the modifications of the Fermi surface and the dispersion relation for fermionic excitations throughout the whole crossover region. We also determine the critical temperature of the second-order phase transition to superfluidity. Our results are in agreement with BCS theory including Gorkov’s correction for a small negative scattering length $a$ and with an interacting Bose gas for a small positive $a$. At the unitarity point the result for the gap at zero temperature agrees well with quantum Monte Carlo simulations, while the critical temperature differs.

Figure 1

Box diagram for the flow of the four-fermion interaction.

Figure 2

Flow of the fermionic wave-function renormalization $Z_{\psi }$ at $T=0$. The solid line is at the unitarity point ($c^{-1}=0$). The long-dashed line corresponds to the BCS side at $c^{-1}=-1$, and the short-dashed line to the BEC side at $c^{-1}=1$.

Figure 3

Fermionic wave-function renormalization $Z_{\psi }$ at the macroscopic scale $k=0$ as a function of the crossover parameter $({\mathit{ak}}_F){}^{-1}$.

Figure 4

Effective Fermi radius $r_F/k_F$ as a function of the crossover parameter $({\mathit{ak}}_F){}^{-1}$ for vanishing temperature (solid line). We find that the Fermi sphere vanishes approximately at the point with $({\mathit{ak}}_F){}^{-1}\approx 0.6$. For comparison we also give the effective Fermi radius in an approximation where we omitted the contribution of the atom-dimer vertex ${\lambda }_{\varphi \psi }$ (dotted line).

Figure 5

Positive branch of the dispersion relation $\omega (q)$ in units of the Fermi energy as given in Eq. (14) for $c^{-1}=-1$ (long-dashed line), $c^{-1}=0$ (solid line), and $c^{-1}=1$ (short-dashed line). The gap and the Fermi radius are evaluated at the macroscopic scale $k=0$.

Figure 6

Gap in units of the Fermi energy $\Delta /E_F$ as a function of $({\mathit{ak}}_F){}^{-1}$ (solid line). For comparison, we also plot the result found by Gorkov and Melik-Bakhudarov (dashed line; left) and extrapolate this to the unitarity point $({\mathit{ak}}_F){}^{-1}=0$, where ${\Delta }_{\mathrm{GMB}}/E_F=0.49$.

Figure 7

Critical temperature $T_c/T_F$ in units of the Fermi temperature as a function of the crossover parameter $({\mathit{ak}}_F){}^{-1}$ (solid line). For comparison, we plot the BCS result with Gorkov’s correction (short-dashed line; left). On the BEC side we show the critical temperature for a gas of interacting bosonic molecules according to [42, 43, 44] (long-dashed line; right). The three filled circles close to and at unitarity show the QMC results of Burovski et al. [4].