Collective mode of homogeneous superfluid Fermi gases in the BEC-BCS crossover

We perform a detailed study of the collective mode across the whole crossover from the Bose-Einstein condensate (BEC)-to the BCS regime in fermionic gases at zero temperature, covering the whole range of energy beyond the linear regime. This is done on the basis of the dynamical BCS model. We recover first the results of the linear regime in a simple form. Then specific attention is paid to the nonlinear part of the dispersion relation and its interplay with the continuum of single-fermionic excitations. In particular we consider in detail the merging of the collective mode into the continuum of single-fermionic excitations. This occurs not only on the BCS side of the crossover, but also slightly beyond unitarity on the BEC side. Another remarkable feature is the very linear behavior of the dispersion relation in the vicinity of unitarity almost up to merging with the continuum. Finally, while on the BEC side the mode is quite analogous to the Bogoliubov mode, a difference appears at high wave vectors. On the basis of our results we determine the Landau critical velocity in the BEC-BCS crossover which is found to be largest close to unitarity. Our investigation has revealed interesting qualitative features which deserve experimental exploration as well as further theoretical studies by more sophisticated means.

Figure 1

Threshold ${\omega }_{\mathit{th}}$ as a function of wave vector $q$ for, from bottom to top, $1∕k_{F}a=-1$, $-0.5$, 0 (unitarity), 0.553 $(\mu =0)$, 1, and 2.

Figure 2

Dispersion relation $\omega ∕E_F$ of the collective mode as a function of $q∕k_F$ for $1∕k_{F}a=-1$ (lower thick line) and $-0.5$ (upper thick line). The location (see Sec. 5A) of the threshold for pair breaking is given in each case by the thinner line.

Figure 3

Wave vector $q_m$, in units of $k_F$, at which the collective mode dispersion relation meets the excitation continuum, as a function of $1∕k_{F}a$ (note that the $q_m$ scale is logarithmic). The dashed line corresponds to $q=2\sqrt{2m\mu }$. It goes to $q∕k_F=2$ when $a\to -\infty$.

Figure 4

Dispersion relation $\omega ∕E_F$ of the collective mode as a function of $q∕k_F$ at unitarity (thick line). The location (see Sec. 5A) of the threshold for pair breaking is given by the thinner line. The collective mode merges into the continuum for $q∕k_F=1.76$, as can be seen from Fig. 3.

Figure 5

Dispersion relation $\omega ∕E_F$ of the collective mode (thick line) as a function of $q∕k_F$ for $\mu =0$, corresponding to $1∕k_{F}a=0.553$. The location (see Sec. 5A) of the threshold for pair breaking is given by the thinner line. The dashed line indicates the result obtained from the Bogoliubov formula for the collective mode for this value of $\mu$.

Figure 6

Dispersion relation $\omega ∕E_F$ of the collective mode as a function of $q∕k_F$ for $1∕k_{F}a=1$ (thick line), for which $\Delta ∕\mid \mu \mid =1.66$. The threshold for pair breaking is given by the thinner line. The dashed line indicates the result obtained from the Bogoliubov formula.

Figure 7

Dispersion relation $\omega ∕E_F$ of the collective mode as a function of $q∕k_F$ for $1∕k_{F}a=2$ (thick line), for which $\Delta ∕\mid \mu \mid =0.474$. The threshold for pair breaking is given by the thinner line. The dashed line indicates the result obtained from the Bogoliubov formula.

Figure 8

Critical velocity $v_c$ (full line) and sound velocity $c_s$ (dashed line) as a function of $1∕k_{F}a$. The inset gives $k_F\xi =v_F∕v_c$.

Figure 9

Two vertices ${\Gamma }_{11}\left(q\right)$ and ${\Gamma }_{12}\left(q\right)$ in superconductive state. $p=(\overline{p},\omega )$ and $q=(\Omega ,\overline{q})$ are four-momenta.

Figure 10

The Bethe-Salpeter equation for the vertex ${\Gamma }_{11}$.

Figure 11

The Bethe-Salppeter equation for ${\Gamma }_{12}$.