Pauli-blocking effects and Cooper triples in three-component Fermi gases

We investigate the effect of Pauli blocking on universal two- and three-body states in the medium. Their corresponding energies are extracted from the poles of two- and three-body in-medium scattering amplitudes. Compared to the vacuum, the binding of dimer and trimer states is reduced by the medium effects. In two-body scattering, the well-known physics of Cooper pairs is recovered. In the three-body sector, we find a class of positive energy poles which can be interpreted as Cooper triples.

Figure 1

Bubble sum for the full interacting dimer propagator (thick line). The double lines correspond to the bare dimer propagator and the single lines indicate particle propagators.

Figure 2

The energy $E$ of the dimer pole at $P=0$ plotted against the inverse scattering length $1/a_k$ for a Fermi momentum $k_F=0.7/l_0$ (solid line) and $k_F=1/l_0$ (dash-dotted line). In addition, four selected points are marked I, II, III, and IV on the solid line. For comparison, the vacuum pole energy is shown by the dashed line. In the inset, the dimer pole energy is displayed as a function of the total momentrum $P$ for $k_F=0.7/l_0$ and $a=l_0$. Curves are as above.

Figure 3

(a) The energy $E$ depicted as function of the total momentum $P$ for $a_k=-3l_0$ and $k_F=0.7/l_0$ (solid line). The vertical dotted line gives $E=k_F^2$. (b) Energy $E$ plotted against the Fermi momentum $k_F$ for $a_k=2l_0$ (solid), $4l_0$ (dashed), $6l_0$ (dashed dotted); $P=0$.

Figure 4

The energy of the poles plotted as a function of the total momentum $P$ for $a=2l_0$ [panel (a)] and $a=3l_0$ [panel (b)] and $k_F=0.7l_0$ (solid line). The dashed line shows the vacuum poles and the dash-dotted line gives the kinetic energy $P^2/4$. The horizontal dotted line in (b) gives $k_F^2$ and the vertical dotted line gives $2k_F$.

Figure 5

Feynman diagrams for the fermion-dimer scattering amplitude for zero total momentum. Momenta $\mathbf{p},\mathbf{q},\mathbf{k}$ and fermion indices $i,j,r,r^{\prime }$ are assigned as in Eq. (17).

Figure 6

Binding energy $E_B$ of two states depicted in dependence of $k_F$: $a_k=l_0$ for $k=0$, 1, 2, and $\Lambda =250/l_0$.

Figure 7

(a) Energy of three-body poles plotted against $k_F$ for $a_0=-1.0l_0$ (dashed), $a_0=-1.25l_0$ (dash-dotted), and $a_0=-1.5l_0$ (solid); $\Lambda =150/l_0$ and the remaining scattering lengths are $a_1=-2l_0$ and $a_2=-2.5l_0$. (b) The pole energy is shown as a function of $1/a_0$ for $\Lambda =150/l_0$ (dash-dotted), $\Lambda =160/l_0$ (solid), and $\Lambda =170/l_0$ (dashed); $k_F=0.7/l_0$ and the remaining scattering lengths are $a_1=-l_0$ and $a_2=-0.99l_0$. The horizontal line gives $E=1.5k_F^2$.

Figure 8

Configurations in the Fermi sphere for two (left) and three particles (right) with total momentum $P=0$.

Figure 9

$\Delta E$ is plotted against the inverse scattering length $1/a_0$ of three-body (solid line) and two-body (dashed line) poles; $\Lambda =160/l_0$, $k_F=0.7/l_0$ and the constant scattering lengths are $a_1=-l_0$ and $a_2=-0.99l_0$. The horizontal dotted line shows $\Delta E$ for the constant scattering length $a_2=-0.99l_0$.