Properties of strongly paired fermionic condensates

We study a gas of fermions undergoing a wide resonance $s$-wave BCS–Bose-Einstein-condensate (BEC) crossover, in the BEC regime at zero temperature. We calculate the chemical potential and the speed of sound of this Bose-Einstein-condensed gas, as well as the condensate depletion, in the low-density approximation. We discuss how higher-order terms in the low-density expansion can be constructed. We demonstrate that the standard BCS-BEC gap equation is invalid in the BEC regime and is inconsistent with the results obtained here. The low-density approximation we employ breaks down in the intermediate BCS-BEC crossover region. Hence our theory is unable to predict how the chemical potential and the speed of sound evolve once the interactions are tuned towards the BCS regime. As a part of our theory, we derive the well-known result for the bosonic scattering length diagrammatically and check that there are no bound states of two bosons.

Figure 1

The lowest-order contribution in the low-density approximation to the anomalous fermion self-energy, ${\Sigma }_a$. The dashed line is a condensate line.

Figure 2

The next-to-leading-order contribution in the low-density approximation to the normal fermion self-energy.

Figure 3

The normal (top) and anomalous (bottom) self-energy contributions to the bosonic propagator. The wavy lines are bosonic propagators.

Figure 4

A possible higher-order contribution to the bosonic self-energy.

Figure 5

The simplest diagram contributing to the gap equation (36).

Figure 6

The simplest diagram contributing to the gap equation at order ${\Delta }_0^3$.

Figure 7

The sum of all diagrams contributing to the gap equation at order ${\Delta }_0^3$.

Figure 8

Diagrams contributing to the $t$ matrix of fermion-boson scattering.

Figure 9

The integral equation for the $t$ matrix of fermion-boson scattering.

Figure 10

The integral equation for the $t$ matrix of boson-boson scattering. $\Gamma$ is the sum of all two boson irreducible diagrams.

Figure 11

The simplest diagrams contributing to the two-boson irreducible diagram $\Gamma$. The first diagram on the right-hand side corresponds to ${\Gamma }^{\left(0\right)}$ defined in Eq. (C11).

Figure 12

Integral equation for the two-boson irreducible diagram with two bosons coming in and one boson and two fermions going out.

Figure 13

The bubble diagrams of fermion scattering.

Figure 14

The bosonic propagator renormalized by fermion loops. The thin wavy lines are unrenormalized boson propagators.

Figure 15

The original and rotated contours of frequency integration used in Eq. (42). The crosses correspond to poles of the bosonic propagators which approach the imaginary axis as $k,q\to 0$. The dots and black lines towards $\pm \infty$ are branch cuts of the bosonic propagators and the grey boxes are possible nonanalytic structure of $\Gamma (q,q_0;p,p_0)$. These do not come closer to the imaginary axis than $\pm b$.