Virial expansion with Feynman diagrams

We present a field theoretic method for the calculation of the second and third virial coefficients $b_2$ and $b_3$ of two-species fermions interacting via a contact interaction. The method is mostly analytic. We find a closed expression for $b_3$ in terms of the two- and three-body $T$ matrices. We recover numerically, at unitarity, and also in the whole Bose-Einstein-condensate–BCS crossover, previous numerical results for the third virial coefficient $b_3$.

Figure 1

The lowest-order diagram.

Figure 2

The two diagrams contributing to the virial coefficient $b_2$. (a) Diagrammatic expansion of the Fermi-Dirac distribution. (b) Beth-Uhlenbeck contribution.

Figure 3

The two diagrams contributing to $b_3$ with one $G^{(0,2)}$ and one $G^{(0,1)}$ (see text).

Figure 4

The diagram contributing to $\delta n^{(3,3,1)}$.

Figure 5

The general structure of a diagram contributing to $\delta n^{(3,3,2)}$.

Figure 6

The six diagrams contributing to $\delta n^{(3,3,2)}$ (see text).

Figure 7

The third virial coefficient $b_3-3^{-5/2}$ as a function of the dimensionless parameter ${\Lambda }_T/a$.

Figure 8

Diagram of Fig. 6a, with $T_3^{\uparrow }$ taken in the Born approximation.