New Field-Theoretic Method for the Virial Expansion

We develop a graphical method for computing the virial expansion coefficients for a nonrelativistic quantum field theory. As an example we compute the third virial coefficient $b_3$ for unitary fermions, a nonperturbative system. By calculating several graphs and performing an extrapolation, we arrive at $b_3=-0.2930$, within 0.7% of a recent computation $b_3=-0.29095295$ by Liu, Hu, and Drummond, which involved summing 10 000 energy levels for three unitary fermions in a harmonic trap.

Figure 1

The free fermion contribution to $\Omega$. The conventional finite temperature Feynman diagram on the left is expanded as a sum over worldline loops about the compact time direction (chronographs) with winding number $\nu$ making a contribution proportional to $z^{\nu }$. The black dot indicates the nontrivial topology, and Euclidian time increases in the counterclockwise direction.

Figure 2

The contour $C$ in Eq. (5) is designed to pick up contributions from all cuts and poles along the real energy axis.

Figure 3

Feynman graphs for dimer mediated two-body scattering, and the integral equation relevant for three-body scattering $\mathrm{\text{dashed line}}=\mathrm{\text{fully dressed dimer}}$, $\mathrm{\text{solid line}}=\mathrm{\text{fermion}}$; $\mathrm{\text{gray}}=K^{-1}$).

Figure 4

Chronograph expansion for the virial coefficients $b_n$. Dashed line: fully dressed dimer propagator; shaded box: summed three-body interaction from Fig. 3. For $b_3$, bosons or multiple fermion species would require introduction of a trimer field as well.

Figure 5

The three propagator subgraph (in black) contributing to $b_3$ and corresponding to the expression $L$ appearing in Eqs. (10, 11).

Figure 6

Results for $I_n/n$, including error estimate from numerical integration for $n=3,\dots ,8$. The solid line is the large-$n$ fit to the last five data points by the function ($c_0/n+c_1/n^2$), as described in the text.