Virial Expansion for a Strongly Correlated Fermi Gas

Using a high temperature virial expansion, we present a controllable study of the thermodynamics of strongly correlated Fermi gases near the BEC-BCS crossover region. We propose a practical way to determine the expansion coefficients for both harmonically trapped and homogeneous cases, and calculate the third order coefficient $b_3(T)$ at finite temperatures $T$. At resonance, a $T$-independent coefficient $b_{3,\infty }^{\mathrm{hom}}\approx -0.29095295$ is determined in free space. These results are compared with a recent thermodynamic measurement of ${}^6\mathrm{Li}$ atoms, at temperatures below the degeneracy temperature, and with Monte Carlo simulations.

Figure 1

Predicted interaction energy as a function of entropy at unitarity, compared with experimental data from Duke University [10]. The solid (dashed) line shows the contribution up to the third (second) order virial coefficient. The arrow indicates the degeneracy point.

Figure 2

Relative energy levels of a three-fermion system at the ground state section ($l=1$).

Figure 3

The second and third virial coefficients as a function of the interaction parameter $1/k_{F}a$. We have used a total number of atoms $N=100$, leading to $\tilde{\omega }=(3N{)}^{-1/3}\approx 0.15$ at $T=T_F$.

Figure 4

Temperature dependence of interaction energy of a homogeneous gas at unitarity, obtained using different virial coefficients. For comparison, two quantum Monte Carlo data reported in Refs. 2, 4 are shown. A finite range of interactions has been used in Ref. 2, which may lead to a systematic downshift in energies. We list also the result calculated by using Rupak’s $b_{3,\infty }^{\mathrm{hom}}\approx 1.11$.