Virial coefficients for trapped Bose and Fermi gases beyond the unitary limit: An $S$-matrix approach

We study the virial expansion for three-dimensional Bose and Fermi gases at finite temperature using an approximation that only considers two-body processes and is valid for high temperatures and low densities. The first virial coefficients are computed and the second is exact. The results are obtained for the full range of values of the scattering length, and the unitary limit is recovered as a particular case. A weak coupling expansion is performed and the free case is also obtained as a proper limit. The influence of an anisotropic harmonic trap is considered using the local density approximation (LDA), analytical results are obtained, and the special case of the isotropic trap is discussed in detail.

Figure 1

Foam diagrams.

Figure 2

Second virial coefficient against the ratio between the thermal wavelength and the scattering length: $b_2$ $\times$ $\alpha =\frac{{\lambda }_T}{a}$ for bosons (black). The values of the unitary limit $\left(\alpha \to 0^{\pm }\right)$ obtained in [19] are represented by the dashed (blue) lines and the value of the free case $\left(g=0\Rightarrow \alpha \to \pm \infty \right)$ is represented by the dotted (red) line.

Figure 3

Second virial coefficient against the ratio between the thermal wavelength and the scattering length: $b_2$ $\times$ $\alpha =\frac{{\lambda }_T}{a}$ for fermions (black). The values of the unitary limit $\left(\alpha \to 0^{\pm }\right)$ obtained in [19] are represented by the dashed (blue) lines and the value of the free case $\left(g=0\Rightarrow \alpha \to \pm \infty \right)$ is represented by the dotted (red) line.

Figure 4

Third virial coefficient against the ratio between the thermal wavelength and the scattering length: $b_3$ $\times$ $\alpha =\frac{{\lambda }_T}{a}$ for bosons (black). The values of the unitary limit $\left(\alpha \to 0^{\pm }\right)$ obtained in [19] are represented by the dashed (blue) lines and the value of the free case $\left(g=0\Rightarrow \alpha \to \pm \infty \right)$ is represented by the dotted (red) line.

Figure 5

Third virial coefficient against the ratio between the thermal wavelength and the scattering length: $b_3$ $\times$ $\alpha =\frac{{\lambda }_T}{a}$ for fermions (black). The values of the unitary limit $\left(\alpha \to 0^{\pm }\right)$ obtained in [19] are represented by the dashed (blue) lines and the value of the free case $\left(g=0\Rightarrow \alpha \to \pm \infty \right)$ is represented by the dotted (red) line. The exact result for the untarity limit with infinite negative scattering length obtained in [24] is represented by a thick dashed (green) line.

Figure 6

Second virial coefficient against the ratio between the thermal wavelength and the scattering length: $b_2$ $\times$ $\alpha =\frac{{\lambda }_T}{a}$ for bosons (black). The dashed (red) line shows the same result obtained with the expression of the weak coupling expansion.

Figure 7

Second virial coefficient against the ratio between the thermal wavelength and the scattering length: $b_2$ $\times$ $\alpha =\frac{{\lambda }_T}{a}$ for fermions (black). The dashed (red) line shows the same result obtained with the expression of the weak coupling expansion.

Figure 8

Third virial coefficient against the ratio between the thermal wavelength and the scattering length: $b_2$ $\times$ $\alpha =\frac{{\lambda }_T}{a}$ for bosons (black). The dashed (red) line shows the same result obtained with the expression of the weak coupling expansion.

Figure 9

Third virial coefficient against the ratio between the thermal wavelength and the scattering length: $b_2$ $\times$ $\alpha =\frac{{\lambda }_T}{a}$ for fermions (black). The dashed (red) line shows the same result obtained with the expression of the weak coupling expansion.

Figure 10

Ratio between the second, third, and fourth virial coefficients in the presence and in the absence of an harmonic isotropic trap against the frequency of the trap for $T=1$ in the three-dimensional case $\left(d=3\right)$, $\frac{b_n^{\prime }}{b_n}\times w$. $n=2$ is the dashed (blue) line, $n=3$ is the thin (red) line, and $n=4$ is the thick (green) line.

Figure 11

Ratio between the second, third, and fourth virial coefficients in the presence and in the absence of an harmonic isotropic trap against the temperature for $w=1$ in the three-dimensional case $\left(d=3\right)$, $\frac{b_n^{\prime }}{b_n}\times T$. $n=2$ is the dashed (blue) line, $n=3$ is the thin (red) line, and $n=4$ is the thick (green) line.