High-temperature limit of the resonant Fermi gas

We use the virial expansion to investigate the behavior of the two-component, attractive Fermi gas in the high-temperature limit, where the system smoothly evolves from weakly attractive fermions to weakly repulsive bosonic dimers as the short-range attraction is increased. We present a formalism for computing the virial coefficients that employs a diagrammatic approach to the grand potential and allows one to easily include an effective range $R^{*}$ in the interaction. In the limit where the thermal wavelength $\lambda \ll R^{*}$, the calculation of the virial coefficients is perturbative even at unitarity and the system resembles a weakly interacting Bose-Fermi mixture for all scattering lengths $a$. By interpolating from the perturbative limits $\lambda /\left|a\right|\gg 1$ and $R^{*}/\lambda \gg 1$, we estimate the value of the fourth virial coefficient at unitarity for $R^{*}=0$ and we find that it is close to the value obtained in recent experiments. We also derive the equations of state for the pressure, density, and entropy, as well as the spectral function at high temperatures.

Figure 1

Different interaction regimes for the high-temperature unpolarized Fermi gas at density $n\ll {\lambda }^{-3}$. When the scattering length $a<0$ and the thermal de Broglie wavelength $\lambda \gg \left|a\right|$, the system corresponds to a weakly attractive Fermi gas. In the opposite limit, where $\lambda \gg a>0$, the system instead resembles a weakly repulsive gas of bosonic dimers once the two-body binding energy ${\varepsilon }_{\mathrm{b}}\gg k_{B}Tlog(\sqrt{2}/n{\lambda }^3)$, where $T$ is the temperature. In the crossover region $\lambda /\left|a\right|\lesssim 1$, the system is strongly interacting in the sense that each coefficient of the virial expansion cannot be obtained from a perturbative calculation around a noninteracting system. This region can be made perturbative by introducing a large effective range parameter $R^{*}>\lambda$.

Figure 2

The interaction part of the second virial coefficient as a function of interaction strength for several values of $R^{*}/\lambda$. As $\lambda /a\to \infty$ the virial coefficient approaches $\sqrt{2}{e}^{\beta {\varepsilon }_{\mathrm{b}}}$ and we normalize by this factor (taking ${\varepsilon }_{\mathrm{b}}=0$ for $a\le 0$).

Figure 3

The interaction part of the third virial coefficient as a function of interaction parameter. In the Bose regime the virial coefficient is proportional to $e^{\beta {\varepsilon }_{\mathrm{b}}}$—see Eq. (42)—and we normalize by this factor (taking ${\varepsilon }_{\mathrm{b}}=0$ for $a\le 0$). (a) In the broad resonance limit, $R^{*}/\lambda =0$, we compare $\Delta b_3$ (solid) with the two-body contribution which dominates in the Fermi regime (short dashed), and the results of the Beth-Uhlenbeck formula (42) including $s$ wave only (dot-dashed), $s$ and $p$ wave (long dashed), and up to $d$ wave (dotted). The circle at $\lambda /a=0$ depicts the experimentally determined virial coefficient from Refs. [26, 27]. (b) The effect of a finite range parameter on $\Delta b_3$ (solid), and the approximation given by including only the diagrams of Eqs. (43a)–(43c) (dashed).

Figure 4

The interaction part of the third virial coefficient (solid line) at unitarity as a function of $R^{*}/\lambda$. The dashed line depicts the result of considering only the diagrams of Eqs. (43a)–(43c), corresponding to replacing the full three-body $T$ matrix by its Born approximation. The dotted curve is the narrow resonance asymptote—see Eq. (44).

Figure 5

The interaction part of the fourth virial coefficient as a function of the interaction parameter $\lambda /a$. In the Bose regime, $\Delta b_4\to e^{2\beta {\varepsilon }_{\mathrm{b}}}/4$ and we normalize by this factor, taking ${\varepsilon }_{\mathrm{b}}=0$ for $a\le 0$. (a) For a broad resonance with $R^{*}/\lambda =0$ we display our approximate value of $\Delta b_4$ (solid), along with the dominant two-body contribution in the Fermi regime (short dashed), and the dominant contribution in the Bose limit, Eq. (45), taking $a_{\mathrm{dd}}\approx 0.6a$ (dotted). The circle represents the experimental measurement [26, 27] and the square the calculation of Ref. [25]. (b) Taking the effective range into account, we display the virial coefficient for $R^{*}/\lambda =0$, 1/2, 1, 2 (bottom to top).

Figure 6

The interaction part of the fourth virial coefficient at unitarity ($1/a=0$) as a function of the effective range $R^{*}/\lambda$. The four-body contribution to the virial coefficient is approximated as explained in the text.

Figure 7

The pressure $P$, density $n$, and entropy density $s$ in the grand canonical ensemble as a function of expansion parameter $z^{*}$ for a broad resonance $R^{*}=0$. We compare the results from the virial expansion at second and fourth order in the fugacity, for three different values of the interaction parameter $\lambda /a$: From top to bottom we consider the Bose regime with $\lambda /a=1.5$, unitarity, and the Fermi regime with $\lambda /a=-2$. For the latter, the virial coefficients are dominated by the two-body contribution (47), the result of which is shown as a thick dashed line.

Figure 8

The pressure, density, and entropy at resonance. We compare the second and fourth orders of the virial expansion for various resonance widths. In the narrow resonance limit, the limiting behavior of the thermodynamic variables is given in Eqs. (53a)–(53c).

Figure 9

Comparison between experiment [27] and various theories for the pressure and density at unitarity for a broad resonance.

Figure 10

Contour plot of the effective fugacity $z^{*}$ in the $T/T_F$ versus $1/k_{F}a$ plane for a broad resonance with $R^{*}=0$. The high-temperature expansion is considered up to the third virial coefficient, since our Born approximation for the fourth virial coefficient overestimates the effective dimer-dimer repulsion in the Bose regime. Note that we observe qualitatively similar behavior for a narrow resonance, and also if we limit ourselves to the virial expansion up to second order in the fugacity.

Figure 11

The crossover behavior of the pressure $P$ and entropy density $s$ for both broad and narrow resonances. The solid lines correspond to the results of the virial expansion where the noninteracting Fermi gas contribution is treated exactly while the interacting part is only calculated up to the second virial coefficient. The solid circles mark the Fermi-Bose crossover point where $n{\lambda }^3\approx \sqrt{2}{e}^{-\beta {\varepsilon }_{\mathrm{b}}}$. The dashed lines illustrate the thermodynamic values expected for an ideal Bose gas of dimers.

Figure 12

The occupied part of the spectral function for various values of the interaction parameter $1/k_{F}a$ (top to bottom) and effective range $k_F{R}^{*}$ (left to right) near the crossover. They all correspond to the expansion parameter $z{e}^{\beta {\varepsilon }_{\mathrm{b}}/2}\approx 0.2$. The short-dashed line corresponds to the free particle dispersion.