Virial expansion for a strongly correlated Fermi gas with imbalanced spin populations

Quantum virial expansion provides an ideal tool to investigate the high-temperature properties of a strongly correlated Fermi gas. Here, we construct the virial expansion in the presence of spin-population imbalance. Up to the third order, we calculate the high-temperature free energy of a unitary Fermi gas as a function of spin imbalance, with infinitely large attractive or repulsive interactions. In the latter repulsive case, we show that there is no itinerant ferromagnetism when quantum virial expansion is applicable. We therefore estimate an upper bound for the ferromagnetic transition temperature $T_c$. For a harmonically trapped Fermi gas at unitarity, we find that $(T_c){}_{\mathrm{upper}}<T_F$, where $T_F$ is the Fermi temperature at the center of the trap. Our result for the high-temperature equations of state may confront future high-precision thermodynamic measurements.

Figure 1

Temperature dependence of the free energy of a homogeneous unitary Fermi gas, calculated up to the second order (dashed lines) or third order (solid lines) using the quantum virial expansion theory. For comparison, we show the ideal noninteracting free energy by a thin dot-dashed line. As anticipated, the free energy of an attractive or a repulsive Fermi gas is always smaller or larger than that of an ideal Fermi gas, respectively.

Figure 2

Free energy of a homogeneous unitary Fermi gas as a function of spin imbalance at a given temperature (as indicated) for attractive interactions or for repulsive interactions. The exact free energy could lie between the second- and third-order virial expansion predictions.

Figure 3

Temperature dependence of the free energy of a trapped unitary Fermi gas. We use the same notations as in Fig. 1. The experimental data for an attractive trapped Fermi gas at unitarity, reported by the Salomon group at ENS [20], are shown by squares.

Figure 4

Free energy of a trapped unitary Fermi gas as a function of spin imbalance at a given temperature as indicated. The same notations as in Fig. 2 have been used.

Figure 5

Spin susceptibility of a homogeneous Fermi gas at unitarity, normalized by the $T=0$ ideal Fermi-gas susceptibility value ${\chi }_0=3n/(2{\epsilon }_F)$, with $n$ as the density and ${\epsilon }_F$ as the Fermi energy. The spin susceptibility of a repulsive or attractive Fermi gas at unitarity lies above or below the ideal Fermi-gas result (dot-dashed line). The inset shows the inverse of the spin susceptibility of a repulsive unitary Fermi gas. A linear extrapolation of the high-$T$ virial expansion result at $T/T_F=4–6$ to low temperatures indicates a ferromagnetic transition temperature of $T_c~0.4T_F$.