Thermodynamic properties of universal Fermi gases

We develop a simple, mean-field-like theory for the normal phase of a unitary Fermi gas by deriving a self-consistent equation for its self-energy via a momentum-dependent coupling constant for both attractive and repulsive universal fermions. For attractive universal fermions in the lower branch of a Feshbach resonance, we use zero-temperature Monte Carlo results as a starting point for one-step iteration in order to derive an analytical expression for the momentum-dependent self-energy. For repulsive universal fermions in the upper branch of a Feshbach resonance, we iteratively calculate the momentum-dependent self-energy via our self-consistent equation. Lastly, for the case of population imbalance, we propose an ansatz for higher-order virial expansion coefficents. Overall, we find that our theory is in good agreement with currently available, high-temperature experimental data.

Figure 1

The self-energy of a unitary Fermi gas in the upper branch of a Feshbach resonance as related to its momentum. Values of $\gamma =0.1$, 1, and 2 are shown as black, dark gray, and light gray lines, respectively. (Plotted quantities are dimensionless.)

Figure 2

The occupation numbers of a unitary Fermi gas in the upper branch of a Feshbach resonance as related to momentum. Values of $\gamma =0.1$, 1, and 2 are shown as black, dark gray, and light gray lines, respectively. (Dashed lines of the same hue are the noninteracting occupation numbers corresponding to these $\gamma$ values, and the plotted quantities are dimensionless.)

Figure 3

The pressure of a unitary Fermi gas as a function of $\xi$ in the upper branch of a Feshbach resonance. Here, $P^{*}=P\left(\xi \right){\lambda }^3/k_{B}T$, where $\xi =\exp (-\mu /k_{B}T)$. (Plotted quantities are dimensionless.)

Figure 4

The pressure of a unitary Fermi gas as a function of $\xi$ in the lower branch of a Feshbach resonance. Here, $h\left(\xi \right)=P(\mu ,T)/P_1(\mu ,T)$, where $\xi =\exp (-\mu /k_{B}T)$. Our theory (black line) vs experimental data [41] (gray points). (Plotted quantities are dimensionless.)

Figure 5

The entropy as a function of energy for harmonically trapped, unitary fermions. Our theory (black line) vs experimental data [42] (gray points). (Plotted quantities are dimensionless.)

Figure 6

The finite-temperature equation of state for a population-imbalanced, unitary Fermi gas. Values of $\eta =1.0$, 0.5, and 0.25 are shown in black, dark gray, and light gray lines, respectively. The gray points represent experimental data [41] for the population-balanced case, which is recovered from Eq. (26) when $\eta =1.0$. (Plotted quantities are dimensionless.)