Interacting preformed Cooper pairs in resonant Fermi gases

We consider the normal phase of a strongly interacting Fermi gas, which can have either an equal or an unequal number of atoms in its two accessible spin states. Due to the unitarity-limited attractive interaction between particles with different spin, noncondensed Cooper pairs are formed. The starting point in treating preformed pairs is the Nozières-Schmitt-Rink (NSR) theory, which approximates the pairs as being noninteracting. Here, we consider the effects of the interactions between the Cooper pairs in a Wilsonian renormalization-group scheme. Starting from the exact bosonic action for the pairs, we calculate the Cooper-pair self-energy by combining the NSR formalism with the Wilsonian approach. We compare our findings with the recent experiments by Harikoshi et al. [Science 327, 442 (2010)] and Nascimbène et al. [Nature (London) 463, 1057 (2010)], and find very good agreement. We also make predictions for the population-imbalanced case, which can be tested in experiments.

Figure 1

Diagrammatic representation of (a) the bare Cooper-pair propagator and (b) the bare Cooper-pair interaction. The Cooper pairs are represented by thick lines, while the thin lines correspond to fermionic propagators.

Figure 2

Diagrammatic representation of the “$\beta$ functions.” (a) Feynman diagram determining the self-energy of the Cooper pairs. (b) Feynman diagrams renormalizing the Cooper-pair interaction. The middle diagram is also called the ladder diagram, the right diagram is called the bubble diagram. Note that the lines are thick and correspond to the Cooper-pair propagator.

Figure 3

Bare many-body scattering length $a_{\mathrm{MB}}^{\mathrm{B}}$ (dashed line) and fully renormalized many-body scattering length $a_{\mathrm{MB}}^{\infty }$ (full line) as a function of temperature $T$ in the unitarity limit for the balanced Fermi gas. The Fermi wave vector $k_{\mathrm{F}}$ and Fermi temperature $T_{\mathrm{F}}$ are directly related to the total particle density through the Fermi energy ${\epsilon }_{\mathrm{F}}=k_{\mathrm{B}}{T}_{\mathrm{F}}={\hslash }^2{k}_{\mathrm{F}}^2/2m={\hslash }^2{\left(3{\pi }^{2}n\right)}^{2/3}/2m$, which is calculated with the RG approach. The many-body scattering length is inversely proportional to the Cooper-pair chemical potential, and the difference between the bare and the renormalized value is due to the repulsive Cooper-pair interaction effects.

Figure 4

Equation of state for the normal phase of a strongly interacting balanced Fermi gas at unitarity in the canonical ensemble. The energy $E$ of the gas is calculated as a function of temperature $T$ with the renormalization-group approach (full line) and the Nozières-Schmitt-Rink approach (dashed line). The squares are the experimental results of Harikoshi et al. [14], the triangles are the results of Nascimbène et al. [15], and the dots are the Monte Carlo results of Burovski et al. [19].

Figure 5

Equation of state for the normal phase of a strongly interacting balanced Fermi gas at unitarity in the grand-canonical ensemble. The pressure $P=-\Omega /V$ of the gas is calculated as a function of the inverse fugacity $e^{-\beta \mu }$ with the renormalization-group approach (full line) and the Nozières-Schmitt-Rink approach (dashed line). The pressure of the ideal two-component Fermi gas is given by $P_{\mathit{ig}}$. The triangles are the experimental results of Nascimbène et al. [15].

Figure 6

Equation of state for the normal phase of a strongly interacting imbalanced Fermi gas at unitarity in the grand-canonical ensemble. The pressure $P=-\omega (T,\mu ,h)$ of the gas is calculated at a temperature $T=0.75\mu$ as a function of the normalized chemical potential difference $h/\mu =({\mu }_{+}-{\mu }_{-})/({\mu }_{+}+{\mu }_{-})$ with the renormalization-group approach (full line), the Nozières-Schmitt-Rink approach (dashed line), and for the ideal Fermi gas (dashed-dotted line). For each curve, the pressure of the imbalanced gas is normalized by the corresponding pressure of the balanced gas $P_0=-\omega (T,\mu ,0)$.