Pairing in a two-component ultracold Fermi gas: Phases with broken-space symmetries

We explore the phase diagram of a two-component ultracold atomic Fermi gas interacting with zero-range forces in the limit of weak coupling. We focus on the dependence of the pairing gap and the free energy on the variations in the number densities of the two species while the total density of the system is held fixed. As the density asymmetry is increased, the system exhibits a transition from a homogenous Bardeen-Cooper-Schrieffer (BCS) phase to phases with spontaneously broken global space symmetries. One such realization is the deformed Fermi surface superfluidity (DFS) which exploits the possibility of deforming the Fermi surfaces of the species into ellipsoidal form at zero total momentum of Cooper pairs. The critical asymmetries at which the transition from DFS to the unpaired state occurs are larger than those for the BCS phase. In this precritical region the DFS phase lowers the pairing energy of the asymmetric BCS state. We compare quantitatively the DFS phase to another realization of superconducting phases with broken translational symmetry: the single-plane-wave Larkin-Ovchinnikov-Fulde-Ferrell phase, which is characterized by a nonvanishing center-of-mass momentum of the Cooper pairs. The possibility of the detection of the DFS phase in the time-of-flight experiments is discussed and quantified for the case of ${}^6\mathrm{Li}$ atoms trapped in two different hyperfine states.

Figure 1

(Color online) The dependence of the pairing gaps in the LOFF phase (upper panel) and the DFS phase (lower panel) on the asymmetry parameter for several values of the the total momentum $P∕k_F$ and deformation parameter $\delta ϵ$ which are indicated in the panels.

Figure 2

The dependence of the quasiparticle spectra of two hyperfine states ${\omega }_1$ and ${\omega }_2$ on the momentum for $\alpha =0=\delta ϵ$ (solid line), $\alpha =0.05$ and $\delta ϵ=0$ (long-dashed lines), $\alpha =0.05$, $\delta ϵ=0.1$, and $x=0$ (short-dashed lines), and $x=1$ (dash-dotted lines). The Fermi momentum $k_F=4.83\times {10}^4{\mathrm{cm}}^{-1}$ is indicated by the vertical line.

Figure 3

The dependence of the critical asymmetry ${\alpha }_c$ of the transition from the superfluid to the normal state on the total momentum in the LOFF phase and the deformation parameter in the DFS phase. The model parameters are as in Fig. 1.

Figure 4

(Color online) The dependence of the free energy of the LOFF (upper panel) and the plane-wave DFS phase (lower panel) on the asymmetry parameter for several values of the deformation parameter $\delta ϵ$ and the total momentum $P∕k_F$ which are indicated in the panels.

Figure 5

(Color online) The dependence of the occupation probabilities of two hyperfine states on the momentum. The Fermi momentum $k_F=4.83\times {10}^4{\mathrm{cm}}^{-1}$ is indicated by the vertical line. The labeling of the lines is as follows: $\alpha =0=\delta ϵ$ (solid line), $\alpha =0.05$ and $\delta ϵ=0$ (long-dashed lines), $\alpha =0.05$, $\delta ϵ=0.1$, and $x=0$ (short-dashed lines), and $x=1$ (dash-dotted lines). The bell-shaped light curves show the anisotropy—the difference between the $x=1$ and $x=0$ occupation numbers—for $\alpha =0.05$, $\delta ϵ=0.1$. The remaining parameters are as in Fig. 1.