Pairing in a three-component Fermi gas

We consider pairing in a three-component gas of degenerate fermions. In particular, we solve the finite-temperature mean-field theory of an interacting gas for a system where both interaction strengths and fermion masses can be unequal. At zero temperature, we find the possibility of a quantum phase transition between states associated with pairing between different pairs of fermions. On the other hand, finite-temperature behavior of the three-component system reveals some qualitative differences from the two-component gas: for a range of parameters it is possible to have two different critical temperatures. The lower one corresponds to a transition between different pairing channels, while the higher one corresponds to the usual superfluid-normal transition. We discuss how these phase transitions could be observed in ultracold gases of fermionic atoms.

Figure 1

(Color online) The free-energy landscape as a function of ${\Delta }_{12}$ and ${\Delta }_{23}$ when all three components have the same mass. The dash-dotted line in (a) and the filled circles in (b)–(d) show the location of the global minimum. We used equal coupling strengths $g_{12}=g_{23}=-0.50$, and (a)–(c) were calculated at zero temperature while (d) was calculated at $k_{B}T∕{ϵ}_F=0.08$, which is above the critical temperature. The chemical potentials were such that in (a) ${\mu }_1={\mu }_2={\mu }_3=1$; in (b) ${\mu }_1=1.01$, ${\mu }_2=1$, ${\mu }_3=0.99$; in (c) ${\mu }_1=0.99$, ${\mu }_2=1$, ${\mu }_3=1.01$; and finally in (d) ${\mu }_1={\mu }_2={\mu }_3=1$.

Figure 2

(Color online) The free-energy landscape as a function of ${\Delta }_{12}$ and ${\Delta }_{23}$ with the mass ratio $m_1∕m_3=0.15$ and ${\mu }_1={\mu }_2=1.0$. The filled circle shows the location of the global minimum. The coupling strengths were $g_{12}=g_{23}=-0.5$, and (a)–(c) were calculated at zero temperature while (d) was calculated at $k_{B}T∕{ϵ}_F=0.07$. In (a) ${\mu }_3=0.07$, in (b) ${\mu }_3=0.15$, in (c) ${\mu }_3=0.25$, and in (d) ${\mu }_3=0.15$. [In (d) we used a logarithmic scale to calculate the contour plot in order to enhance the relevant features.] The results show two types of transitions: a zero-temperature quantum phase transition between the two pairing channels [from (a) to (b) and from (b) to (c)] and a finite-temperature second-order transition between the pairing channels [from (b) to (d)].

Figure 3

The gap parameters as a function of temperature $T$ and of the third-component chemical potential ${\mu }_3$. We used the mass ratio $m_1∕m_3=0.15$, ${\mu }_1={\mu }_2=1$, and the coupling strengths were $g_{12}=g_{23}=-0.5$. Panel (c) shows $\Delta =\sqrt{{\Delta }_{12}^2+{\Delta }_{23}^2}$ and demonstrates clearly the sharp change in the order parameter at zero temperature (the quantum phase transition) as well as the smoother transition from the ${\Delta }_{23}$ superfluid to ${\Delta }_{12}$ superfluid at a finite temperature.