Comment on “Superfluid stability in the BEC-BCS crossover”

We point out an error in recent work by Pao, Wu, and Yip [Phys. Rev. B 73, 132506 (2006)], that stems from their use of a necessary but not sufficient condition [positive compressibility (magnetic susceptibility) and superfluid stiffness] for the stability of the ground state of a polarized Fermi gas. As a result, for a range of detunings their proposed ground-state solution is a local maximum rather than a minimum of the ground state energy, which thereby invalidates their proposed phase diagram for resonantly interacting fermions under an imposed population difference.

Figure 1

(Color online) Polarization $\mathrm{\Delta }N∕N$ vs detuning $-1∕k_F{a}_s$ phase diagram of the one-channel model (appropriate for width $\gamma ⪢1$) showing regimes of FFLO, superfluid (confined to the $\mathrm{\Delta }N=0$ axis), magnetized superfluid $\left({\mathrm{SF}}_M\right)$, and phase separation. The dashed lines are the purported phase boundaries reported in Ref. 3. As we show (Ref. 5), for the entire shaded region single-component (uniform) solutions to the stationarity conditions Eqs. (7a, 7b, 7c) are local maxima of $E_G\left(\Delta \right)$ (and therefore unstable, leading to phase separation), shown for the point marked with the “x” in Fig. 2.

Figure 2

(Color online) Plot of $E_G\left(\Delta \right)$ at fixed $\mu =-0.013{ϵ}_F$ and $h=1.16{ϵ}_F$, normalized to the Fermi energy ${ϵ}_F$, for coupling $g=1∕k_F{a}_s=0.63$, so that Eqs. (7a, 7b, 7c) yield a stationary solution with $\mathrm{\Delta }N∕N=0.2$ and $\Delta =1.02$. As seen in Fig. 2, however, this solution is a local maximum (saddle point) of $E_G\left(\Delta \right)$. At this point ($1∕k_F{a}_s=0.63$ and $\mathrm{\Delta }N∕N=0.2$) in the phase diagram (illustrated by an “x” in Fig. 1) the actual mean-field ground state is a phase-separated mixture of a superfluid and a normal state.