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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 {\bf 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.

There has been considerable recent interest in paired superfluidity of fermionic atomic gases under an imposed spin polarization 1,2 , i.e., when the numbers N ↑ and N ↓ of the two atomic species undergoing pairing are different. Along with the detuning δ of the Feshbach resonance (controlling the strength of the interatomic attraction), the population difference ∆N = N ↑ − N ↓ is an experimentally-adjustable "knob" that allows the study of novel regimes of strongly-interacting fermions.
A crucial question concerns the phase diagram of resonantly interacting fermions as a function of δ and imposed ∆N . Two early theoretical studies that have addressed this issue are the work by Pao, Wu, and Yip 3 on the one-channel model of interacting fermions and our work 4 on the two-channel model of interacting fermions, with the details and extensions of the latter presented in our recent preprint. 5 Apart from the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase appearing over a thin sliver of the phase diagram in Ref. 4 (explicitly not considered by Pao, Wu, and Yip), for the positive-detuning BCS and crossover regimes the phase diagrams in these manuscripts are in qualitative agreement 6 . However, at negative detuning in the BEC regime, the phase diagrams are qualitatively different. In particular, in Fig. 4 of Ref. 3, stable superfluidity is claimed to exist above and to the left of a nearly vertical and positively sloped phase boundary that we re-plot here in Fig. 1 as a dashed line at negative detuning. In qualitative contrast, in our work, Ref. 4, we found that stable superfluidity exists only below and to the left of a negatively-sloped phase boundary (see Fig. 1).
What is the source of this qualitative discrepancy? Although Ref. 3 and Ref. 4 use different models of resonantly-interacting fermions, the close relationship between the one-and two-channel models (particularly within the mean-field approximation) implies that they should yield qualitatively similar phase diagrams. Furthermore, we have extended our original two-channel model study 4 to that of a one-channel model 5 and, as expected, found results in qualitative agreement with those in our Letter 4 , but in disagreement with that of Ref. 3.
Indeed, as we explicitly show here, the origin of the discrepancy is that the criterion for stability of the superfluid phase used in Ref. 3 (based on positivity of magnetic susceptibilities) is a necessary but not sufficient condition for stability and does not ensure 7,8,9 that the state is even a local minimum of the energy. This thereby leads to incorrect phase boundaries in both the BEC and BCS regimes, although below we shall focus on the phase boundary inside the BEC regime, where the error is qualitative and most pronounced.
Thus, much of what is claimed to be a "stable superfluid" in Ref. 3 (in the negative-detuning BEC regime) is actually unstable to phase separation. This is illustrated in the correct T = 0 mean-field phase diagram, Fig. 1, for the one-channel model (quantitatively consistent with other recent work 10,11 and derived in detail in Ref. 5), plotted as a function of the dimensionless parameter −(k F a s ) −1 (proportional to the Feshbach resonance detuning δ, with k F the Fermi wavevector and a s the s-wave scattering length) and the polarization Our starting point is the single-channel model Hamiltonian (studied in Ref. 3) for two resonantly-interacting species of fermionĉ kσ (σ =↑, ↓): where ǫ k = k 2 /2m, m is the fermion mass, and V is the system volume. A broad Feshbach resonance is modeled by an attractive interaction λ < 0, the magnitude of which increases with decreasing Feshbach resonance detuning.
The equilibrium ground state of a many particle system in the grand-canonical ensemble (with chemical potentials µ ↑ and µ ↓ ) at T = 0 is characterized by the grand thermodynamic potential Ω(µ ↑ , µ ↓ ) defined by 12 with the grand-canonical Hamiltonian Here,N σ ≡ kĉ † kσĉkσ is the number operator for fermion species σ and the minimization in Eq. (2) is over all possible ground states.
The standard (BCS-type) mean-field approximation that we shall utilize (as done in Ref. 3) amounts to assuming a restricted class of possible many-body groundstates self-consistently parametrized by the ground-state expectation value For pairing amplitude ∆ = 0, this class includes both weakly-paired BCS-type and strongly-paired molecular BEC-type pairing order. For ∆ = 0 it is an unpaired Fermi gas. Once this mean-field approximation has been made, it is straightforward 4,5 to compute the variational groundstate energy, which is the expectation value E G (∆) = Ĥ . Evaluating this expectation value, and converting momentum sums to integrals, we find where ξ k ≡ ǫ k − µ, E k ≡ ξ 2 k + ∆ 2 , and we have defined the chemical potential µ = 1 2 (µ ↑ + µ ↓ ) and the chemical potential difference h = 1 2 (µ ↑ − µ ↓ ). We have also exchanged the bare interaction parameter λ for the vacuum s-wave scattering length a s given by The determination of the mean-field phase diagram using Eq. (5) is conceptually quite simple. According to Eq. (2), the ground state at a particular µ ↑ and µ ↓ (or, equivalently, µ and h) is given by the minimization of E G (∆) with respect to the pairing amplitude ∆ that can be taken to be real. Any such minima of course satisfy the stationarity constraint or gap equation [equivalent to Eq. (4)] where we emphasize that the derivative is taken at fixed µ and h. Since experiments are conducted at fixed atom number, we must augment Eq. (7a) with the number constraint equations N σ = N σ . Examining Eq. (3), we see that the constraints can be rewritten as with the total particle number N = N ↑ + N ↓ and population difference ∆N = N ↑ − N ↓ .
Our key point (apparently missed by the authors of Ref. 3) is that not every simultaneous solution of the gap and number equations, Eqs. (7), corresponds to a physical ground state of the system; the additional criterion is that the solution ∆ must also be a minimum of E G (∆) at fixed µ σ . The verification that an extremum solution is indeed a minimum is particularly essential when there is the possibility of a first-order transition, with E G (∆) exhibiting a local maximum that separates local minima, as is the case for a polarized Fermi gas, studied here and in Refs. 3,4,5.
Analyzing Eqs. (7), we find that for sufficiently large ∆N in the positive-detuning BCS regime and for ∆N = N in the BEC regime, a solution to Eqs. (7) may be found that minimizes E G (∆) at ∆ = 0, indicating a polarized normal phase. Also, for sufficiently low detuning in the BEC regime, a polarized molecular superfluid (SF M ) solution exists that minimizes E G (∆) at ∆ = 0. A more general analysis 4,5,13 shows that a periodicallypaired FFLO solution is the ground state over a thin range of polarization values at sufficiently large positive detuning.
However, over the large shaded portion of the phase diagram, at intermediate detuning and polarization, we find that it is not possible to satisfy Eqs. (7) with a (homogeneous, single component) minimum of E G (∆). For the corresponding range of parameters the system phase-separates 14 into two coexisting ground states (that are degenerate minima of E G [∆]). The resulting phaseseparated state can be explicitly accounted for by generalizing the ground-state ansatz to include the possibility of such an inhomogeneous mixture 5 .
The contrasting strategy of Pao, Wu, and Yip 3 is to find solutions of Eqs. (thereby determining the phase boundaries plotted in Ref. 3) based on criteria of the positivity of magnetic susceptibilities (atomic compressibilities) and the superfluid stiffness. However, as we now discuss, these stability criteria are necessary but not sufficient (i.e., not generally restrictive enough) to ensure that a solution to Eqs. (7) is indeed a minimum of E G (∆). In contrast, it can be shown that the converse is true, namely that the minimization of E G (∆) ensures that the local susceptibilities are positive definite. 15 The stability criteria 16 used by Pao, Wu, and Yip can be understood by examining Fig. 3 Fig. 3 of Pao, Wu, and Yip), they find solutions satisfying ∂h ∂n d n < 0 and correctly conclude that such solutions (having a negative magnetic susceptibility) are unstable. However, at sufficiently negative detuning (g > ∼ 0.5) Pao, Wu, and Yip. find solutions to Eqs. (7) with a positive susceptibility ∂h ∂n d n > 0, and based on their criterion (erroneously) conclude that these solutions indicate a stable magnetic superfluid ground state. They then define a phase boundary in the BEC regime (the leftmost dashed curve of Fig. 1), to a stable magnetized superfluid, by where ∂h ∂n d n changes sign. However, our explicit calculation of E G (∆) (plotted in Fig. 2) for one such solution (with 1 kFas = g = 0.63 and n d /n = 0.2, indicated with an "x" in Fig. 1 and corresponding to a point on the uppermost solid curve of Fig. 3 of Ref. 3), purported by Pao, Wu, and Yip to be stable (to the left of their proposed stability boundary), shows that in fact this solution (indicated with a dot in Fig. 2) is a local maximum and therefore does not represent a ground state. This solution was obtained by numerically solving Eqs. (7) at g = 0.63 and ∆N/N = 0.2, yielding µ = −0.013ǫ F , h = 1.16ǫ F and ∆ = 1.02ǫ F , the latter two values consistent with Figs. 2 and 3 of Pao, Wu, and Yip, showing that we are indeed reproducing a solution claimed to be stable by Pao, Wu, and Yip. Thus, the method used by Pao, Wu, and Yip has not correctly located the global minimum of the ground-state energy; indeed, it has not even found a local minimum.
Although it might appear from the plot of E G (∆, µ, h) (Fig. 2) that the true ground state is an unpaired (∆ = 0) normal state, this state does not satisfy Eqs. (7b) and (7c); thus, it is also not the ground state at this coupling and polarization. Indeed, as noted above, we find that it is impossible to minimize E G (∆) while satisfying Eqs. (7b) and (7c) at this coupling and polarization (marked by an "x" in Fig. 1), nor anywhere inside the shaded region in the phase diagram Fig. 1, indicating the absence of a uniform solution. The true mean-field ground state everywhere in the shaded region is a phaseseparated mixture of two phases of different densities in chemical equilibrium such that the total number and polarization constraints are satisfied 4,5 .
We note that a ground state determined by minimizing E G (∆) at a particular µ ↑ and µ ↓ automatically satisfies the condition of having a positive magnetic susceptibility (compressibility). Indeed, it is straightforward to generally show 15,17 that, since Ω(µ ↑ , µ ↓ ) is concave downwards 12 , the eigenvalues of the susceptibility matrix are positive in the ground state. An equivalent stability criterion was derived in Ref. 18 by considering stability against local density variations. The procedure used by Pao, Wu, and Yip, however, did not amount to analyzing Eq. (8); as discussed above, the phase diagram of Pao, Wu, and Yip was obtained by computing ∂h ∂n d n which is not an equivalent condition. If the authors had instead studied ∂n d ∂h µ , they would have found that the solution plotted in Fig. 2 has a negative magnetic susceptibility in the grand-canonical ensemble (and therefore is unstable).
However, we must emphasize the important point that any particular extremum solution may have a positive magnetic susceptibility and still not be the ground state. The simplest example of this is the normal Fermi gas state (∆ = 0), which satisfies the gap and numberconstraint equations everywhere in the phase diagram (including ∆N = 0) and has a positive magnetic susceptibility, but is only the actual ground state (a minimum of E G (∆)) at sufficiently large ∆N . Thus, if the authors of Ref. 3 had computed the eigenvalues of Eq. (8) instead of ∂h ∂n d n , they would have been able to discard some of the erroneous solutions plotted in Fig. 3 of Ref. 3. However, in general , Eq. (8) is still not sufficient and the most correct scheme is to use Eq. (2), i.e., to find the global minimum, in the grand-canonical ensemble, of the mean-field ground-state energy. We conclude by noting that, although a mean-field analysis of the one-channel model is not expected to be quantitatively accurate near the resonance position where k F |a s | → ∞, it is expected to yield a qualitatively correct description of a polarized resonantly-interacting Fermi gas. To summarize, we have shown that while for equal species number (∆N = h = 0) such analysis can simply proceed by solving the gap and number equations [Eqs. (7)], the existence of first-order transitions at h = 0 implies that the ground-state energy E G (∆) exhibits local maxima as a function of ∆, yielding solutions to Eq. (7a) that do not represent physical ground states.