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The BCS Functional for General Pair Interactions

The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.


Introduction
Over the last few years, significant attention has been devoted to ultra-cold gases of fermionic atoms, both from the experimental and the theoretical point of view. Because of the ability to magnetically tune Feshbach resonances it has become possible to vary interatomic interactions via external magnetic fields, and hence to control the strength of the interactions among the atoms. This method was used in remarkable experiments to study the crossover from the regime of Bose-Einstein condensation (BEC) of tightly bound diatomic molecules to the Bardeen-Cooper-Schrieffer (BCS) regime of weakly bound Cooper pairs in ultra-cold gases of fermionic atoms. For a recent review on this topic we refer to [5,7].
The main purpose of this paper is to give a mathematically precise study of the BCS regime. This regime is usually described by the BCS functional, derived by Leggett [10] based on the original work of Bardeen-Cooper-Schrieffer [3]. We note that while in the original BCS model simple phonon-induced non-local interaction potentials were considered, the interaction potentials are local for atomic Fermi gases. In order to allow for a wide range of applications, we will consider here the BCS functional for general pair interaction potentials V .
One purpose of our paper is to obtain necessary and sufficient conditions on V for the system to display superfluid behavior. Recall that in the BCS model superfluidity (or, rather, superconductivity in the original BCS case, where electrons in a crystal structure have been considered) is related to the non-vanishing of the pairing wave function, describing particle pairs with opposite spin and zero total momentum. That is, superfluidity occurs if the paired state is energetically favored over the normal state described by the Fermi-Dirac distribution.
More precisely, for arbitrary temperature T = 1/β ≥ 0 a system is in a superfluid state if there exists a non-trivial solution ∆ to the BCS gap equation with E(p) = (p 2 − µ) 2 + |∆(p)| 2 . Here, µ denotes the chemical potential. Notice that the gap equation is highly nonlinear. In Theorem 1 below we will show that the existence of a non-trivial solution to the gap equation is equivalent to the existence of a negative eigenvalue of the corresponding linear operator K β,µ + V , K β,µ = (p 2 − µ) e β(p 2 −µ) + 1 e β(p 2 −µ) − 1 . (1. 2) The operator K β,µ is understood as a multiplication operator in momentum space, i.e., p 2 equals −∆ in configuration space. Observe that in the limit T → 0 this operator reduces to the Schrödinger type operator This linear characterization of the existence of solutions to the nonlinear BCS equation represents a considerable simplification of the analysis. In particular, it proves the existence of solutions to (1.1) for a wide range of interaction potentials V , and hence generalizes previous results [4,13,18,19] valid only for non-local V 's under suitable assumptions. Moreover, the monotonicity of K β,µ in β guarantees the existence of a critical temperature T c , with 0 ≤ T c < ∞, such that superfluidity occurs whenever T < T c and normal behavior for T ≥ T c . In particular, there is a phase transition at T c . For positive values of µ, we shall show that T c is strictly positive whenever the negative part of V , denoted by V − , is non-vanishing and the positive part V + is small enough in a suitable sense. Moreover, we prove that T c is exponentially small in the strength of the interaction potential. More precisely, if the potential is given by λV for some fixed V ∈ L 1 ∩L 3/2 , then T c e −1/λ as λ → 0. This generalizes the corresponding result in [3,Eq. (3.29)] to a very large class of interaction potentials. For negative potentials, also a corresponding lower bound will be shown to hold. In a forthcoming work [9], the precise asymptotics of T c for λ → 0 will be investigated.
At zero temperature, the existence of a non-trivial solution of the BCS gap equation is also related to the non-vanishing of the energy gap between the ground state and the first excited state in the effective quadratic BCS Hamiltonian on Fock space. For very specific attractive interaction potentials, it has been argued that this gap is non-vanishing, see e.g. [10,14,15]. We note, however, that the existence of such an energy gap is not necessarily implied by the non-vanishing of the solution to the BCS equation. In fact, the presence of a gap is not necessary for our analysis. We consider general potentials where such an energy gap need not be present, a priori. We do prove that the gap is non-vanishing for a certain class of interaction potentials, however.
We note that our results are relevant for actual experiments on cold atomic gases as long as the description via the BCS functional is applicable. There is an extensive literature on this subject, see e.g. [1,5,6,7,14,15,16,17]. According to Leggett [10] this is the case for weak interactions as long as the range of the potential V is much smaller than µ −1/2 (in appropriate units), i.e., for weakly coupled dilute gases. In particular, "weakly coupled" means that the corresponding Schrödinger operator p 2 + V typically does not allow for a negative energy bound state since otherwise the system is in the BEC regime of tightly bound bosonic molecules. We refer to [10] for a detailed discussion on this question.

Model and Main Results
The precise definition of the BCS functional considered in this paper is given as follows. For the convenience of reader, we describe in the appendix the motivation and physical background; the discussion there follows Leggett's derivation in [10]. See also [2] for a detailed study of the BCS approximation to the Hubbard model. We use the standard conventionα(p) = (2π) −3/2 R 3 α(x)e −ipx dx for the Fourier transform. DEFINITION 1. Let D denote the set of pairs of functions (γ, α), with γ ∈ L 1 (R 3 , (1 + p 2 )dp), 0 ≤ γ(p) ≤ 1, and α ∈ H 1 (R 3 , dx), satisfying |α(p)| 2 ≤ γ(p)(1 − γ(p)). Let V ∈ L 3/2 (R 3 , dx) be real-valued and µ ∈ R. For T = 1/β ≥ 0 and (γ, α) ∈ D, the energy functional F β is defined as As explained in the Appendix, (2.1) has the physical interpretation of − 1 2 (2π) 3/2 times the pressure of a system of spin 1/2 fermions at temperature T interacting via a pair potential given by 2V .
It is natural to require V ∈ L 3/2 (R 3 ), as we do here, since this guarantees that V is relatively form-bounded with respect to p 2 (see, e.g., Sect. 11 in [11]). In fact, it would be enough to require the negative part of V to be in L 3/2 + L ∞ ǫ , and allow for more general positive parts of V . For simplicity of the presentation, we will not pursue this generalization here.
In the absence of a potential V , it is easy to see that F β is minimized on D by the choice α ≡ 0 and γ(p) = γ 0 (p) = [e β(p 2 −µ) + 1] −1 . We refer to this state as the normal state of the system. We will be concerned with the question of whether a non-vanishing α can lower the energy for given potential V .
Note that at T = 0 it is natural to formulate the model in terms of a functional that depends only on α and not on γ. Namely, for fixed |α(p)| 2 , the optimal choice of γ(p) is in this case. Similarly, also at T > 0 it is enough to minimize F β only over pairs (γ, α) satisfying a certain relation independent of V , which is displayed in Eq. (3.4) below.
According to the usual interpretation of the BCS theory, the normal state is unstable if the energy can be lowered by the formation of Cooper pairs, i.e., if F β can be lowered by choosing α = 0. The following Theorem, which is the main result of this paper, shows that this property is equivalent to the existence of a negative eigenvalue of a certain linear operator.
Then the following statements are equivalent: (ii) There exists a pair (γ, α) ∈ D, with α = 0, such that satisfies the BCS gap equation has at least one negative eigenvalue.
Here, * denotes convolution in the BCS gap equation (2.4). In order to avoid having to write factors of 2π, we find it convenient to define convolution with a factor (2π) −3/2 in front, i.e., (f * g)(p) := (2π) −3/2 R 3 f (p − q)g(q)dq. Remark 1. The operator (2.5) is obtained by taking the second derivative of the functional F β with respect to α at the normal state (γ 0 , 0) (see Eq. (3.10) below). Since the normal state is a critical point of F β , it is then easy to prove that (iii) implies (i). The opposite direction is not immediate, however. It says that if (γ 0 , 0) is not a minimizer, it can also not be locally stable.

Remark 2.
As mentioned above, a minimizer of F β satisfies the V -independent relation (2.2) at T = 0, and (3.4) at T > 0. It is not necessary to assume that these equations hold in part (ii) of Theorem 1, however.
Note that in the zero temperature case T = 0 the gap equation (2.4) takes the form (compare with [3,8,10,12]) and the operator in (2.5) is given by For any potential V we shall prove the existence of a critical temperature 0 ≤ T c = 1 β c < ∞ at which a phase transition from a normal to superfluid state takes place. More precisely, we shall show that the gap equation (2.4) has a non-trivial solution for all T < T c , whereas it has no non-trivial solution if T ≥ T c . This follows from the next Theorem, in combination with Theorem 1.

THEOREM 2. For any
With the aid of the Birman-Schwinger principle, it is possible to characterize the value of the critical temperature T c , at least if T c > 0. In order to do this, it is convenient to decompose V into its positive and negative parts, i.e., We shall show in Proposition 4 that T c > 0 satisfies the equation Note that since K β,µ ≥ 2/β, this leads immediately to the general rough upper bound (2.7) For negative potentials V and a positive chemical potential µ, we shall show that the critical temperature is always positive, i.e., that |p 2 − µ| + V has a negative energy bound state. We can even say more, namely we can guarantee superfluidity for small temperature whenever the positive part V + is not too large in a suitable sense.
be not identically zero, and let µ > 0.
(i) If V ≤ 0, then the corresponding critical temperature is positive, In particular, this theorem states that for any negative potential one can add a sufficiently small positive part such that the critical temperature T c is still strictly positive.
The next question concerns the magnitude of T c . The rough upper bound given in (2.7) turns out to be much too rough, in general. In fact, the following Theorem shows that T c is always exponentially small in the strength of V . To be able to state the bound precisely, let a := inf t>0 t(e t + 1)/((t + 2)(e t − 1)). Numerically, a ≈ 0.654. Let further Note that f is a positive and monotone decreasing function, with f (t) ∼ ln(1/t) as t → 0. We obtain the following upper bound on the critical temperature T c .

Then the critical temperature satisfies the upper bound
Hence the critical temperature is exponentially small in the potential for For negative potentials, one can show that the bound given in Theorem 4 is indeed optimal. More precisely, if the potential is given by λV for some fixed negative V ∈ L 3/2 , we shall show in Proposition 5 the lower bound where g µ,V is some positive function depending only on µ and V satisfying g µ,V (t) ∼ e −1/t for small t. Combining both bounds then implies that for negative potentials V the critical temperature is exponentially small in the coupling parameter. This generalizes well-known calculations in the physics literature, see e.g. [3, Eq. (3.29)], to a very large class of interaction potentials. Note that, in particular, this result implies that it is not possible to calculate the critical temperature via a perturbative expansion in V .
Finally, we comment on the continuity properties of minimizers of F β . In fact, for positive temperature T > 0 the minimizing γ andα can be shown to be continuous functions. (We show this in Prop. 3 below.) This may not be surprising, since already the minimizer without potential, γ 0 (p) = 1/(e β(p 2 −µ) + 1), is continuous. But in the case of T = 0 the situation is not so clear, since the normal state is a step function with jump at p 2 = µ for µ > 0.
The continuity of a minimizer γ at T = 0 is closely related to the existence of an energy gap, given by with E defined in (2.4). As explained in the Appendix, E(p) has the interpretation of the dispersion relation for quasi-particle excitations. The following shows that in case V has a strictly negative Fourier transform, γ is continuous even at T = 0 and the gap Ξ is non-vanishing.
This result implies, in particular, that Ξ > 0 for negative V with strictly negative Fourier transform. We note that this property need not necessarily be true for all potentials V , however.
We conclude this section with two remarks.
Remark 3. Our results can be generalized to non-local potentials given by an integral kernel V (x, y). Exactly such a non-local potential was used in the original paper of BCS [3] This models an effective potential arising from electron-phonon interactions in crystals. In this case, the minimizer of F β can be evaluated explicitly (see, e.g., [12]). In particular, our analysis can be used to generalize previous results [4,13,18,19] on the existence of solutions of the gap equation, as already mentioned in the Introduction. None of this previous work allows for an interaction kernelV (k, k ′ ) of the formV (k − k ′ ), however, since such a kernel does not satisfy appropriate L p conditions. Such conditions were necessary for utilization of fixed point arguments in the previous investigations of this problem.
Remark 4. In [6,15] the BCS-BEC crossover is studied numerically for certain potentials V that are negative and have negative Fourier transform. These potentials satisfy all the assumptions of our results stated above.
In the following Sections 3 and 4, the proofs of the claims of this section will be given.

Properties of the Minimizer of F β
In this section we shall give the proof of Theorem 1 and Proposition 1 stated above. We start by showing the existence of a minimizer of F β .

Proposition 2.
There exists a minimizer of F β in D.
To show that a minimizer of F β exists in D, we pick a minimizing sequence (γ n , α n ) ∈ D, with F β (γ n , α n ) ≤ 0. From (3.1) we conclude that α n 2 H 1 ≤ 8A, and hence we can find a subsequence that converges weakly to someα ∈ H 1 . Since 11,Thm. 11.4].
It remains to show that the remaining part of the functional, 2) is weakly lower semicontinuous. Note that F 0 β is jointly convex in (γ, α), and that its domain D is a convex set. We already know that α n ⇀α weakly in H 1 (R 3 ). Moreover, since γ n is uniformly bounded in L 1 (R 3 ) ∩ L ∞ (R 3 ), we can find a subsequence such that γ n ⇀γ weakly in L p (R 3 ) for some 1 < p < ∞. We can then apply Mazur's theorem [11,Theorem 2.13] to construct a new sequence as convex combinations of the old one, which now converges strongly to (γ,α) in L p (R 3 ) × L 2 (R 3 ). By going to a subsequence, we can also assume that γ n → γ andα n →α pointwise [11,Theorem 2.7]. Because of convexity of F 0 β , this new sequence is again a minimizing sequence.
Next, we establish a few auxiliary results that will be needed below in the proof of Theorem 1. Lemma 1. Let (γ, α) ∈ D be a minimizer of F β for 0 < β < ∞. Then, for a.e. p ∈ R 3 ,
, with Fourier transform supported in A ǫ . Then (γ, α+tg) ∈ D for small enough |t|. By the Lebesgue dominated convergence theorem, it follows that Similarly, forĝ real, (γ + tĝ, α) ∈ D for small |t|, and we get Since (γ, α) minimize F β by assumption, the expressions in (3.5) and (3.6) vanish. It follows that for a.e. p ∈ A ǫ we have To conclude that (3.7) holds a.e. in R 3 , it remains to show that B := R 3 \∪ ǫ>0 A ǫ has zero measure. By definition, if p ∈ B we have γ(p)(1 − γ(p)) = |α(p)| 2 , i.e. s(p) s(p) − 1 = 0. It is then easy to see that if B has non-zero measure, F β (γ, α) can be lowered by modifying γ andα on B (since t → t ln t has a diverging derivative at zero). This contradicts the fact the (γ, α) is a minimizer, and hence B has zero measure.

Remark 5.
A similar strategy as in the proof of Lemma 1 leads to the conclusion that Eq. (3.3) is also valid at T = 0. The relation between γ and α, given in Eq. (3.4) for T > 0, has to be replaced by (2.2) at T = 0, however. We omit the details.
Note that Eq. (3.4) and the positivity of f immediately imply that γ(p) ≥ 1/2 for p 2 < µ, and γ(p) ≤ 1/2 for p 2 > µ. Of greater importance below will be the following monotonicity property, which bounds γ in the opposite direction.
Proof. Let p 2 > µ such thatα(p) = 0 and assume by way of contradiction that γ(p) ≤ γ 0 (p). We then have Since f is a strictly monotone increasing function, we obtain This, however, contradicts the fact that both (γ 0 , 0) and (γ, α) are solutions of (3.4), and hence both sides of (3.8) are equal to β(p 2 − µ). The case p 2 < µ is treated similarly.
We now have all the necessary prerequisites at our disposal to give the proof of Theorem 1.
[iii] ⇒ [i]: Consider first the case T > 0. Showing that the existence of a negative eigenvalue of the operator (2.5) implies that the minimizing α is not identically zero is equivalent to proving that α ≡ 0 implies that inf spec (K β,µ + V ) ≥ 0. First, we note that, for any δ ∈ C ∞ 0 , (tδ) 2 + (γ 0 − 1 2 ) 2 ∞ < 1 4 for all small |t|. With the aid of the Lebesgue dominated convergence theorem it is easy to see that d 2 dt 2 F β (γ 0 , tδ) exists for small t, and is given by dp .
Assuming that the minimizing α is identically zero, it follows that the corresponding γ = γ 0 and hence d 2 and hence δ, (K β,µ + V )δ ≥ 0 for all δ ∈ C ∞ 0 . This proves the statement.
In the case T = 0, it is sufficient to minimize F β over (γ, α) ∈ D with γ of the form (2.2), as remarked earlier. For simplicity, we abuse the notation slightly and denote the resulting functional by F ∞ (α). Subtracting its value for α = 0, it is given by Assume that |p 2 −µ|+V has a negative eigenvalue. We can then find an α ∈ C ∞ 0 (R 3 ) such that α, (|p 2 − µ| + V )α < 0. For 0 < ǫ ≪ 1 small enough, we then have This implies that a minimizer α does not vanish identically.
In the following, we investigate the continuity properties of a minimizing pair (γ, α) ∈ D.
The right side of this equation is continuous and does not vanish, not even when E(p) = 0 (which could happen, in principle, for p 2 = µ). Hence γ is continuous. Using again Eq. (3.3), this also implies continuity ofα.
The situation is different at zero temperature, however. In the limit β → ∞, the right side of Eq. (3.13) vanishes in case E(p) = 0. Hence γ could possibly be discontinuous if p 2 = µ. In fact, this is what happens in the case of vanishing potential. On the other hand, a non-vanishing gap Ξ = inf p E(p) > 0 implies continuity of γ. At T = 0, the gap equation (2.4) takes the form with Hence the gap Ξ is guaranteed to be non-zero if the right side of (3.14) vanishes nowhere on the sphere p 2 = µ. This can be easily shown in caseV is strictly negative, which is the content of our Proposition 1.
Proof of Proposition 1. SinceV ≤ 0, Moreover, sinceV is assumed to be strictly negative, there can be equality in (3.15) only ifα(p) = e iκ |α(p)| for some constant κ ∈ R. This implies that this is the case for a minimizer of F ∞ , and henceV * α vanishes nowhere. The statement then follows from Eq. (3.14).

The Critical Temperature
In this section we investigate the relation between properties of the interaction potential V and the critical temperature T c at which the phase transition takes place. In particular, we shall prove Theorems 2-4.
From the definition of K β,µ in (2.5) one easily sees that It follows that inf spec(K β,µ + V ) < 0 for all β > β c . Moreover, since K β,µ depends continuously on β (in a norm resolvent sense), we have that K βc,µ + V ≥ 0. Setting β c = ∞ in the case inf spec(K β,µ + V ) ≥ 0 for all β and letting T c = 1/β c finishes the proof.
Next, we prove the first assertion of Theorem 3, namely that for any negative potential the critical temperature is strictly positive.

Proof of Theorem 3(i).
In light of Theorem 2 it suffices to prove that inf spec(|p 2 − µ| + V ) < 0. According to the Birman-Schwinger principle, this is equivalent to the fact that for some e > 0, the Birman-Schwinger operator has an eigenvalue 1. Notice that lim e→∞ B e = 0 since V ∈ L 3/2 is relatively bounded with respect to p 2 (in the sense of quadratic forms). Because of continuity in e, it thus suffices to find an e > 0 such that B e > 1.
Hence we have to show that there exists a φ ∈ L 2 (R 3 ) with φ 2 = 1 such that φ, B e φ ≥ 1 for an appropriate e. For this purpose, we choose a normalized function φ in such a way that φ|V | 1/2 does not vanish in a neighborhood of some point p with p 2 = µ. We can, for instance, choose φ to be rapidly decaying, in which case | φ|V | 1/2 | is continuous and hence strictly positive on some small ball. This ball can then be made to overlap with the sphere p 2 = µ by simply multiplying φ by a factor e iqx for appropriate q, which translates the function φ|V | 1/2 by q. In this way, Thus there exists e > 0 such that φ, B e φ > 1.
Before proceeding with the proof of the second part of Theorem 3, we introduce an alternative characterization for the critical temperature T c in the case T c > 0, which we already referred to in Eq. (2.6).

Proposition 4. For all
whenever T c = 0.
Proof. Let the operator B β e be defined as This operator is compact (in fact, Hilbert-Schmidt) and depends continuously on β and e. Moreover, B β e is strictly decreasing in e and 1/β. Since T c > 0 by assumption, we know from Theorem 2 that K β,µ + V has a negative eigenvalue for T < T c . Hence the Birman-Schwinger principle implies that B β e > 1 for 1/β and e small enough. On the other hand, B β 0 → 0 as β → 0. Thus there exists a uniqueβ such that Bβ 0 = 1.
From Prop. 4 it is then easy to conclude that the critical temperature stays strictly positive under small positive perturbations to negative potentials.

Proof of Theorem 3(ii).
According to Prop. 4, it suffices to prove that for β > β c (−V − ) and for λ > 0 small enough, V In order to simplify the notation let Since V + ∈ L 3/2 by assumption, it is relatively form-bounded with respect to K β,µ , and hence K is finite. Therefore B λ > 1 if B 0 > 1 and λ is small enough.
Next, we shall derive upper and lower bounds on the critical temperature. These bounds are similar to the ones found in the literature, but generalize these results from simple rank 1 potentials as in [3] to very general pair interactions.
Using Lemmas 3 and 4 as well as V + ≥ 0, we thus have Since f is monotone decreasing, this implies the statement.
Finally, we turn to a lower bound for the critical temperature. We assume that the interaction potential is given by λV , with V ≤ 0 not identically zero, and λ > 0 a coupling parameter.
The latter integral in monotone increasing in β c and diverges logarithmically as β c → ∞ for appropriate ψ such that ψ|V | 1/2 (p) does not vanish identically if p 2 = µ. (See the proof of Thm. 3(i) for an argument concerning the existence of such a ψ.) This implies the statement.
The energy expectation value can be conveniently expressed in terms of these quantities: ρ(H) = 2 k k 2 γ(k) + (2π) 3/2 |Ω| k,pV (k)ϕ(k − p)ϕ(p) The first term in the second line is just a constant times the square of the number of particles. The last term is the exchange terms. Both these terms will be dropped in the sequel. This truncations can be justified on physical grounds for interaction potentials V that are of short range such that s-wave scattering is dominant [10].
Finally, we will briefly explain in which sense E(p) is an effective dispersion relation and how it is related to an energy gap between the ground state and the first excited state of the system.
(A.4) The Hamiltonian H qu − µN can be diagonalized in a standard way (see, e.g., [12]) via a Bogoliubov transformation. Using the fact that ϕ satisfies the gap equation E(k)ϕ(k) = − 1 2 (2π) 3/2 k ′V (k − k ′ )ϕ(k ′ ), where E(k) = (k 2 − µ) 2 + |∆(k)| 2 and ∆ is determined from ϕ via (2.2)-(2.4), this leads to The operators c k , c † k , d k and d † k satisfy the canonical anticommutation relations, and both c k and d k annihilate the BCS ground state. Equation (A.5) thus explains why E(k) is interpreted as the quasi-particle dispersion relation and Ξ = inf k E(k) as the energy gap of the system.