Boundary Superconductivity in the BCS Model

We consider the linear BCS equation, determining the BCS critical temperature, in the presence of a boundary, where Dirichlet boundary conditions are imposed. In the one-dimensional case with point interactions, we prove that the critical temperature is strictly larger than the bulk value, at least at weak coupling. In particular, the Cooper-pair wave function localizes near the boundary, an effect that cannot be modeled by effective Neumann boundary conditions on the order parameter as often imposed in Ginzburg-Landau theory. We also show that the relative shift in critical temperature vanishes if the coupling constant either goes to zero or to infinity.


Introduction and Main Result
We study how a boundary influences the critical temperature of a superconductor in the Bardeen-Cooper-Schrieffer (BCS) model. At superconductor-insulator (or superconductor-vacuum) boundaries, it is natural to impose Dirichlet boundary conditions on the Cooper-pair wave function. In several works [1,5,6] it was concluded that the presence of the boundary only affects the Cooper-pair wave function on microscopic scales; in particular, on larger scales described by Ginzburg-Landau theory (GL), the effect of the Dirichlet boundary conditions disappears and consequently the GL order parameter should satisfy Neumann boundary conditions [11,Ch. 7.3], [14,Ch. 6]. This seems to implicitly assume that the effect of the boundary on the critical temperature is negligible. Recent computations [3,4,16] indicate, however, that the Cooper-pair wave function can localize near the boundary, leading to an increase in the critical temperature compared to its bulk value. In this paper, we shall give a rigorous proof of the occurrence of this phenomenon in the simplest setting of one dimension, with δ-interactions among the particles. We consider a system on the half-line, where the boundary is then just a point.
The increase of the critical temperature in the presence of a boundary has some far-reaching implications. First of all, it implies that boundary superconductivity in the BCS model sets in already above the bulk value of the critical temperature. Second, it questions the validity of the often employed phenomenological GL theory in the presence of boundaries, as detailed in [17]. Note that GL theory has so far only been rigorously derived from the BCS model for periodic systems without boundaries [9]. (In the low-density BEC limit at zero temperature it was shown in [10] that the effective Gross-Pitaevskii theory inherits the microscopic Dirichlet boundary conditions.) In mathematical terms, the presence of a boundary manifests itself in a compact perturbation of a translation-invariant operator, and we shall show that at weak coupling this leads to the appearance of discrete eigenvalues outside the continuous spectrum. In particular, there is an effective attraction to the boundary, which is strong enough to create bound states.
In the following, we shall consider a superconductor on a domain Ω, with either Ω " R or Ω " R`" p0, 8q. The main quantity of interest is the linear two-particle operator acting in L 2 symm pΩ 2 q " tψ P L 2 pΩ 2 q|ψpx, yq " ψpy, xq for all x, y P Ωu, where ∆ denotes the Dirichlet Laplacian on Ω, and the subscripts x and y, respectively, indicate the variable on which ∆ acts. The first term is defined through functional calculus. In the second term, δ is the Dirac delta distribution, and v ą 0 is a coupling constant. Moreover, T ą 0 denotes the temperature, and µ P R is the chemical potential.
As explained in [8], H Ω T characterizes the local stability of the normal state in BCS theory. If H Ω T has spectrum below zero, i.e. inf σpH Ω T q ă 0, the normal state is unstable and the system in Ω is superconducting. If inf σpH Ω T q ě 0, the normal state is locally stable. We define the critical temperatures T Ω c as T Ω c pvq :" inf T P p0, 8q| inf σpH Ω T q ě 0 The sample is thus superconducting for T ă T Ω c . In the translation-invariant case, i.e. Ω " R, it is also known that local stability of the normal state implies global stability [12]; in particular, the sample is always in a normal state for T ě T R c in this case, i.e. T R c separates the superconducting and the normal phases. For the point interactions considered in (1.1), one can derive the explicit relation 1 2π Because of translation invariance, H R T has purely essential spectrum. Moreover, H RT has the same essential spectrum and possibly additional eigenvalues below it. In particular, for all v ą 0 the critical temperatures satisfy Our main result states that this inequality is actually strict, at least for small v, proving that the boundary increases the critical temperature. Moreover, the relative difference between the two critical temperatures vanishes both in the weak and in the strong coupling limit.
(ii) In the weak coupling limit (iii) In the strong coupling limit This result can be viewed as a rigorous justification of the observations in [16]. Numerics shows that the ratio T Rc pvq{T R c pvq can be as large as 1.06, see [16,Fig. 2]. Moreover, numerics also suggests that T Rc pvq and T R c pvq actually agree for v large enough, but it remains an open problem to show this.
Part (i) of Theorem 1.1 follows from the existence of an eigenvalue of H RT below the spectrum of H R T . It is quite remarkable that a Dirichlet boundary can decrease the ground state energy and create bound states. In contrast, for two-particle Schrödinger operators of the form´∆ x∆ y`V px´yq, only Neumann boundaries can bind states [7,15].
While we restrict our attention in this article to the one-dimensional setting with point interactions, we expect that our methods can be generalized to a larger class of interaction potentials, as well as to higher dimensions and the corresponding more complicated geometries possible. We shall leave these generalizations for future investigations, however. Remark 1.2. Our techniques can also be applied in case of Neumann boundary conditions for ∆ on R`. In this case one obtains the following results instead.
(iii) In the strong coupling limit In the remainder of this article we shall give the proof of Theorem 1.1. In the next Section 2, we shall use the Birman-Schwinger principle to conveniently reformulate the problem in terms of bounded operators and compact perturbations. Section 3 contains the proof of part (i), the existence of boundary superconductivity. The analysis of the weak and strong coupling limits in parts (ii) and (iii) is the content of Sections 4 and 5, respectively. Finally, Section 6 contains the proofs of some auxiliary Lemmas.

Preliminaries
Let us fix the notation Using the partial fraction expansion for tanh (Mittag-Leffler series), one can obtain the series representation [8] L T,µ pp, qq " 2T ÿ for w n " πp2n`1qT . Moreover, let F T,µ ppq :" L T,µ pp, pq " In order to control the kinetic energy in H Ω T the following bounds turn out to be useful. We shall prove them in Section 6.1.
Since vδpx´yq is infinitesimally form bounded with respect to´∆ x´∆y , it follows that the H Ω T are self-adjoint operators defined via the KLMN theorem. Moreover, the operators H Ω T become positive for T large enough. In particular, the critical temperatures defined in (1.2) are finite in both cases Ω " R and Ω " R`. Let L Ω T,µ denote the operator L T,µ p´i∇ x ,´i∇ y q defined through functional calculus. Of course, L Ω T,µ depends on the domain Ω and on the boundary conditions imposed on ∆. Its integral kernel is given by where for the problem on the full real line t R pxq " 1 ?
2π e´i x and on the half-line with Dirichlet boundary condition t R`p xq " 1 ? π sinpxq. For Neumann boundary conditions, one would have t R`p xq " 1 ? π cospxq instead. It is convenient to switch to the Birman-Schwinger formulation of the problem. For a more regular interaction V instead of δ, the Birman-Schwinger operator would be V 1{2 L Ω T,µ V 1{2 . For the δ-case, it turns out that V 1{2 has to be understood as restriction of a two-body wave function to its diagonal. Hence, the Birman-Schwinger operator has kernel L Ω T,µ px, x; x 1 , x 1 q and acts on functions of one variable only. For the two domains under consideration, the Birman-Schwinger operators A RT ,µ : L 2 pp0, 8qq Ñ L 2 pp0, 8qq and A R T,µ : L 2 pRq Ñ L 2 pRq are explicitly given by dy sinppxq sinpqxqL T,µ pp, qq sinppyq sinpqyqαpyq (2.8) and pA R T,µ βqpxq " for either Ω " R or Ω " R`.
From now on we will work with the operators A Ω T,µ rather than H Ω T . In momentum space, the operator A R T,µ is multiplication by the function where B is defined in (2.4).

Lemma 2.3 (Momentum representation of
The following Lemma shows that adding the boundary to the system effectively introduces the perturbation 1 4π B T,µ , where B T,µ is short for the operator with integral kernel B T,µ pp, qq. Note that here we work with the cosine transform and not the sine transform as might be expected from (2.8). This is because α is the diagonal of a function which is antisymmetric under both x Ñ´x and y Ñ´y and hence symmetric under px, yq Ñ p´x,´yq.

Lemma 2.3 follows from an analogous computation.
Since the operator A R T,µ is multiplication by the function (2.14), it has purely essential spectrum. The perturbation B T,µ in A RT ,µ is Hilbert-Schmidt and thus compact. Hence, σpA R T,µ q " σ ess pA RT ,µ q. It follows that for all T ă T R c pvq we have sup σpA RT ,µ q ě sup σpA R T,µ q ą 1{v, which implies (1.4).
Remark 2.5. Choosing Neumann instead of Dirichlet boundary conditions amounts to changing the minus sign in (2.16) into a plus sign.
It is possible to give a more explicit expression for sup σpA R T,µ q. The following is proved in Section 6.1. Consequently, Hence, in the translation invariant case superconductivity is equivalent to a T,µ ą 1 v and the critical temperature is determined by (1.3). Note that a T,µ is decreasing in T . Therefore, T R c pvq is a monotonically increasing function of v.

Existence of Boundary Superconductivity
From now on we assume that µ ą 0. In this Section, we show that for weak coupling the half-line critical temperature is higher than the bulk critical temperature. The idea is to prove that for T below a threshold T 0 ą 0 we have sup σpA RT ,µ q ą a T,µ . (3.1) Then consider v ăṽ :" a´1 T 0 ,µ . We must have T R c pvq ă T 0 by the monotonicity of T R c pvq. By definition and continuity of inf σpH RT q in T , sup σˆA T Rc pvq, we would get a contradiction to (3.1). Thus, T R c pvq ‰ T Rc pvq and, together with (1.4), part (i) of Theorem 1.1 follows. To prove (3.1), we use the variational principle with a trial function mimicking the ground state found in [16]. We choose ψ λ ǫ pxq " e´ǫ |x|`λ gpxq, where λ P R and the cosine Fourier transform p gppq " 1 ? π ş 8 0 gpxq cosppxqdx is real, continuous and centered at 2 ? µ.
Proof. Let h ǫ pxq " e´ǫ |x| . The cosine Fourier transform of the trial state is p  We now compute the two limits. Note that for bounded continuous functions f , we have lim ǫÑ0 ş In the first summand, we want to interchange limit and integration using dominated convergence.
The following Lemma is proved below.
(ii) There is a g P L 1 pRq X L 8 pRq such that |f pp, qq| ď gpqq for all p and q.
By dominated convergence the first term on the right hand side of (3.4) vanishes and thus lim ǫÑ0 xh ǫ |A RT ,µ´a T,µ I|h ǫ y "´1 4 B T,µ p0, 0q. Combining this with (3.2) and (3.3) yields For T Ñ 0 the term B T,µ p0, 0q is bounded while the second summand diverges logarithmically, which is content of the following Lemma.
Therefore, the last term in (3.5) dominates for small T and makes the right hand side positive. This completes the proof of Prop. 3.1.
The logarithmic divergence in Lemma 3.3 originates from the following asymptotics proved in Section 6.2. pp{2q 2´µ dp, (3.12) where we used p gpkq ď 1 and tanhpxq ď 1. The last summand is some constant independent of T . Using that B T,µ p0, pq " F T,µ pp{2q and Lemma 3.5 the asymptotic behavior for T Ñ 0 is and the claim follows. Part (ii). Recall that xg|A RT ,µ´a T,µ |gy " ż R dp p gppq 2 pA T,µ ppq´a T,µ q´ż R dp p gppq ż R dq 1 4π B T,µ pp, qqp gpqq. (3.14) By Lemma 2.6, the first summand is negative and thus also xg|A RT ,µ´a T,µ |gy ă 0. Moreover, using Lemma 2.6 and 0 ă p gppq ď 1 we have |xg|A RT ,µ´a T,µ |gy| ď ż R dp p gppq 2 a T,µ`ż R dp p gppq In both terms, the integral over p gives a finite constant independent of T . The claim follows from the asymptotics in Lemma 3.5.

Weak Coupling Limit
In [16] it was observed by numerical and non-rigorous analytical computations that the effect of boundary superconductivity disappears in the weak coupling limit, in the sense that In this section we shall verify this claim.
Recall that the bulk critical temperature T R c pvq is the unique T ą 0 such that a T,µ " 1 v . For the system on the half-line, we have by continuity of inf σpH RT q in T T Rc pvq " mintT P r0, 8q| sup σpA RT ,µ q " v´1u. (4.1) We want to invert this function and view v as function of T Rc . We define vpT q :" psup σpA RT ,µ qq´1. Note that v˝T Rc " id and for all T ą 0 we have T Rc pvpT qq ď T .
The claim can be reformulated in terms of the operator A RT ,µ and a T,µ in the following way.
Proof. By definition, we have sup σpA where in the last equality we used Lemma 3.5 and that T ě T Rc pvpT qq ě T R c pvpT qq ě 0 and thus lim T Ñ0 T R c pvpT qq " 0. Therefore, There exists a sequence pT n q such that T n Ñ 0 as n Ñ 8 and T Rc pvpT n qq " T n for all n. Therefore, Since lim T Ñ0 T R c pvpT qq " 0, also lim T Ñ0 vpT q " 0. Thus, and the claim follows.
Recall the definition of A T,µ in (2.14). With the notation E T,µ ppq " 4π pa T,µ´AT,µ ppqq (4.6) we have for all ψ P L 2 pp0, 8qq  where ¨ HS denotes the Hilbert-Schmidt norm. for constants c 1 , c 2 ą 0 and T small enough.
Proof of Theorem 1.1 (ii). By Lemma 4.1 it suffices to prove 0 " lim T Ñ0 inf σpa T,µ I´A RT ,µ q " lim T Ñ0 1 4π inf σpE T,µ`BT,µ q. By (1.4), we only need to show that lim T Ñ0 inf σpE T,µ`BT,µ q ě 0. For δ ą 0 we can write since E T,µ ppq ě 0 by Lemma 2.6. We shall show that for all δ ą 0 Hence, the operator in the bracket in (4.11) is positive for small T . This implies that for all δ ą 0 for T small enough we have inf σpE T,µ`BT,µ`δ q ą 0. Since δ can be arbitrarily small, the theorem follows.
Remark 4.5. In the case of Neumann boundary conditions, the same argument proves (1.9).

Proofs of Intermediate Results
Proof of Lemma 4.2. In order to bound B T,µ pp, qq we apply the following inequality proved in Section 6.3.

Lemma 4.6.
For all x, y P R and T ą 0 it holds that tanhpx{T q`tanhpy{T q x`y ă 2 |x|`|y| . (4.16) Hence, B T,µ pp, qq is bounded above by The function f has singularities at the four points where t|p|, |q|u " t0, 2 ? µu. Since f diverges linearly at those points, the idea is to do a Schur test with a test function of the form dppq α , where dppq is the distance from the singularities in variable p and α P p0, 1q. We choose the function hppq " mint|p|, |2 ? µ´|p||u 1{2 . The Schur test gives In order to estimate hppq ş 8 0 f pp,qq hpqq dq, we split the domain into nine regions as indicated in Figure 1. The finiteness of the right hand side of (4.18) follows from the bounds listed in Table 1. In the following, we prove the bounds in Table 1.
In region 4, we have p ă ? µ and where we used p ă ? µ in the last inequality.

Strong Coupling Limit
The goal of this section is to prove part (iii) of Theorem 1.1. As for the weak coupling limit, we first translate the question about the relative temperature difference into a condition on A RT ,µ and a T,µ . While the weak coupling limit turned out to be equivalent to a low temperature limit, the strong coupling limit corresponds to a high temperature limit. In this limit, the relevant quantities behave as follows.
The proof is provided in Section 5.1. We can reformulate Theorem 1.1(iii) as follows.
,0 q " a 1,0 (5.1) Proof. By Lemma 5.1(iv) and the definition of vpT q we have where we used Lemma 5.1(i) and vpT Rc pvqq " v for the second equality. Since a 1,0 ą 0, the claim follows. ż R |ψppq| 2 dp ż R B 1,0 p0, qqdq " ψ 2 2 a 1,0 . (5.5) In order to show this, we shall bound B 1,0 by a positive definite kernel K, in such a way that the right hand side of (5.5) does not change.
Lemma 5.4. Let K be the operator on L 2 pR 2 q with integral kernel Kpp, qq " mintB 1,0 pp, 0q, B 1,0 pq, 0qu (5.6) Then K satisfies (i) B 1,0 pp, qq ď Kpp, qq for all p, q P R (ii) Kpp, qq " Kpq, pq for all p, q P R This implies (5.5) and hence part (iii) of Theorem 1.1 since Proof of Lemma 5.4. Property (ii) is obvious. Properties (iv) and (v) follow from the fact that Kpp, qq " mintF 1,0 pp{2q, F 1,0 pq{2qu " F 1,0 pmaxt|p|, |q|u{2q, has a maximum at p " 0 and is monotonously decreasing for p ą 0. For (i) consider the following inequality, which is proved in Section 6.4. Together with the monotonicity of tanhppq{p for p ě 0, it implies (i). For property (iii) it suffices to show that there is a real-valued function g such that Since a T R c pvpT qq,µ " sup σpA RT , µ q this is equivalent to lim T Ñ8 a T R c pvpT qq,µ " 0. (5.14) Using that a T,µ is strictly decreasing in T with lim T Ñ8 a T,µ " 0, this in turn is equivalent to Fix some T 0 ą 0. Since tanhpxq{x is decreasing for x ě 0 and bounded by 1, the integrand is bounded by 1 2 χ |q|ă2 for T ą T 0 . This is an L 1 function, so by dominated convergence we can pull the limit into the integral and arrive at the claim.
(iv) Let U T denote the unitary transformation U T ψppq " T 1{4 ψpT 1{2 pq on L 2 pR 2 q. We shall prove that lim T Ñ8 U T T 1{2 A RT ,µ U :
Proof of Lemma 2.6. First, we show that for every x, y P R tanhpxq`tanhpyq x`y ď 1 2ˆt anhpxq x`t anhpyq y˙(

6.3)
Since changing x Ñ´x, y Ñ´y does not change the expressions, we may assume without loss of generality that x ě |y|. Note that tanhpxq`tanhpyq x`y " 1 2px`yq " px`yqˆt anhpxq x`t anhpyq y˙`p x´yqˆt anhpxq x´t anhpyq y˙ (6.4) Since tanhpxq{x ď tanhpyq{y, the last term is not positive and the inequality (6.3) follows.
(6.11) Therefore, this term is of order one for T Ñ 0 and ş R F T,µ ppqdp " ş ?

From Section 4
Proof of Lemma 4.6. In the case xy ą 0, the inequality follows immediately from the fact that | tanhpzq| ă 1 for all z P R. In the case xy ă 0, let us replace y Ñ´y and assume without loss of generality that x ą y ą 0. Since the function s Þ Ñ e´2 s is convex, we have e´2 y´e´2x x´y ď´d ds e´2 sˇs "y " 2e´2 y (6.12) We estimate x`y x´y ptanhpxq´tanhpyqq " 2px`yq 1`e´2 y e´2 y´e´2x px´yqp1`e´2 x q ď 2px`yqe´2 y 1`e´2 y min " 2, 1 x´y * ď 4p2y`1{2qe´2 y 1`e´2 y , (6.13) where we maximized over x in the last step. The maximum of the last expression over y is attained at the value y "ỹ satisfying e´2ỹ " 2ỹ´1{2. Therefore, we get x`y x´y ptanhpxq´tanhpyqq ď 4p2ỹ´1{2q. (6.14) The function e´2 y is decreasing in y and 2y´1{2 is increasing. For y " 1{2 we have e´1 ă 1{2, hence the intersection pointỹ satisfies 0 ăỹ ă 1{2 . Thus, x`y x´y ptanhpxq´tanhpyqq ă 2, which proves the claim.
Proof of Lemma 4.7. Without loss of generality, we may assume that y ă x. We have tanhpxq´tanhpyq " e x´e´x e x`e´x´e y´e´y e y`e´y " 2 e x´y´ey´x pe x`e´x qpe y`e´y q ď 2 e x´y´ey´x e x`y " 2pe´2 y´e´2x q Applying (6.12) the claim follows.