An operator-theoretical study of the specific heat and the critical magnetic field in the BCS-Bogoliubov model of superconductivity

In the preceding paper, introducing a cutoff, the present author gave a proof of the statement that the transition to a superconducting state is a second-order phase transition in the BCS-Bogoliubov model of superconductivity on the basis of fixed-point theorems, and solved the long-standing problem of the second-order phase transition from the viewpoint of operator theory. In this paper we study the temperature dependence of the specific heat and the critical magnetic field in the model from the viewpoint of operator theory. We first show some properties of the solution to the BCS-Bogoliubov gap equation with respect to the temperature, and give the exact and explicit expression for the gap in the specific heat divided by the specific heat. We then show that it does not depend on superconductors and is a universal constant. Moreover, we show that the critical magnetic field is smooth with respect to the temperature, and point out the behavior of both the critical magnetic field and its derivative. Mathematics Subject Classification 2010. 45G10, 47H10, 47N50, 82D55.

preliminaries. In the physics literature, one differentiates the thermodynamic potential with respect to the temperature twice in order to show that the transition from a normal conducting state to a superconducting state is a second-order phase transition in the BCS-Bogoliubov model of superconductivity. Since the thermodynamic potential has the solution to the BCS-Bogoliubov gap equation in its form, one differentiates the solution with respect to the temperature twice without showing that the solution is differentiable with respect to the temperature. Therefore, if the solution were not differentiable with respect to the temperature, then one could not differentiate the solution with respect to the temperature, and hence one could not show that the transition is a second-order phase transition. This is why we need to show that the solution is differentiable with respect to the temperature twice as well as its existence and uniqueness.
Actually, as far as the present author knows, no one (except for the present author) showed that the solution is differentiable with respect to the temperature twice. Then, on the basis of fixed-point theorems, the present author 1 [Theorems 2.3 and 2.4] introduced a cutoff and showed that the solution is indeed partially differentiable with respect to the temperature twice, and gave an operator-theoretical proof of the statement that the transition from a normal conducting state to a superconducting state is a second-order phase transition. In this way, from the viewpoint of operator theory, the present author solved the long-standing problem of the second-order phase transition left unsolved for sixty-two years since the discovery of the BCS-Bogoliubov model.
In this paper we introduce a cutoff and study the temperature dependence both of the specific heat at constant volume and of the critical magnetic field in the BCS-Bogoliubov model of superconductivity from the viewpoint of operator theory. On the basis of fixed-point theorems, we first show some properties of the solution with respect to the absolute temperature T both at sufficiently small T and at T in the neighborhood of the transition temperature T c . We then give the exact and explicit expression for author knows, one obtains the same results only when the potential ⋅ ⋅ U( , ) in (1.1) below is a constant in the physics literature. But we obtain the results even when the potential ⋅ ⋅ U( , ) is not a constant but a function. Moreover, we show that the critical magnetic field applied to type-I superconductors is of class C 1 both with respect to sufficiently small T and with respect to T in the neighborhood of the transition temperature T c , and point out the behavior of the critical magnetic field and its derivative. We carry out their proofs on the basis of fixed-point theorems. As far as the present author knows, no one (except for the present author) showed that the critical magnetic field is differentiable with respect to T.
Here the BCS-Bogoliubov gap equation 2,3 is a nonlinear integral equation where the solution u is a function of the absolute temperature T and the energy x, and ω D stands for the Debye angular frequency and is a positive constant. The potential ⋅ ⋅ U( , ) satisfies U(x, ξ) > 0 at all  ξ ε ω ∈ x ( , ) [ , ] D 2 . Throughout this paper we use the unit where the Boltzmann constant k B is equal to 1.
Remark 1.1. In (1.1), we introduce a cutoff ε > 0 and fix it. In the original BCS-Bogoliubov gap equation, one sets ε = 0 and does not introduce the cutoff ε > 0 since the effect of the region around the Fermi surface is very important in superconductivity (see, e.g. 4 ). But, if we do not introduce the cutoff ε > 0, then the first-order derivative of the thermodynamic potential with respect to T diverges logarithmically only at the transition temperature T c , and hence the entropy also diverges only at T c . Therefore, the transition from a normal conducting state to a superconducting state at T = T c is not a second-order phase transition. This contradicts a lot of experimental results that the transition is a second-order phase transition without an external magnetic field. Therefore, we introduce the cutoff ε > 0 and fix it. For more details, see Remarks 1.10 and 1.11 below.
We consider the solution u to the BCS-Bogoliubov gap equation as a function of T and x, and deal with the integral with respect to the energy ξ in (1.1). Sometimes one considers the solution u as a function of the absolute temperature and the wave vector, and accordingly deals with the integral with respect to the wave vector over the three dimensional Euclidean space  3 . In this situation, the existence and uniqueness of the solution were established and studied in [5][6][7][8][9][10][11][12][13][14][15][16][17] . For interdisciplinary reviews of the BCS-Bogoliubov model of superconductivity, see Kuzemsky 18 [Chapters 26 and 29] and 19,20 . From the viewpoint operator theory, the present author studied the temperature dependence of the solution and showed the second-order phase transition in the BCS-Bogoliubov model of superconductivity (see 1,[21][22][23][24]. In this connection, the BCS-Bogoliubov gap equation plays a role similar to that of the Maskawa-Nakajima equation 25,26 . If there is a nonnegative solution to the Maskawa-Nakajima equation (resp. to the BCS-Bogoliubov gap equation), then the massless abelian gluon model (resp. the BCS-Bogoliubov model) exhibits the spontaneous breaking of the chiral symmetry (resp. the U(1) symmetry). If there is a unique solution 0 to the Maskawa-Nakajima equation (resp. to the BCS-Bogoliubov gap equation), then the massless abelian gluon model (resp. the BCS-Bogoliubov model) realizes the chiral symmetry (resp. the U(1) symmetry). In fact, the Maskawa-Nakajima equation has attracted considerable interest in elementary particle physics, and is applied to many models such as a massless abelian gluon model, a massive abelian gluon model, a quantum chromodynamics (QCD)-like model, a technicolor model and a top quark condensation model. In Professor Maskawa's Nobel lecture, he stated the reason why he reconsidered the spontaneous chiral symmetry breaking in a renormalizable model of strong interaction. See the present author's paper 23 where the temperature τ 1 > 0 is defined by (see 2 and 27,28 ) Here the solution becomes a function of the temperature T only, and so we denote the solution by Δ 1 .
Physicists and engineers studying superconductivity always assume that there is a unique nonnegative solution Δ 1 to the simple gap equation (1.2) and that the solution Δ 1 is of class C 2 with respect to T. And they differentiate the solution with respect to T without showing that it is differentiable with respect to T. As far as the present author knows, no one except for the present author gave a mathematical proof for these assumptions; the present author 1,21 applied the implicit function theorem to (1.2) and gave a mathematical proof: Proposition 1.2 ([Proposition 1.2 [21] ]). Let U 1 > 0 be a positive constant and set U( We introduce another positive constant U 2 > 0. Let 0 < U 1 < U 2 and set U( . Then a similar discussion implies that for U 2 , there is a unique nonnegative solution Δ 2 :[0, Here the temperature τ 2 > 0 is defined by Note that the solution Δ 2 to (1.3) has properties similar to those of the solution Δ 1 to (1.2).
We next turn to the BCS-Bogoliubov gap equation (1.1). We assume the following condition on the potential.
consisting of continuous functions of the energy x only, and deal with the following temperature dependent subset V T : with T fixed is continuous with respect to the energy x and varies with the temperature as follows: The existence and uniqueness of the transition temperature T c were pointed out previously (see 8,11,13,17 ). In our case, we can define it as follows.
be as in Theorem 1.7. Then the transition temperature T c is defined by Actually, Theorem 1.7 tells us nothing about continuity (or smoothness) of the solution u 0 with respect to the temperature T. From the viewpoint of operator theory, the present author 22 [Theorem 1.2] showed that u 0 is indeed continuous both with respect to T and with respect to x under the restriction that T is sufficiently small. Moreover, under a similar restriction, the present author and Kuriyama 24 [Theorem 1.10] showed that the solution u 0 is partially differentiable with respect to T twice, that the first-order and second-order partial derivatives of u 0 are both continuous with respect to (T, x), and that u 0 is monotone decreasing with respect to T from the viewpoint of operator theory. As mentioned before, the present author 1 [Theorems 2.3 and 2.4] showed that the solution is partially differentiable with respect to T (in the neighborhood of the transition temperature T c ) twice, and gave a proof of the statement that the transition from a normal conducting state to a superconducting state is a second-order phase transition from the viewpoint of operator theory.
Let us turn to the thermodynamic potential. The thermodynamic potential Ω is given by the partition function Z: As mentioned before, we use the unit where the Boltzmann constant k B is equal to 1 throughout this paper. We fix both the chemical potential and the volume of our physical system, and so we consider the thermodynamic potential Ω as a function of the temperature T only. Let T c be the transition temperature (see Definition 1.8), and let u 0 be the solution to the BCS-Bogoliubov gap equation (1.1). Then the thermodynamic potential Ω in the BCS-Bogoliubov model becomes Here, μ > 0 is the chemical potential and is a positive constant, N(ξ) ≥ 0 stands for the density of states per unit energy at the energy ξ (−μ ≤ ξ < ∞). We assume that Remark 1.9. If the solution u 0 to the BCS-Bogoliubov gap equation (1.1) is partially differentiable with respect to the temperature T twice, then the thermodynamic potential Ω is differentiable with respect to T twice. Then the specific heat at constant volume at T is given by Therefore the gap ΔC V in the specific heat at constant volume at the transition temperature T c is given by Remark 1.10. When we try to show the second-order phase transition, we need to differentiate the thermodynamic potential with respect to the temperature T twice. The first-order derivative (∂Ψ/∂T) of the thermodynamic potential Ψ (see (1.6)) with respect to T has the following term in it: www.nature.com/scientificreports www.nature.com/scientificreports/

Here the function v(·) is that in Condition (C) of Section 2 and is continuous on
This means that if ε = 0, then the first-order derivative of the thermodynamic potential with respect to T diverges logarithmically only at the transition temperature T c , and that the entropy also diverges only at T c . Therefore, if ε = 0, then the transition from a normal conducting state to a superconducting state at T = T c is not a second-order phase transition. This contradicts a lot of experimental results that the transition is a second-order phase transition without an external magnetic field. This is why we introduce ε > 0 in the thermodynamic potential. This means that the solution u 0 (T, x) to the BCS-Bogoliubov gap equation (1.1) is defined at x ≥ ε, and hence that the range of integration in the right side of (1.1) is from ε to ω D . This is why we introduce the cutoff ε > 0 also in the BCS-Bogoliubov gap equation (1.1).
If ε = 0, then the integral (1.9) again diverges logarithmically. Therfore, if ε = 0, then the first-order derivative of the thermodynamic potential with respect to T again diverges logarithmically only at the transition temperature T c , and the transition from a normal conducting state to a superconducting state at T = T c is not a second-order phase transition. We again reach a contradiction. This is why we introduce the cutoff ε > 0 both in the thermodynamic potential and in the BCS-Bogoliubov gap equation (1.1).

Main Results
In the physics literature, one differentiates the solution to the BCS-Bogoliubov gap equation, the thermodynamic potential and the critical magnetic field with respect to the temperature without showing that they are differentiable with respect to the temperature. So we need to show that they are differentiable with respect to the temperature, as mentioned in the preceding section.
We introduce the cutoff ε > 0 and assume that the potential ⋅ ⋅ U( , ) satisfies (1.4) throughout this paper. We denote by z 0 > 0 a unique solution to the equation = z tanh z 2 (z > 0). The value of z 0 is nearly equal to 2.07, and the inequality ≤ z tanh  We then define our operator A (see (1.1)) on V: . The following is one of our main results.

Theorem 2.2. Let us introduce the cutoff ε > 0 and assume (1.4). Let V be as above. Then our operator
 . Moreover, u 0 is monotone decreasing and Lipschitz continuous with respect to T, and satisfies Furthermore, if u 0 ∈ V, then u 0 is partially differentiable with respect to T twice, and the first-order and second-order partial derivatives of u 0 are both continuous on then the solution in the thermodynamic potential Ψ(T) (see (1.6)) is nothing but this u 0 ∈ V, and hence the solution in Ψ(T) is partially differentiable with respect to the temperature T twice. So we can differentiate the thermodynamic potential Ψ(T) with respect to the temperature T twice. On the other hand, is approximated by a suitably chosen element u 1 ∈ V. In such a case, we replace the solution in Ψ(T) by this element u 1 ∈ V. Let us remind here that the element u 1 ∈ V is partially differentiable with respect to the temperature T twice. Once we replace the solution in Ψ(T) by this element u 1 ∈ V, we can again differentiate the thermodynamic potential Ψ(T) with respect to the temperature T twice. In this way, in both cases, we can differentiate the thermodynamic potential Ψ(T), and hence Ω(T) with respect to the temperature T twice.
Remark 2.4. The behavior of the solution u 0 given by Theorem 2.2 is in good agreement with the experimental data.
The function  (1.4)). Note that the function is also continuous. Here, 0 < τ < T c . We then consider the sum of the two continuous functions above: Note that the second term just above tends to zero as Δ 2 (τ)/ε goes to zero. Let τ be very close to T c and let Δ 2 (τ)/ε be very small so that the inequality We then fix τ and ε, and we deal with the set . Note that the left side of the inequality just above is a continuous function of Remark 2.5. We let τ be very close to T c , and we let Δ 2 (τ)/ε be very small so that (2.3) holds true. Let us consider the following condition. Condition (C). Let τ and ε be as above. An element is partially differentiable with respect to the temperature T ∈ [τ, T c ) twice, and the partial derivatives (∂u/∂T) and (∂ 2 u/∂T 2 ) both belong to D satisfying the following: Here, δ does not depend on We denote by W the following subset of the Banach space τ ε ω and we define our operator A (see (1.1)) on W: If u 0 ∈ W, then the solution in the thermodynamic potential Ψ(T) (see (1.6)) is nothing but this u 0 ∈ W, and hence the solution in Ψ(T) satisfies Condition (C). So we can differentiate the thermodynamic potential Ψ(T) with respect to the temperature T twice. On the other hand, if  is approximated by a suitably chosen element u 1 ∈ W. In such a case, we replace the solution in Ψ(T) by this element u 1 ∈ W. Let us remind here that the element u 1 ∈ W satisfies Condition (C). Once we replace the solution in Ψ(T) by this element u 1 ∈ W, we can again differentiate the thermodynamic potential Ψ(T) with respect to the temperature T twice. In this way, in both cases, we can differentiate the thermodynamic potential Ψ(T), and hence Ω(T) with respect to the temperature T twice.
Let www.nature.com/scientificreports www.nature.com/scientificreports/ Note that g(η) < 0. As mentioned before, if the solution u 0 to the BCS-Bogoliubov gap equation (1.1) is partially differentiable with respect to the temperature T twice, then the thermodynamic potential Ω is differentiable with respect to T twice, and the specific heat at constant volume at T is given by Therefore the gap ΔC V in the specific heat at constant volume at the transition temperature T c is given by (see Remark 1.9) Remark 2.13. In the physics literature, one differentiates the thermodynamic potential to obtain the specific heat at constant volume without showing that the thermodynamic potential is differentiable with respect to T. Note that the thermodynamic potential has the solution to the BCS-Bogoliubov gap equation (1.1) in its form. In other words, one differentiates the thermodynamic potential with respect to T without showing that the solution is differentiable with respect to T. But Combining Theorems 2.2 and 2.10 with Remarks 2.3 and 2.12 implies that we can differentiate the solution u 0 , and hence the thermodynamic potential Ω with respect to T twice. .
c is explicitly and exactly given by the expression  is very large in many superconductors. The following then gives that the expression just above does not depend on superconductors and is a universal constant.
which does not depend on superconductors and is a universal constant.
in the BCS-Bogoliubov model of superconductivity. Here, ζ  s s ( ) is the Riemann zeta function. Therefore, Corollary 2.16 gives another expression for c . Note that we use the unit where k B = 1. Let us turn to the critical magnetic field applied to type-I superconductors. It is well known that superconductivity is destroyed even at a temperature T less than the transition temperature T c when the sufficiently strong magnetic field is applied to type-I superconductors. It is also known that, at a fixed temperature T, superconductivity is destroyed when the applied magnetic field is stronger that the critical magnetic field H c (T), and that superconductivity is not destroyed when the magnetic field is weaker than H c (T). The critical magnetic field H c (·) is a function of the temperature T, and H c (T) ≥ 0 at T ≤ T c . The critical magnetic field is related to Ψ(T) (see (1.6)) as follows: Remark 2.18. In the physics literature, one differentiates the thermodynamic potential, and hence the critical magnetic field with respect to T without showing that they are differentiable with respect to T. Note that the thermodynamic potential has the solution to the BCS-Bogoliubov gap equation (1.1) in its form. In other words, one differentiates the critical magnetic field with respect to T without showing that the solution is differentiable with respect to T. But Combining Theorems 2.2 and 2.10 with Remarks 2.3 and 2.12 implies that we can differentiate the solution u 0 , and hence the critical magnetic field H c (·) with respect to T.
The following gives the smoothness of the critical magnetic field with respect to T and some of its properties. (

Proof of Theorem 2.2
We prove Theorem 2.2 in this section. Our proof is similar to that of 24  since T ≤ τ 3 < τ 0 . Therefore, 1 > U 1 a. Here, a is that in (3.1). Let us choose U 2 (>U 1 ) such that 1 > U 2 a holds true. Set Lemma 3.1. The subset V is bounded, closed, convex and nonempty.

Proof.
We have only to show that the subset V is convex.
n n n n n n n n it follows that n n is partially differentiable with respect to T twice, and  www.nature.com/scientificreports www.nature.com/scientificreports/ Step 2. We next show + − ∈ tu tv V (1 ) . Since . Thus the subset V is convex. ◻ A proof similar to that of 24 [Lemma 2.5] gives the following.
A proof similar to that of 24 [Lemma 2.4] gives the following.
A proof similar to that of 24 [Lemma 2.6] gives the following.
A straightforward calculation gives the following.

Lemma 3.5. Let u ∈ V. Then Au is partially differentiable with respect to T twice
Proof. By the preceding lemma, Au is partially differentiable with respect to T twice.
Step 1. We first show at T = 0. Thus Step 2. We next show at T = 0. Thus We thus have the following.
A proof similar to that of 24 [Lemma 2.9] gives the following.

Lemma 3.8. The set AV is relatively compact.
A proof similar to that of 24  .
A proof similar to that of 24  , and . Lemma 3.13. The set AV is uniformly bounded and equicontinuous, and hence the set AV is relatively compact.
2 for ∈ u V , the set AV is uniformly bounded. By an argument similar to that in the proof of Lemma 3.4, the set AV is equicontinuous. Hence the set AV is relatively compact.
◻ By an argument similar to that in the proof of Lemma 3.9 gives the following.
is continuous. Lemmas 3.13 and 3.14 immediately imply the following.
. A proof similar to that of Lemma 3.2 gives the following.

Proof.
A straightforward calculation gives the result. ◻ Let u ∈ W and let v be as in Condition (C). Here, v depends on the u. We set Since R > 0 is arbitrarily large, the result follows ◻. The lemmas above immediately give the following.

Proofs of Theorem 2.14 and Corollary 2.16
Proof of Theorem 2.14 We first give a proof of Theorem 2.14. The thermodynamic potential Ω N corresponding to normal conductivity is given by (1.5). The specific heat at constant volume at the temperature T is defined by