Another operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity

In the preceding papers, imposing certain complicated and strong conditions, the present author showed that the solution to the BCS-Bogoliubov gap equation in superconductivity is twice differentiable only on the neighborhoods of absolute zero temperature and the transition temperature so as to show that the phase transition is of the second order from the viewpoint of operator theory. Instead, we impose a certain simple and weak condition in this paper, and show that there is a unique nonnegative solution and that the solution is indeed twice differentiable on a closed interval from a certain positive temperature to the transition temperature as well as pointing out several properties of the solution. We then give another operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model. Since the thermodynamic potential has the squared solution in its form, we deal with the squared BCS-Bogoliubov gap equation. Here, the potential in the BCS-Bogoliubov gap equation is a function and need not be a constant.

In 1-3,6-22 , the existence, the uniqueness and several properties of the solution to the BCS-Bogoliubov gap equation were established and studied. See also Kuzemsky 23 [Chapters 26 and29] and 24,25 . Anghel and Nemnes 26 and Anghel 27,28 showed that if the physical quantity µ in the BCS-Bogoliubov model is not equal to the chemical potential, then the phase transition from a normal conducting state to a superconducting state is of the first order under a certain condition without any external magnetic field. Introducing imaginary magnetic field,  pointed out that the phase transition is of the second-order if and only if a certain value is greater than 17 − 12 √ 2 and that the phase transition is of order 4n + 2 if and only if the value above is less than or equal to 17 − 12 √ 2 . Here. n is an arbitrary positive integer. In this connection, the BCS-Bogoliubov gap equation in superconductivity plays a role similar to that of the Maskawa-Nakajima equation 33,34 in elementary particle physics. In Professor Maskawa's Nobel lecture, he stated the reason why he considered the Maskawa-Nakajima equation. See the present author's paper 35  where f ∈ W (see (2.2) below for the subset W). We define our operator A on the subset W and look for a fixed point of our operator A. Note that a fixed point of A becomes a solution to the squared BCS-Bogoliubov gap equation, and that its square root becomes a solution to the BCS-Bogoliubov gap equation (1.1).
Let U 1 and U 2 be positive constants, where (0 <) U 1 ≤ U 2 . If the potential U(·, ·) is a positive constant and U(x, ξ) = U 1 at all (x, ξ) ∈ I 2 , then the solution to the BCS-Bogoliubov gap equation (1.1) becomes a function of the temperature T only. Denoting the solution by T → � 1 (T) , we have (see 4 ) Here, the temperature τ 1 > 0 is defined by (see 4

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The solution T → � 1 (T) is continuous and strictly decreasing with respect to T, and moreover, the solution is of class C 2 with respect to T. For more details, see 20 We set � 1 (T) = 0 at all T ≥ τ 1 . Then (1.

Main results
Suppose that the potential U(·, ·) in the BCS-Bogoliubov gap equation (1.1) satisfies the following conditions: Let W be a subset of the Banach space C(D) satisfying Here, the norm of the Banach space C(D) is given by and Note that

Remark 2.3
The conditions imposed in the previous papers of the present author 1 [Condition (C)] and 2 [Condition (C)] were very complicated, and so it was very tough to show the existence, uniqueness and smoothness of the solution to the BCS-Bogoliubov gap equation (1.1). Instead, we impose the simple condition that f ∈ C 2 (D) and f (T c , x) = 0 in the definition of the subset W (see (2.2)). Thanks to this simple condition, it is straightforward to show the existence, uniqueness and smoothness of the solution.
We denote by W the closure of W with respect to the norm � · � mentioned above. The following are our main results.
In order to show that the transition from a normal conducting state to a superconducting state at T = T c is of the second-order, we need to deal with the thermodynamic potential and differentiate it with respect to the temperature T twice. Note that the thermodynamic potential has the fixed point f 0 ∈ W given by Theorem 2.  Proof (1) Note that the function T → � 2 (T) 2 is strictly decreasing (see 20 In order to show that W is convex, it suffices to show that www.nature.com/scientificreports/ Next let T = T c . We remind Remark 2.2 here. Then Therefore, W is convex. as n → ∞ . Thus tf + (1 − t)g ∈ W , and hence W is convex.
We next show that A : W → W.
Proof Let (T, x), (T 1 , x 1 ) ∈ D , and suppose T < T 1 < T c . We can deal with the case where T 1 = T c similarly . Then Step 1. A straightforward calculation gives where The function f is continuous since f ∈ W . Therefore, for an arbitrary where |T − T 1 | < δ with some ξ 1 ∈ I . Note that (see (1.3)) and that , , where Thus where |T − T 1 | < min(δ, δ 1 ).

Lemma 3.3 Let f ∈ W .
(1) Af is partially differentiable with respect to both T and x. Its first-order partial derivatives (Af ) T and (Af ) x are both continuous on D. Therefore, Af ∈ C 1 (D). (2) Af is twice partially differentiable with respect to both T and x. Its second-order partial derivatives (Af ) TT , (Af ) Tx = (Af ) xT and (Af ) xx are all continuous on D. Therefore, Af ∈ C 2 (D).

Proof (1) Let us show that
It follows from f (T c , ξ) = 0 (see (2.2)) that for some T 1 (T < T 1 < T c ) . Then Since T is in a neighborhood of T c , we let T ≥ T c /2 . Therefore, where the right side is independent of T and is Lebesgue integrable on I. Thus, as T ↑ T c , Therefore, Af is partially differentiable with respect to T at (T c , x 0 ) , and We next show that (Af ) T is continuous at (T c , x 0 ) . Here, where Note that Here, we assumed T ≥ T c /2 . The right side of this inequality is independent of T and is Lebesgue integrable on I 2 , and so (as T ↑ T c ) Similarly we can show that as T ↑ T c . Moreover, we have similarly that as x → x 0 . It thus follows from (3.3) that (Af ) T is continuous at (T c , x 0 ) . Similarly we can show the rest of (1), and (2). Note that (Af ) TT is given as follows.
where A proof similar to that of 20 [Lemma 3.4] gives the following. , Vol First let T < T c . Then B < A since tanh z > z cosh z at z > 0 , f (T, ξ) > 0 and ξ ≥ ε . Therefore, √ a AB < (a 1/4 A) 2 ≤ 1 by (3.4). Thus as long as Here,  (1.5)). Therefore, the preceding lemma holds true not only when the potential U(·, ·) in the BCS-Bogoliubov gap equation (1.1) is a positive constant, but also when U(·, ·) is a function.

Lemma 3.9
The set AW is equicontinuous.
Proof Let f ∈ W . Let (T, x), (T 1 , x 1 ) ∈ D and suppose T < T 1 < T c . We can deal with the case where T 1 = T c similarly. Then The preceding lemma gives Here, T < T 2 < T 1 and ξ ∈ I . Therefore, a proof similar to that of Lemm 3.2 gives x) . We next extend the domain W of our operator A to its closure W with respect to the norm � · � of the Banach space C(D).

Lemma 3.12 A : W → W.
Proof For f ∈ W , there is a sequence {f n } ∞ n=1 ⊂ W satisfying �f − f n � → 0 as n → ∞ . By the preceding lemma, Therefore, the sequence {Af n } ∞ n=1 ⊂ W is a Cauchy sequence. Hence there is an element F ∈ W satisfying �F − Af n � → 0 as n → ∞ . Note that the element F does not depend on how to choose the sequence {f n } ∞ n=1 ⊂ W , as shown below. Suppose that there is another sequence {g n } ∞ n=1 ⊂ W satisfying �f − g n � → 0 as n → ∞ . Similarly, the sequence {Ag n } ∞ n=1 ⊂ W becomes a Cauchy sequence, and hence there is an element G ∈ W satisfying �G − Ag n � → 0 as n → ∞ . Then as n → ∞ . Therefore, F = G , and hence F does not depend on how to choose the sequence in W. Thus we define F = Af . The result thus follows. www.nature.com/scientificreports/ fixed point f 0 ∈ W given by Theorem 2.5 in its form, not the solution f 0 to the BCS-Bogoliubov gap equation. Suppose that the fixed point f 0 is an element of the subset W. It then follows immediately from Theorem 2.5 that f 0 ∈ C 2 (D) . Hence the thermodynamic potential with the fixed point f 0 satisfies all the conditions in the operator-theoretical definition of the second-order phase transition (see 2 [Definition 1.10]). We thus apply a proof similar to that of 2 [Theorem 2.4] to have Theorem 2.8. Suppose that the fixed point f 0 is an accumulating point of the subset W. We then replace the fixed point f 0 ∈ W \ W in the form of the thermodynamic potential by a suitably chosen element of f ∈ W since the fixed point f 0 is an accumulating point of the subset W. Thanks to Theorem 2.5, we find that the suitably chosen element f is in C 2 (D) . Then we can differentiate the suitably chosen element f with respect to the temperature T twice. Therefore, once we replace the fixed point f 0 ∈ W \ W in the form of the thermodynamic potential by a suitably chosen element of f ∈ W , we can again show that the thermodynamic potential with this f ∈ W satisfies all the conditions in the operator-theoretical definition of the second-order phase transition. We can again apply a proof similar to that of 2