Cluster Decomposition Principle and Two-Electron Wave Function of the Cooper Pair in the BCS Superconducting State

We present the explicit forms of the maximum eigenvalue and the corresponding eigenfunction for the second-order reduced density matrix (RDM2) of the BCS superconducting state (SS). Using these quantities, we deal with two topics in the present paper. As the first topic, it is shown that the cluster decomposition principle holds in the BCS-SS. This proof gives a theoretical foundation that the abnormal density can be chosen as the order parameter of the SS. As the second topic, it is shown that such an eigenfunction is spin singlet and spatially extends isotopically, and further that the mean distance of two electrons which consists of the above eigenfunction is in a good agreement with Pippard's coherence length. This means that maximum geminal of the RDM2 of the BCS-SS can be regarded as the Cooper pair itself which are condensed to the same energy level in a number of O(N).

The definition of the superconductivity is given on the basis of the Bose-Einstein condensation (BEC) of the fermion system [17,18]. According to this definition, when the system in the SS, the magnitude of the maximum eigenvalue of the second-order reduced density matrix (RDM2) is O N ( ) and the maximum geminal belongs to type (b) defined in the previous paper [19]. Here note that the maximum geminal of the RDM2 is defined as the eigenfunction of the RDM2 with the maximum eigenvalue. In the previous paper, it has been pointed out that the abnormal density can be an OP of the SS only if the cluster decomposition principle holds [19]. However, it is not obvious whether the cluster decomposition principle holds or not in the SS. In other words, the cluster decomposition principle has not well argued in the SSs including the BCS-SS, so far. Therefore, the suspicion whether the abnormal density is appropriate as the order parameter of the SS has remained persistent up to the present even at the level of BCS-SS. In this paper, we clear this persistent suspicion at the level of BCS-SS. That is to say, the cluster decomposition principle is proven true in the BCS-SS. Using this proof, we can say that the abnormal density corresponds to the maximum geminal of the RDM2 of the BCS-SS multiplied by the square root of twice the number of maximum geminal in the SS.
In the previous paper [19], it has been also shown that the maximum geminal is condensed to one level in a number of O N ( ) when the system is in the SS. In this paper, it is shown that the maximum geminal is spin singlet and has s-wave symmetry, and further that the mean distance of electrons which consist of the maximum geminal is in a good agreement with Pippard's coherence length of the BCS theory [20][21][22][23]. This means that the maximum geminal of the RDM2 of the BCS-SS corresponds to the Cooper pair itself of the BCS-SS.
Thus, we give answers to the fundamental questions ' Is the abnormal density appropriate for the OP of the SS?', and 'what is the relation between the maximum geminal and Cooper pair?' in the case of the BCS-SS, which is the most typical case of the SS's.

Cluster decomposition principle in the BCS superconducting state
First, we shall prove that the cluster decomposition principle holds in the BCS-SS. The proof is performed in the following 5 steps.
(1) Notations for various physical quantities are given for the later convenience.
(2) The cluster decomposition principle is reviewed.
(3) The RDM2 of the BCS-SS is calculated.
(4) The eigenvalue and eigenfunction for the RDM2 of the BCS-SS are found.
(5) The expectation value of the abnormal density with respect to the BCS-SS is calculated. The result of Step (4) and that of Step (5) are compared to each other, so that it is confirmed that the cluster decomposition principle holds in the BCS-SS [19].
Step (1) We shall define the notations of various physical quantities as a preparation. A free electron state with the wavenumber k and spin s =  , Using the creation operator s C , The coordinate representation of equation ( Step (2) If there does not exist any correlation between two systems which separate from each other in an infinite distance, the cluster decomposition principle holds [19,25]. Specifically, we consider applying the following limits to the RDM2 [19]: , a n d . 6  (7) is called the cluster decomposition principle of the RDM2. It should be noted that the cluster decomposition principle is different from Wick's theorem. The cluster decomposition principle means that distant experiments yield uncorrelated results [25]. As mentioned above, it is not well argued whether equation (7) holds or not when Fñ | is the SS.
Step (3) An explicit form of the BCS wave function Q ñ BCS | is given in Appendix A [22]. Let us consider the expectation value of equation (5) with respect to Q ñ.

|
It is calculated as where v k represents the probability amplitude such that v k 2 | | gives the probability of two electron state  -ñ k k , | being occupied in the BCS-SS, and where the state Q ñ k BCS |˜which appears in equation (8), is calculated as where u k represents the probability amplitude such that u k 2 | | gives the probability of two electron state  -ñ k k , | being unoccupied in the BCS-SS. Details of derivations of equations (8) and (9) are given in Appendix A.
Step (4) We shall consider the eigenvalue and eigenfunction of equation (8). The eigenvalue equation of equation (8) is written as [19,24] ò ò respectively. The maximum eigenvalue and the corresponding eigenfunction can be found by utilizing equation (3). The result obtained is is the eigenfunction which yields the maximum eigenvalue, which is hereafter called the maximum geminal of the RDM2 of the BCS-SS. As shown below, it can be confirmed that equation (11) actually gives the maximum eigenvalue. Substituting equations (8), (9) and (11) into the LHS of equation (10), we get where equation (A-6) shown in Appendix A is used. In the RHS of equation (12), the magnitude of v k 4 | | is less than or equal to unity while that of ( ) The proof is given in Appendix B. Therefore equation (12) is reasonably approximated as where equation (11) is used. Since the magnitude of the eigenvalue for the RDM2 means the occupation number of two-particle states [17,18], and since the magnitude of the eigenvalue for equation (13) | | corresponds to the maximum eigenvalue of the RDM2 of the BCS-SS. Namely, equation (13) is an eigenvalue equation of RDM2 of the BCS-SS with the maximum eigenvalue and the corresponding eigenstate.
Considering the normalization constant of the eigenfunction, the eigenvalue equation (13) is rewritten as Step (5) In this step, we shall show that the BCS-SS satisfies the cluster decomposition principle. Using equations (A-1) and (A-4) shown in Appendix A, the expectation value of the abnormal density with respect to the BCS-SS is given by Using the commutation relations of the operators B k and B , k † equation (17) is rigorously calculated to be Further using equations (11) and (15), the above equation finally becomes On the other hand, if the cluster decomposition principle holds for the state Fñ, | then equation (7) holds. Using this fact and further using the spectrum decomposition of the RDM2 [19], it has been shown in the previous paper that if the cluster decomposition principle holds for the state Fñ, | then the following relation is established [19]: where Fñ | is the general SS not limited to the BCS-SS, and where n z z ¢ ¢ F r r , max ( )is the maximum geminal of the RDM2 of Fñ | [19]. By employing this result, it is concluded from equation (19) that the cluster decomposition principle holds in the BCS-SS Q ñ.

Features of the maximum geminal
We shall rewrite the maximum geminal n z z ¢ ¢ Q r r , max BCS ( )by using the relative coordinates, r = -¢ r r , and center of mass coordinates, = + ¢ R r r 2. where W is the volume of the system. Note that n z z ¢ ¢ Q r r , max BCS ( )is not dependent on R but is dependent only on the relative coordinates r, which means that the maximum geminal extend homogeneously in the system.
If we consider the case where the attractive interaction between electrons is isotropic, then the gap parameter is independent of the wavevector, i.e., the gap parameter is approximated as D 0 (=constant) [22,23]. In this case, it can be shown that the maximum geminal n z z ¢ ¢ Q r r ,

Mean distance of two electrons in the maximum geminal
The mean-square-distance between two electrons which consist of the maximum geminal n ñ Substituting equation (21) into equation (24), we get the following result after careful calculations: where v F is the Fermi velocity. In the derivation of equation (25), we consider the case where the attractive interaction between electrons is isotropic, and employ an approximation Therefore, the root-mean-square-distance (which is abbreviated as mean distance) between two electrons for the maximum geminal is given by Using the result of the BCS theory D = k T 3.53 2 B c 0 [22,23], equation (26) is rewritten as where T c is the critical temperature of the SS. As is well known, Pippard's coherence length is given by l = v k T 0.1803 P F Bc ( )in the BCS theory [20,22]. Equation (27) is in a good agreement with l . P Pippard's coherence length has been considered to be the mean distance of two electrons which consist of the Cooper pair [20,21,23]. Therefore, the maximum geminal of the BCS-SS can be reasonably regarded as the Cooper pair itself from the viewpoint of the present scheme using the RDM2 of the BCS-SS. Thus, we can say that the SS is the state such that the maximum geminal of the RDM2 of the SS is condensed as the Cooper pair in a number of O N .
( ) Furthermore, it can be also confirmed that the BCS-SS actually meets the condition of the superconductivity defined in the previous paper [19] because the maximum geminal belongs to type (b) which is localized in a smaller region than the whole system in relative coordinates [19].

Discussions and concluding remarks
In this paper we deal with a long-standing suspicion as to whether the abnormal density is appropriate as the order parameter of SS or not. We prove that the cluster decomposition principle holds in the BCS-SS. Specifically, equation (19) can be proved true in the BCS-SS, so that the long-standing suspicion is dispelled at the level of BCS-SS. Although we discuss only the BCS state that is a special case but the most typical case of the superconductivity, this result means that the abnormal density may be an OP of the SS [19]. In other words, the present result may give a theoretical foundation of the facts that the abnormal density has been commonly used as the OP of the SS in the first-principles theory including the SC-DFT [5][6][7][8][9][10][11][12][13][14][15][16], and that the calculation results can explain the experiments, especially the critical temperature, reasonably well in the SC-DFT [5][6][7][8][9][10][11][12][13][14][15][16]. Superficially regarded, it seems to be strange to adopt the abnormal density as the OP of SS, because the abnormal density takes nonzero value only if the number of electrons in the system of interest is not invariant. However, it is not strange because the system of interest is an electron system that consists of superconducting electrons and may change the number of superconducting electrons.
In addition to the above, the cluster decomposition principle can be proven true from the other viewpoint. As is well known, the fluctuation of the particle number is given by in the BCS theory [22,23]. Here we use the present result given by equation (15). On the other hand, in the previous paper [19], we have discussed the fluctuation of particle number in a general state not limited to the BCS state. Within the Hartree-Fock approximation it is calculated as [19] ò åå y z y z y z y z Difference between factors 4 and 2, which are seen in equations (28) and (30), seems to be caused by the Hartree-Fock approximation adopted in the previous paper [19]. This difference is not essential to the estimation of the fluctuation of particle number. The crucial point is that the fluctuation of the particle number is O n max 2 ( ) ( ) if the system is in the SS. A scale agreement between equations (28) and (30) also means that the cluster decomposition principle holds in the BCS-SS.
In the present paper, we also prove that the maximum geminal of the RDM2 of the BCS-SS can be regarded as the Cooper pair itself. Specifically, the maximum geminal is the spin singlet and spatially extends like s-wave, and the mean distance of electrons of the maximum geminal is in a good accordance with Pippard's coherence length of the BCS theory. It can be said that the SS is a many-electron state where thus-defined Cooper pair is condensed in a number of O N .
( ) In what follows, we shall reiterate the importance of this achievement. Generally, there exist two kinds of pictures for the SS. The first one is to see the SS as the set of the Cooper pairs. In this picture, the superconductor is considered to be a set of the Cooper pairs which are condensed to the same energy level in a number of O N .
( ) Then, the Cooper pair in the case of the BCS-SS is spin singlet and extends like s-wave within the range of about Pippard's coherence length. The second picture is to describe the SS by means of the many-body wave function such as the BCS wave function. In this picture, the condensation of electron pairs to the same energy level is not explicitly expressed, but occupations of electron states within the free-electron Fermi surface are simply expressed. Unfortunately, the correspondence between these two pictures is not always trivial. This is because it is generally difficult in the many-electron system to define not only a single state (one-electron state) but also a pair state (two-electron state) due to the interaction between electrons. That is to say, it is generally difficult to construct one-electron picture (two-electron picture) in which many-electron state is described by a single configuration of a set of one-electron states (two-electron states). Therefore, it is difficult to define the Cooper pair from the SS that is of course a kind of many-electron state. In this paper, the problem is overcome by investigating the properties of the maximum geminal of the RDM2 of the SS.
Thus, the present work gives an answer to the issues 'what is the OP of SS?', and 'what is the Cooper pair in the SS?' in the BCS-SS. Although the state to handle in this paper is the BCS-SS, this state is the most typical SS and therefore seems to be meaningful as a prototype of the various types of SSs.

| | ( -)
where D ¢ k is the gap parameter of the BCS theory and where x ¢ k is the free-electron energy shifted by the chemical potential m, i.e.,