Entanglement entropy of the maximum geminal of the BCS ground state

From the viewpoint of the Bose–Einstein condensation (BEC) of the fermion system, the maximum geminal of the second-order reduced density matrix of the superconducting state exactly corresponds to the Cooper pair. In this paper the entanglement entropy (EE) for the maximum geminal of the BCS ground state is evaluated. The EE behaves logarithmically with respect to the number of the maximum geminal. Furthermore, the disappearance point of superconductivity is defined on the basis of the fermion BEC. In the superconducting ground state, almost all electrons in the energy width of the gap parameter near the Fermi level are condensed as a maximum geminal. They suddenly change to normal electrons with a finite gap of the EE at the disappearance point like a first-order phase transition.

If we attempt to argue the electron correlation in the SS by means of the degree of the entanglement, the EE between electrons composing the Cooper pair should be treated directly.The most appropriate way to give the definition of the Cooper pair is to use the concept of the BEC of a fermion system [39][40][41][42][43].Although the details are presented in section 2, outline of the previous works is that the eigenstate corresponding to the maximum eigenvalue for the second-order reduced density matrix (RDM2) of the SS can be exactly regarded as the Cooper pair [39][40][41][42][43].This eigenstate was named as maximum geminal in the previous paper [42,43], and we shall use this terminology also in this paper.An explicit form of the maximum geminal of the BCS ground state [44] has already been derived, which is the spin singlet and s-wave state that is spatially extended isotopically [43].Furthermore, the mean distance between the two electrons composing the maximum geminal is in a good agreement with Pippard's coherence length [43,45,46].To be exact, it is approximately 10 percent larger than Pippard's coherence length [43], but it well matches the commonly used mean distance of the Cooper pair.These results support that the maximum geminal can be exactly regarded as Cooper pair itself 4 .
So far, there have been several works concerning the EE of the SS [25][26][27][28][29][30].However, in these previous works [25][26][27][28][29][30], the entanglement between the electrons composing the Cooper pair has not been directly addressed.For example, the entanglement between the up-spin and down-spin has been discussed for the BCS ground state [25], but unfortunately it never deals with the maximum geminal of the BCS ground state.As mentioned above, the most interesting entanglement in the SS is the entanglement between the electrons composing the Cooper pair.In the SS the attractive force works substantially between electrons and forms a Cooper pair, which results in a change in the entanglement between electrons compared to the normal state.We should investigate the entanglement that newly occurs in the Cooper pair by treating the maximum geminal directly.In this paper, we shall calculate the EE of the maximum geminal of the BCS ground state.
The remainder of this paper is organized as follows.In section 2, we first verify the definition of the Cooper pair of the SS on the basis of the fermion BEC, which is specifically given by the maximum geminal of the RDM2 of the BCS ground state [42,43].In addition, the maximum geminal of a composite system is defined so as to discuss the EE for the maximum geminal of RDM2 of the BCS ground state.In section 3, the EE of the Cooper pair is calculated in cases of Al, In, Sn and Pb metals.It is shown that the EE exhibits the logarithmic behavior in terms of the number of the maximum geminal.In section 4, the disappearance of the SS is discussed using the mean distance between the two electrons composing the maximum geminal.It is shown that the EE discontinuously changes at the disappearance point like a first-order phase transition.Finally, in section 5, we reflect on the importance of the present work, and discuss its future application [47,48].

Geminal with maximum eigenvalue of the RDM2 of the BCS ground state
First, we shall give the definitions and notations of various physical quantities for preparation.Let a free electron state with the wavenumber k and spin ( ) As a special case of equation (2), the two-electron state |  -ñ k k , is written as The coordinate representation of equation (3) is given by the inner product with the state | ¢ z z¢ñ r r , ,which is denoted as , is also a two-electron state which is defined as where ( ) y z r and ( ) † y z r are the field operators of electrons, and r and z are the spatial and spin coordinates, respectively.

The maximum geminal of the BCS ground state
Next, we shall confirm the geminal with the maximum eigenvalue of RDM2 of the BCS ground state.It has already been derived in the previous work [43].As is well known, the BCS wave function is given by [44] The RDM2 of the BCS ground state can be obtained by taking the expectation value of equation (6) with respect to equation (5).This is given by [43] 4 To be precise, the mean distance of two electrons composing the maximum geminal is ¯( ) r = v k T 0.200315 F B C [43], while Pippard's coherence length is given by [44].Pippard's coherence length has been considered to be the mean distance of two electrons composing the Cooper pair [44][45][46].Therefore, the maximum geminal of the BCS-SS can be reasonably regarded as the Cooper pair itself from the viewpoint of the mean distance of two particles.(7), is calculated as [43] ˜| ˜( ) The maximum eigenvalue and corresponding eigenstate for equation (7) can be found by utilizing equation (8).
If these physical quantities are denoted as ( ) n max 2 and BCS respectively, then the eigenvalue equation for equation (7) is written as [43] and where V denotes the volume of the system.As mentioned in section 1, the two-particle state BCS has been called the maximum geminal of RDM2 of the BCS ground state [43].
The eigenvalue ( ) n max 2 corresponds to the occupation number of the maximum geminal in the BCS ground state [39][40][41][42].This becomes ( ) O N when the SS appears in the system [39-42].As briefly reviewed in section 1, the maximum geminal of the BCS ground state is the spin singlet state and spatially s-wave, and furthermore has a mean distance between constituent electrons which is comparable to Pippard's coherence length [43].These features for the maximum geminal coincide with those of the Cooper pair of the BCS ground state [43,51].In other words, the maximum geminal can be regarded as the Cooper pair itself.Equation (11) is a concrete expression for the Cooper pair of the BCS ground state.

Definition of the maximum geminal of a composite system
In order to discuss the EE for the maximum geminal of RDM2 of the BCS ground state, we shall introduce a geminal |n ñ Q max AB BCS of a composite system AB by assigning particles 1 and 2 of ( )

BCS
to particles belonging to the A and B systems, respectively.Specifically, we introduce the following geminal: , respectively.Using these equations, equation ( 12) is associated with equation (11) under the assignment that particles 1 and 2 of the maximum geminal belong to the A and B systems, respectively.It is analogous to the case of spin-singlet type Bell state of a composite system [47].
As is well known, the spin singlet consisting of spins 1 and 2 is explicitly written as By assigning spins 1 and 2 of ( ) z z F , singlet 1 2 to the spins belonging to systems A and B, respectively, the spinsinglet-type Bell state of a composite system AB has been introduced as follows [47]: Equation ( 14) has been used when discussing the entanglement between two spins that are components of the spin-singlet state [47].Just like introducing equation ( 14) from equation ( 13), here we introduce equation (12) from equation (11) so as to discuss the entanglement of the Cooper pair of the BCS ground state.
Using equation (12), we can evaluate the entanglement between the two electrons composing the maximum geminal of the BCS ground state.As a degree of entanglement, the von Neumann entropy [48,50] that is so called the EE is adopted in the present work.In the subsequent section, we shall calculate the EE of the Cooper pair using equation (12).

Entanglement entropy of the maximum geminal
In this section, the EE for the maximum geminal of RDM2 of the BCS ground state is calculated.First the reduced density operator for A system is derived on the basis of equation (12).Using this, we derive the expression for the EE.It is found that the present EE exhibits the logarithmical behavior in terms of the number of the maximum geminal.Furthermore, numerical calculations are performed for the present EE.It is much larger than that of the well-known singlet-type Bell state [48].

Reduced density operator for A system
The reduced density operator for the maximum geminal of a composite systems AB is defined as Substituting equation (12) into equation (15), we have The reduced density operator for A system is defined as max where Tr B is the partial trace over B system.Using equation ( 16), the r A max is calculated as where we assume that the system has the time reversal symmetry and/or spatial inversion symmetry; that is, it is assumed that the relations = - u u

Entanglement entropy
Using the reduced density operator for A system, the EE is defined as This S E max corresponds to the von Neumann entropy for r A max [50].Substituting equation (17) into equation (18), we obtain the EE for the maximum geminal of RDM2 of the BCS ground state: The occupation number of the maximum geminal ( ) Although u k and v k are not dependent on spin s, we shall formally denote Let the first term on the RHS of equation ( 22) be written as an integration in terms of the energy e.Using the density of states ( ) e N , we obtain 2 is used; m and D 0 are the chemical potential and gap parameter of the BCS ground state, respectively [44,51], and x is defined as x e m = -.As is shown in appendix A, the integrand in equation ( 23) exhibits a sharp peak in the vicinity of x = 0. Hence equation (23) can be approximated as Furthermore, the integration of the first term of equation ( 24) can be approximately calculated as p -D log 2, 0 which is analogous to the integration evaluated in the precedent work [43].In order to confirm the appropriateness of the approximations used here, numerical calculations of equation ( 23) are also performed.It is shown in appendix B that the approximations used above are appropriate.As a result, equation ( 23) is expressed as Using the previous result [43], ( ) n max 2 is expressed as Substituting equation (26) into equation (25), and using the fact that the chemical potential m can be approximately regarded as the Fermi energy e F at low temperature, the EE for the maximum geminal of the BCS ground state is finally given by where k F is the Fermi wavenumber, and where the density of states of free electrons is used as ( ) e N , the explicit form of which is ( ) ( )( 23212 and where the system is assumed to be a cubic with side L. As can be easily seen from equation (27), the EE shows the logarithmical behavior in terms of the gap parameter that is proportional to the number of the maximum geminal via equation (26).Such a logarithmical behavior is similar to those of many systems possessing long-range order, as mentioned in section 1 [19][20][21][22][23][24][25][26][27][28][29][30].In addition, the present EE has a volume dependence of the system.Specifically, the third term of RHS of equation ( 27) is proportional to the logarithm of ( ) = V L . 3 The former feature will be discussed in the next section along with the topic of the disappearance of superconductivity.Concerning the later feature, it would not be surprising if considering the fact that the EE is a quantity defined by taking the sum of the wave number vectors as shown in equation (22).

Numerical evaluations of S E
max for Al, In, Sn and Pb We shall calculate the EE of the maximum geminal of RDM2 of the BCS ground state for cases of Al, In, Sn and Pb metals.For simplicity, let the system be a cube with side [ ] = L 1 cm .The volume of the system is then given by ( ) 3 3 The physical quantities used in the numerical calculations, such as superconducting critical temperature, Fermi temperature, and so on, are listed in table 1 [48].This fact would be a merit for the application of superconductors in quantum teleportation.This topic may be a future work.
As mentioned in section 1, the present EE is different from spin EE [25].In the case of spin EE, the BCS ground state was regarded as the state of composite system AB by considering that the up-spin and down-spin states belong to the A and B systems, respectively [25].The spin EE is only measuring the entanglement between the up-spin and down-spin in the BCS ground state, and never measuring the entanglement of the Cooper pair itself.The formation of the Cooper pair causes changes in the entanglement between the two electrons composing the maximum geminal of the system.In this work, we directly calculate such an entanglement.

Occupation number of the maximum geminal
Besides the EE, the corresponding occupation number of the maximum geminal is also calculated and is listed in table 2. From appendix C, it is confirmed that almost all electrons within the energy width of the D 0 around the Fermi energy are condensed as the maximum geminal when the system is in the BCS ground state.It turns out that the intuitive picture of the fermion BEC [41,42] is actually realized.

Discussion
In this section, the disappearance of the superconductivity is discussed.First, we review the definition of the SS on the basis of the fermion BEC [42].The eigenvalue of the maximum geminal, i.e., ( )  n , max 2 generally corresponds to the occupation number of the Cooper pair [42].According to the previous work [42], we can say that the SS appears in the system when ( )  n max 2 becomes ( ) N O , and further when the maximum geminal belongs to type (b) which was defined in [42].Here note that the spatial broadening of the geminal is classified into two types ((a) and (b)) in [42].The geminal in type (a) is spatially extended with respect to both R and ρ, where R and ρ are the coordinate of center of gravity and relative coordinate, respectively.In case (a), the system is not under the superconducting state but under the normal state [42].On the other hand, the geminal in type (b) is extended with respect to R throughout the whole system, but is localized with respect to ρ in some limited region ( )  w V [42].In case (b), the system can be under the superconducting state [42].Thus, the condition for the system to be in the SS |Yñ is and r = - r r.
1 2 According to this definition, Cooper pairs occupy the same twoelectron state, the number of which is ( ) , and such two-electron state is spatially localized due to some attractive interaction between the two electrons [42].The SS is analogous to the BEC of ideal bosons, but the condition on the mean distance between two electrons composing the Cooper pair is added [42].

Disappearance of the superconductivity
It is difficult to define the disappearance of the superconductivity by means of the number of the maximum geminal.Specifically, the concrete value of ( ) in equation ( 28) is unknown.There is no way to determine how small ( ) max becomes before the superconductivity disappears.Therefore, we shall newly give a definition of the disappearance of the superconductivity as follows: The superconductivity disappears when the mean distance between two electrons which are components of the maximum geminal coincides with the system size.
Table 1.The physical quantities used in the calculation of the EE for cases of Al, In, Sn and Pb metals.T C is the superconducting critical temperature [52] that is related to the gap parameter D 0 via the equation D = k T 1.764 0 BC [44].e T k , , F F F are the Fermi temperature, Fermi wavenumber and Fermi energy, respectively, that are calculated using the electron density n [53].The ratio of the gap parameter to the Fermi energy is calculated using equation  The mean distance between the two electrons composing the maximum geminal has been derived in the previous work [43].This is given by where v F is the Fermi velocity [43].Specifically, using the experimental values of the electron density and D 0 shown in table 1, this can be calculated as the ¯[ ] r = ´-2.71 10 cm 4 for Al metal.In the process of the disappearance of the superconductivity, the gap parameter becomes smaller [44,51].
Here we assume that the form of equation ( 11) is invariant except for the gap parameter contained in u , k v k and n max 2 in the process of the disappearance of superconductivity.The reduced gap parameter is defined as where a is a parameter which reduces the gap parameter in the process of the disappearance of the superconductivity.The occupation number of the maximum geminal when the gap parameter is reduced is given by Furthermore, we assume that the form of the mean distance is also invariant when the gap parameter is reduced.Namely, when the gap parameter is D a , the mean distance between the two electrons composing the maximum geminal is expressed as Using equations ( 29) and (30), equation ( 32) is rewritten as As in the case of section 2, let the system be a cube with side [ ] = L 1 cm .According to the definition of the disappearance of the superconductivity mentioned above, the disappearance point is given by the equation ¯[ ] r = a 1 cm .Substituting equation (33) into this equation, the reduction parameter ã at the disappearance point for Al metal is obtained as ´-2.71 10 Al .34 4 At the disappearance point, the occupation number of the maximum geminal is calculated as where we use equation (34) and the value for ( shown in table 2. When the number of the maximum geminal becomes equation (35), the superconductivity disappears in the case of Al metal.Further substituting equation (35) into equation (25), the EE at the disappearance point is calculated as where the explicit form of the geminal of RDM2 of the free N-electron ground-state and the derivation process of equation (37) are presented in appendix D. As can be seen in figure 1, the EE jumps discontinuously like a fistorder phase transition at the disappearance point.We shall comment on this discontinuity in the EE.This discontinuity means that the ground state of the system does not vary continuously at the disappearance point of superconductivity.In other words, the normal and superconducting states are not connected continuously at the disappearance point of superconductivity.Actually, when approaching the disappearance point from the SS, the gap parameter at the disappearance point is by no means zero ( D = ´´D a -2.71 10 4 0 ), which means that the number of particles fluctuates at the disappearance point because both v k and u k are never zero.On the other hand, when approaching the disappearance point from the normal state, the ground state remains in a state with a fixed number of particles (N-electron Slater determinant).When the system changes across the disappearance point, the ground state changes suddenly with a finite gap of the EE.
As already mentioned in preceding section 3, almost all electrons within the energy width of the D 0 around the Fermi energy are condensed as the maximum geminal when the system is in the SS with gap parameter D .0 When the number of the maximum geminal decreases by about four orders of magnitude, the superconductivity disappears.The concrete condition that violates the former of equation ( 28) is that the number of superconducting electrons decreases by about four orders of magnitude.This is the first work to show a reduced amount of the Cooper pair at the disappearance point of superconductivity.At the end of this section, we shall list in table 3 the EE, gap parameter and the number of maximum geminal when superconductivity disappears.

Concluding remarks
In order to discuss the degree of entanglement between two constituent electrons in the Cooper pair, it seems to be most appropriate to treat the maximum geminal of RDM2 of the SS in a direct way.In this paper we calculate the EE of the maximum geminal of RDM2 of the BCS ground state.The EE of the maximum geminal shows the  logarithmical behavior with respect to the number of the Cooper pair, as shown in figure 1.The SS has a longrange order which is so called off-diagonal long-range order [39,41], so that the behavior of the EE is similar to those of the other long-range ordered systems [19][20][21][22][23][24].The EE discontinuously changes at the disappearance point of superconductivity.This implies that the many-electron state suddenly changes from the SS, where the number of particles fluctuates, to a normal state with a constant number of particles at the disappearance point of superconductivity.Although it is the next issue, the external magnetic field would be expected to serve as a control parameter for the EE of the Cooper pair of the SS.It seems to be interesting to research how the EE for the Cooper pair of the superconductor immersed in a magnetic field changes under the influence of a magnetic field [54][55][56][57].
The EE may also be useful as an indicator of electron correlation in the system.As can be seen in figure 1, the EE varies discontinuously at the disappearance point of superconductivity.Across this point the Cooper pair cannot exist stably, so that the electron correlation obviously varies at this point.Not only the pair density that is a conventional indicator of electron correlation [58][59][60][61][62][63] but also the EE may be useful for describing electron correlation.
As an example of the use of the EE, we can illustrate the quantum teleportation [47,48].The entangled pair state is indispensable for quantum teleportation.The maximum geminal of the SS would be a strong candidate for such an entangled pair state for quantum teleportation because the present EE of the maximum geminal is much larger than that of the singlet-type Bell state [48].The quantum teleportation using superconductors would be a promising research topic in the future [64][65][66][67].The knowledge obtained in the present work will also be useful in terms of the applications.
It is shown that the function has a peak at x = 0, and the full width at half maximum (FWHM) is D 2 3 .0 As is well known, a usual Lorentz-type function 2 has a sharp peak the FWHM of which is D 2 .0 This is comparable with that of ( ) x f .Therefore, it is reasonable to adopt the approximation used in equation (24), that is, the density of states ( ) x m + N can be approximated as a constant value at the Fermi level given by ( ) m N .

Appendix B. Numerical calculation of equation (23)
In order to confirm the appropriateness of the approximation used in equation (24), we shall perform a numerical integration of equation (23) without any approximations.Let the integral in equation (23) be denoted as F, that is, The appropriateness of the approximation is dependent on whether the first term of RHS of equation (B4) is close to that of equation (27).The first term of RHS of equation (B4) is calculated numerically, and the result is shown in figure B1.As can be seen in figure B1, the first term of equation (B4) is almost equal to 4 log 2 when the value of e D 0 F is less than - 10 . 4Using the values of e D 0 F for Al, In, Sn and Pb metals shown in table 1, the approximation used in equation ( 24) is reasonably well for every metal.
Appendix C. Discussion on the value of ( ) ( ) D n max 2 0 from the viewpoint of the fermion BEC It is shown in this appendix that almost all electrons within the energy width of D 0 around the Fermi energy are condensed as the maximum geminal when the system is in the SS.It is assumed that the normal state of the system is the free N-electron ground state which is given by the Slater determinant consisting of free singleelectron states inside the Fermi sphere.The number of electrons with energy from e -D  Because the gap parameter is roughly two orders of magnitude smaller than the Fermi energy (  e D 0 F ), DW can be approximated as  As mentioned in section 3, the volume of the system, i.e., ( ) = V L 3 is assumed to be 1 cm . 3The number of electrons per volume 1 cm 3 for Al metal is = Ń 1.806 10 23 [53].Substituting it into equation (C3), and using the values of D 0 and e F listed in table 1 of section 3, we get the number of electrons with energy from e -D F 0 to e F in the system the volume of which is = V 1 cm . 3The calculation result is ( ) D = Ẃ 4.038 10 .C4

.
12), the state | ñ k is considered as the direct product of the free-particle state with wave number k and up-spin state.i.e., In addition, the coordinate representations of the states | ñ k and |c ñ  are written as |

Figure 1 .
Figure 1.The EE as a function of the occupation number of the maximum geminal for case of Al metal.The occupation number of the maximum geminal is related to the gap parameter via equation (31).

F
Transforming the variable from x to ( ( ) )x m e = + x , F and using the relation ( ) ( where we use the fact that m can be approximately regarded as e F at low temperature.The EE without the approximation used in equation (24) is expressed as

FF
where e F is the Fermi energy, and ( ) e D is the density of states, which is given by

Figure B1 .
Figure B1.Numerical calculations on the first term of RHS of equation (B4).
Since we have selected metals that can be explained well by the BCS theory, they have similar electron densities and energy gaps as shown in table 1.As a result, these metals have similar values of the EE.It means that the EE of the BCS-type metals have a value of this order shown in table2.The EE for the Cooper pair of the SS is much larger than that of the spin-singlet Bell state, that is,

Table 2 .
The

Table 3 .
The EE, gap parameter and the number of maximum geminal at the disappearance point of the superconductivity for case of Al metal.They are denoted as ( )