Tuning of quantum entanglement of a superconductor by Transition-metal and Rare-earth impurity effect and the role of potential scattering on quantum phase transition

By considering transition-metal (Shiba-Rusinov model) and rare-earth metal impurities (Abrikosov-Gor'kov theory) effect on a many-body system, i.e., a BCS s-wave superconductor, quantum bipartite entanglement of two electrons of the Cooper pairs in terms of the exchange interaction, J, the potential scattering, V (contrary to expectations playing an important role), and the distance of two electron spins of the Cooper pair is calculated at zero temperature by using two-electron spin-space density matrix (Werner state). In transition-metal case, we found new quantum phase transitions (QPTs). The changes of J, which causes to have localized excited state, V and the pair interaction (via energy gap) lead to the displacement of the QPTs (interactions act in the same direction, however sometimes the pair interaction causes the competition with other interactions), regardless of their effects on the value of concurrence. To have the turning point, which is a QPT point, by the reduction of |J|, the system doesn't need to have the large V. For non-magnetic and magnetic (rare-earth) impurity cases, the concurrence versus the distance and collision times is discussed for all finite and infinite Debye frequency. The quantum correlation, instability of the system and what's more important QPT can be tuned by impurity.


Introduction
During the last years, it became evident that the quantum entanglement (QE) is one of the most important resource in quantum information (QI) and quantum computation, especially, the use of the extraction of the QE in a many-body system. The extensive research efforts on QE were revealed both its qualitative and quantitative aspects of QI [1][2][3][4][5][6] .The method of the measuring of the entanglement is a broad field of research in its own.
The consideration of the QI and QE in the many-body systems causes to reveal of the new properties 7 . It should be noted that in the many-body systems, entanglement is much more complex than other systems, and also, multipartite entanglement influences in the ground state at zero temperature. The generation and manipulation of the bipartite and multipartite entangled states through many-body Hamiltonians, such as the quantum computer's Hamiltonian were investigated [7][8][9][10][11][12] . Other investigations on many body systems such as the entanglement of electron spins are as follows. The entanglement of two electron spins of a noninteracting electron gas based on the Green's function approach was discussed 13-14 ; also, multipartite entanglement in a non-interacting Fermi gas was studied 15 . Furthermore, few researches have been done on superconductors using space-spin density matrix approach. One of them is about bipartite entanglement of two electron spins forming Cooper pairs in a BCS s-wave superconductor 8 ; another one is about bipartite and tripartite entanglement and quantum correlation of s-wave and d-wave bulk and nano grain superconductors, which was given by some of our authors 9,10 .
It was found in numerous works that the entanglement, referring to quantum correlations between subsystems can be a good indicator of QPTs, which can be found in the many body systems 4,16 . A lot of work has revealed that the bipartite or the pairwise entanglement itself or its derivatives display local extremes close to quantum critical points (QCPs). It should be noted that generally the investigation of the entanglement itself is not necessary and sufficient condition for the occurrence of QPT; necessarily one-to-one correspondence between QPT and the appearance of the critical point on concurrence does not exist, unless under some conditions, which was usually used in some spin model such as Isingmodel [16][17][18][19] . However, one-to-one correspondence between QPT (with accompanied to discontinuities of the ground state) and the behavior of the matrix elements of density matrix always exist. Nevertheless, in our present work there is one-to-one correspondence between QPT and entanglement.
Here, we bring about some theories related to the impurity effect on superconductors. Over the past decades, researchers and scientists have paid to the investigation of the impurities effect on the properties of the superconductors [20][21][22] . Shiba, Rusinov, Abrikosov, and Gor'kov have provided theories about the Green's functions and some physical properties of the superconductors in the presence of two kinds of the impurities [23][24][25][26][27] . Shiba and Rusinov independently, have given the theory about the presence of the low concentration of the uncorrelated transition-metal impurities into a superconductor. In Shiba-Rusinov (SR) model, the scattering is calculated exactly for a single impurity problem by treating the impurity spin classically and it is shown that there exists a localized excited state in the energy gap [25][26][27] . The use of the SR model on the isotropic s-wave superconductors doped with the magnetic atoms is identified that the information on potential scattering and exchange interaction cannot be separately obtained; also, the energy of bound states is a function of them 28 . Also, the effect of the impurities on the anisotropic superconductors, Josephson current and several other properties of the superconducting alloy was studied by SR approach [29][30][31][32][33] . In the several valuable theoretical and experimental articles by using the lattice model, Bogoliubov-de-Gennes Hamiltonian, the Zeeman field and Majorana fermion approach, have been addressed to the investigation of the QPT in the superconductors in the presence of the magnetic impurity [34][35][36][37][38][39][40] . The robustness to the non-magnetic impurities of the isotropic s-wave superconductors based on the symmetry of the system approach was studied by Anderson 23 . The role of the rareearth metal impurities case in the conventional superconductors was studied by Abrikosov and Gor'kov 24 . The Markowitz and Kadanoff anisotropic superconductivity is weakly suppressed by the paramagnetic impurity scattering effect 41 .
The purpose of this paper is to investigate of QPTs (based on our knowledge, these are new QPTs) and the bipartite entanglement properties of an BCS s-wave superconductor, in the presence of the non-magnetic impurity and magnetic impurities (transition-metal accompanied with SR model and the rare-earth metal accompanied with AG theory) at zero temperature by using the space-spin density matrix, up to the first-order approximation. Meanwhile, the role of the potential scattering, which is exactly the same important role that than exchange interaction, on the competition among interactions is investigated. It merit mentioning that the increase of the absolute value of the exchange interaction in the occurrence of QPTs always causes the increase of the potential scattering and the effects of | | and in the concurrence are always coupled. The values of the all available parameters such as the coefficients of the interactions strongly depend on the assumptions, which are tuned by the numerical calculations. The assumptions are first, the smallness of the perturbed (in the presence of the impurity) Green's functions in comparison with the unperturbed Green's functions ( in all distance of the electrons of the Cooper pair ) and second, the smallness of the appeared parts due to the impurities in the renormalized order parameter (or energy gap), and the single-particle energy and frequency. First, we address the investigation of the role of the exchange interaction, potential scattering, and normalized transition-metal impurity concentration in the SR model by considering the entanglement approach. Furthermore, the appearance, the determination and the displacement of QPTs based on the values of the parameters especially potential scattering regardless of the influence of the parameters on the value of concurrence, are investigated. We show that potential scattering plays an important role in the SR model contrary to expectations. Second, We use the results of the AG theory for the rare-earth metal (except cerium) and non-magnetic impurities in the superconductors up to the first order approximation. One of the most important assumptions in the AG theory is that the exchange interaction between a conduction electron and a magnetic impurity spin is weak and the lowest-order Born approximation is used to treat the scattering. For both rare-earth metal and non-magnetic impurity, we pay attention to investigate both infinite and finite Debye frequencies; it should be mentioned that previously, Green's functions were considered with infinite Debye frequency 21 . Furthermore, the comparison among the finite Debye frequency, the order parameter (energy gap) and the frequency gives different conditions on their appropriate values for the numerical analysis of concurrence.

Results and discussion
A. The transition-metal impurity case First, we consider the effect of the transition-metal impurity, which is described by the SR model, on the s-wave superconductor, by taking care of the critical concentration impurity and considering the potential scattering that has an important role to quantum correlation such as QE. It merits mentioning that unexpectedly, not only the potential scattering as well as the exchange interaction plays the important role in concurrence, but also, the existence of the QPTs depends on the presence of the potential scattering.
Furthermore, of course, if exchange interaction is nonzero, then, the potential scattering becomes important term; otherwise, the potential scattering acts like non-magnetic impurity in the AG theory. The concurrence ( ≡ ( , , , ) ) in the presence of the impurity is calculated as a function of the external and internal parameters, i.e., , which is related to the potential scattering, , via ≡ / (0), , which is defined by the exchange interaction via ≡ 2 / (0), , which is defined as the normalized impurity concentration,and . Meanwhile, (0) , , and are the density of single-particle states at the Fermi surface, the spin of the impurities, the distance of two electrons of the Cooper pairs and Fermi wave number, respectively. We study of the effect of the variations of these parameters on concurrence and QPTs for the transition-metal impurity case (throughout the paper, we use ℏ = 1 ).
Concurrence  should be mentioned that this disappearance is influenced by the approximation used in all calculations and the range used for the potential scattering. In addition, it can be seen that by increasing the absolute value of , the concurrence decreases.
It should be noted that in the presence of the magnetic impurity, the energy gap is reduced 25 ; this reduction depends on the normalized impurity concentration, the exchange interaction, the potential scattering and density of states. Our calculation shows that the change of the energy gap due to the magnetic impurity is very small, so that we can neglect the effect of this reduction on the energy gap and thereby, the maximum error occurred in the calculation is about few percent. Therefore, we have used a typical energy gap such as that than given in Ref. [8]. However, for s-wave superconductors with different energy gap (and also Debye frequency), we can show that how the change of the energy gap can affect not only the value of concurrence but also the occurrence of the critical point at different value of the exchange and potential scattering. Concurrence versus at different fixed  regimes, at a fixed exchange interaction, from lower to the higher value of the potential scattering, the first regime appears up to receiving the turning point (first the concurrence reduces with the sharp slope, then the slope of the curves becomes reverse), the second regime is started from this point and is contained the local maximum point, finally, the third regime is included the curves that are quite ascending with respect to potential scattering. Analytical and numerical studies on the concurrence (also via energy of the system) and its first and second derivatives sign to have QPTs. By changing the values of and ∆, these critical points (turning point and local maximum point) occurat different ; also, the normalized impurity concentration and the relative distance between two electrons of a Cooper pair of a conventional superconductor do not change the value of of QPT points, which will be discussed in the following. Now, we proceed to interpret the change of the bipartite QE of two electron spins forming Cooper pairs. It should be mentioned that the concurrence (also via energy of system) shows the existence of QPTs; there are the necessary and sufficient conditions for QCPs existing in Figs. 1 and 2 to become QPT points; we exactly investigate the behavior of these points also by the change of energy of system, and its first and second derivatives.
There is one-to-one correspondence between QPT and entanglement. The turning point and the local maximum point produced due to the potential scattering are new QPT points.
It worth mentioning that when the cooper pair exists and bipartite entanglement is nonzero, the relative distance between the electrons of the Cooper pair is not influenced to displace the characteristics of the QPT points such as and . Previously, at the point = 0, where the system goes from classical spins model to quantum spin model, the system has been shown a QPT; also, the point = −1, where the bound state is induced, is a QPT point [25][26][27] .However, our investigation about QPTs via QE is related to the value of located between 0 and 1with taking into account the potential scattering (the increase of the , corresponding to the increase of the absolute value of ) up to the first-order approximation. It is worth mentioning that we know exactly what happens in the system and how we can see the interaction influences on the electron spins' correlation of the Cooper pair. In the other words, it is very interesting that to follow the effect of the all real interaction (and not the manipulating the virtual interaction in the virtual system) in the real interesting many-body system, i.e., the interacting Fermi system, so-called the superconductor. At a fixed potential scattering, the concurrence in terms of the potential By considering the fixed value of (= 0.1), the relative distance between two electrons of a Cooper pair of the s-wave superconductor, (= 1 ⁄ ), and a fixed normalized impurity concentration, (= 0.2), the first and second partial derivatives of the concurrence versus are depicted in Fig.6. It can be seen that at ≈ −0.94, has a local minimum and the second partial derivative is equal zero.  It can be seen that from the overall view of Fig.8 (a), at a fixed (= 10), by increasing the value of , concurrence decreases. For the purpose of comparison, we bring concurrence without any impurity 8,9 . The curves of the concurrence in the presence of the non-magnetic impurity with finite case and non-impurity case are overlapped, however, at a fixed value of , for the infinite Debye frequency case, concurrence is less than other cases. In Fig.8 (b), we focus on the investigation of the concurrence versus inverse collision time at a fixed value of (= 1 × 10 ). By increasing the value of the inverse collision time, for the infinite (finite) Debye frequency case, the concurrence decreases linearly (doesn't change significantly). For a fixed 1/ ,for the infinite Debye frequency case, the concurrence is smaller than that of the finite Debye frequency case.

Methods
The entanglement gives rise from two ways; one of them is particle statistics, so the noninteracting system shows entanglement, another way, which leads to an increase or decrease of the bipartite or tripartite or even multipartite entanglement, is due to external or internal interaction, which is found in the few-or many-body systems 9 . For the calculation of the reduced space-spin density matrix and required Green's functions of our system, which is an interacting Fermi gas, we have considered the trick that causes the procedure calculating of the concurrence in the noninteracting systems can be used for the interacting systems. Previously, for s-wave and d-wave superconductors as the interacting systems, the trick was used and the only change was to apply Green's functions for superconductors instead of noninteracting Fermi gases [8][9]13 . The reason of the extraction and the use of the formulation of the interacting system based on the noninteracting system can be explained from other viewpoints as follows; first, a superconducting ground state can be writing as the product of the ground state of a noninteracting Fermi system to an extended Jastrow function or Feenberg factor 9,42-43 , which is related to the interactions and caused to modify bipartite entanglement in the noninteracting part of the state. Second, by Bogoliubov transformation, Hamiltonian of the system can be diagonal as a noninteracting system but with different energy 9 . Of course, the existence of impurity on superconductors will produce additional interaction, which can be considered as a small perturbation on superconductors. Finally, Green's functions of the system can be written similarly to that those of the noninteracting system, like as the ground state of the interacting system under investigation, i.e., the superconductor in the presence of the impurity, which is written via that those of the noninteracting system. Thereby, the formulation of the concurrence in terms of the noninteracting Green's functions can be survived. It should be mentioned that at least the new Green's functions (in the presence of the impurity) has the same form as the absence of the impurity, but only by transforming such as to . Another important point is about the structure of the BCS theory, in which mean-field approximation is used.
We know that this approximation can change the entanglement so that the results of bipartite entanglement maybe incorrect unless fluctuation related to all interactions can be considered to be small (in all paper, smallness of the fluctuation is considered). By these assumptions and explanation, first, we bring the relation between reduced space-spin density matrix (and its elements are written in the spin coordinate and each of the elements depends on the space coordinate) and Green's functions (fermionic case) of a superconductor. We ignore anomalous Green's function of the system, because unperturbed anomalous Green's function in comparison to Green's function can be neglected and according to that, the small modification due to the impurity on anomalous Green's function becomes so small. Then, we have Transition-metal case. We considered the transition-metal impurity as a perturbation in Green's functions, thereby, in the reduced density matrix and concurrence of the system. The ratio of the perturbed Green's functions to the primary Green's functions is tuned at each step of the calculation up to the first-order approximation, so that its value becomes small. This approach limits the range of the parameters (like the exchange interaction, the potential scattering, and the renormalized impurity concentration) existing in the concurrence, which has been considered in the numerical calculation by choosing the value of each parameter and then fixing other parameters to meet the above condition. We start to write the exact Green's functions [25][26][27][28] , with the following notation where ( , , ) = − − ̃ → , = , = ̃ → and = . Also, , ̃ → , and are the renormalized frequency, renormalized kinetic energy with respect to chemical potential, and renormalized order parameter, respectively. It should be mentioned that = + × , , ( , Δ ⁄ , , ) where is collisiontime and = ( , ) 4 ⁄ . By considering that the function ( , ) is always less than one and 0 < < 1, then, can be chosen to be less than one, so in , the additional term to , which is due to the existence of the impurity, can be considered to be small and therefore, we would like to do the calculation up to first-order approximation. However, in the numerical calculation, we aware to check all quantities for satisfying our approximation, for example a part range of is acceptable. It should be noted that can be approximated, when −1 < < 0,0 < < 1, ≪ 1 and ( , Δ ⁄ , , ) ≈ Δ ⁄ < 1. The perturbed Green's functions as follows: where → , = Then, for ( ), after integration on angles, we have Infinite Debye frequency case: At zero temperature, up to the first-order approximation, the Green's function in terms of the non-impurity Green's function is obtained by where ( ⃗) was given in Refs. [8][9]. Also, by considering the rare-earth metal impurity, By doing Fourier transform between momentum and space, which is obtained the perturbed Green's function at space coordinate, we find that the perturbed Green's function under the above conditions does not affect on the concurrence.
Furthermore, under assumptions > and ≫ √ − , we obtain (12) Then, the perturbed Green's function at space coordinate is obtained and again we find that it is not influenced on the concurrence.
Under assumptions < and ≫ √ − , we obtain  ) with no effect on the concurrence.
Other assumptions, for example, < accompanied with < √ − , does not show the physical result, because it is in contrast with the statement that expresses always order parameter is less than the Debye frequency.
The perturbed Green's function with assumptions > and ≫ √ − is