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Physical review. B./Physical review. B

Tuning of quantum entanglement in a superconductor with transition-metal and rare-earth impurities: Effect of potential scattering on quantum phase transitions

N. Ebrahimian, Mehran Khosrojerdi, R. Afzali

Published Jul 20, 2022• 0 citations
Publisher

Abstract

Bipartite quantum entanglement of electron spins of a Cooper pair in a dirty $s$-wave superconductor is investigated at absolute zero temperature in terms of the exchange interaction and potential scattering as well as the relative distance between the two electrons. In the case of transition-metal impurities, we utilize the T-matrix approach to obtain the relevant Green's functions. We employ the two-electron spin-space density matrix, which is associated with the perturbed Green's functions. We show that a two-spin state can be described by the Werner state for dirty superconductors as well as for clean ones. We find that both the first and second quantum phase transitions (QPTs) are generated by the competition between potential scattering and other interactions. Meanwhile, it is not possible to determine the allowable value for each interaction individually. In the case of rare-earth impurities, concurrence, as a measure of bipartite entanglement, is discussed in terms of the relative distance and collision times for all finite and infinite Debye frequencies. The results confirm that quantum correlation and QPTs can be tuned by impurities.

DOI
10.1103/physrevb.106.024512

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Bipartite quantum entanglement of electron spins of a Cooper pair in a dirty sss-wave superconductor is investigated at absolute zero temperature in terms of the exchange interaction and potential scattering as well as the relative distance between the two electrons. In the case of transition-metal impurities, we utilize the T-matrix approach to obtain the relevant Green's functions. We employ the two-electron spin-space density matrix, which is associated with the perturbed Green's functions. We show that a two-spin state can be described by the Werner state for dirty superconductors as well as for clean ones. We find that both the first and second quantum phase transitions (QPTs) are generated by the competition between potential scattering and other interactions. Meanwhile, it is not possible to determine the allowable value for each interaction individually. In the case of rare-earth impurities, concurrence, as a measure of bipartite entanglement, is discussed in terms of the relative distance and collision times for all finite and infinite Debye frequencies. The results confirm that quantum correlation and QPTs can be tuned by impurities.
Abstract By considering transition-metal (Shiba-Rusinov model) and rare-earth metal impurities (Abrikosov-Gorkov 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(playing an important role, unexpectedly), 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 find new quantum phase transitions (QPTs). The changes of J, which causes to have localized excited state, V and the pairing interaction (via energy gap) lead to the displacement of the QPTs (interactions act in the same direction, however sometimes the pairing interaction causes the competition with other interactions), regardless of their effects on the value of concurrence. Determining the allowable values of all interactions by itself is not possible, due to the smallness of the perturbed Green’s functions (appearing in the density matrix). For non-magnetic and magnetic (rare-earth) impurity cases, 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 the impurity.
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We investigate entanglement of two electron spins forming Cooper pairs in an sss-wave superconductor. The two-electron space-spin density matrix is obtained from the BCS ground state using a two-particle Green's function. It is demonstrated that a two spin state is not given by a spin singlet state but by a Werner state. It is found that the entanglement length, within which two spins are entangled, is not of the order of the coherence length but of the order of the Fermi wave length.
Nanotechnology has made it possible to fabricate electronic devices in which it is possible to manipulate and tune quantum states. The paper studies quantum entanglement in Kondo impurity systems. Experimental realization occurs in pairs of quantum dots and in endohedral fullerenes inside carbon nanotubes. Competition of many-body effects plays a very important role in determining quantum entanglement of two localized impurity spins. The reduced density matrix shows that the two spins are in a Werner state with probability coefficients being a function of spin-spin correlation. It is found that to be entangled the two spins need a minimum nonzero anti-ferromagnetic correlation. How the entanglement in the whole system is shared was discussed.
The insertion of a magnetic impurity in a superconductor induces a first order quantum phase transition as the coupling to the electronic spin density increases. As the transition is crossed, a discontinuity is exhibited by various quantities, such as the total spin density, the total gap function, and the gap function at the impurity location. The location of other quantum phase transitions have been detected by singularities in entanglement measures of the system. In this work we show that the single-site and two-site von Neumann entropies, the mutual information, and the Meyer-Wallach measure show discontinuities at the quantum phase transition. The negativity is less sensitive to the transition. We study in detail these quantities as a function of spin coupling and distance from the impurity center.
We study entanglement of electron spins in many-body systems based on the Green's function approach. As an application we obtain the two-particle density matrix of a noninteracting electron gas and identify its two-spin density matrix as a Werner state. We calculate entanglement measures, a classical correlation, mutual information, and a pair-distribution function of two electrons at zero and finite temperatures. We find that changes of entanglement measures are proportional to T2{T}^{2}T2 at low temperatures.