Abstract
Asymmetry in the Josephson couplings between two superconductors through a quantum dot is studied based on a single impurity Anderson model using the numerical renormalization group (NRG). Specifically, we examine how the difference between the couplings ΓL and ΓR affects the ground state, which is known to show a quantum phase transition between a nonmagnetic singlet and a magnetic doublet depending on the various parameters; the Coulomb interaction U, onsite potential , level width caused by the hybridization, and superconducting (SC) gap for the leads on the left and right (). Our results show that whether the local moment is fully screened or not depends substantially on the asymmetry in the couplings . It tends to make the singlet ground state stable, while the amplitude and phase difference of the SC gaps tend to suppress the screening. We also discuss some general symmetry properties of the system and their relation to the current conservation.
Highlights
The Kondo effect in superconductors has been studied over three decades.[1,2,3,4] It was shown in early years that the competition between the superconducting (SC) gap and Kondo energy scale TK determines the lowtemperature properties
Asymmetry in the Josephson couplings between two superconductors through a quantum dot is studied based on a single impurity Anderson model using the numerical renormalization group (NRG)
We examine how the difference between the couplings ΓL and ΓR affects the ground state, which is known to show a quantum phase transition between a nonmagnetic singlet and a magnetic doublet depending on the various parameters; the Coulomb interaction U, onsite potential ǫd, level width Γν caused by the hybridization, and superconducting gap ∆ν ≡ |∆ν |eiθν for the leads on the left and right (ν = L, R)
Summary
The Kondo effect in superconductors has been studied over three decades.[1,2,3,4] It was shown in early years that the competition between the superconducting (SC) gap and Kondo energy scale TK determines the lowtemperature properties. It can drive a quantum phase transition (QPT) between the singlet and doublet ground states, and at the critical point the direction of the current changes discontinuously These properties of the SC-dot-SC systems have been studied with various theoretical approaches; such as the noncrossing approximation,[17,18] slave-boson mean-field theory,[19,20] perturbation theory in the Coulomb interaction U ,21 QMC,[22,23] and NRG.[24,25] So far, most of the calculations have been carried out assuming a highly symmetric condition as |∆L| = |∆R| and ΓL = ΓR.[17,18,19,20,21,22,24] Here, ∆ν ≡ |∆ν | eiθν is the SC gap of the leads on the left and right (ν = L, R), and Γν is the bare level width due to the hybridization. We describe some general properties deduced from the current conservation in the appendix
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