Kondo effect in asymmetric Josephson couplings through a quantum dot

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 \Gamma_L and \Gamma_R 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 \epsilon_d, level width \Gamma_L, \Gamma_R, caused by the hybridization, and superconducting gaps \Delta_L and \Delta_R 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 \Gamma_L \neq \Gamma_R. It tends to make the singlet ground state stable, while the size and phase difference of the two superconducting gaps tend to suppress the screening. We also discuss some general symmetry properties of the system and their relation to the current conservation.


I. INTRODUCTION
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 T K determines the lowtemperature properties. The SC energy gap ∆ SC disturbs the conduction-electron screening of the local moment. The ground state becomes a magnetic doublet for ∆ SC ≫ T K , while the ground state is still a nonmagnetic singlet for ∆ SC ≪ T K . These aspects of the Kondo physics in superconductors have been re-examined precisely by efficient numerical methods such as the quantum Monte Carlo (QMC) 5 and numerical renormalization group (NRG) approaches. 6,7,8,9 In quantum dots embedded in a Josephson junction, new interesting features have been predicted to be observed 10,11,12 and some experiments have been reported. 13,14,15,16 Particularly, the screening of the local moment in the quantum dot is affected not only by the size of the two SC gaps but also by the Josephson phase φ between the two SC leads on the left and right. The phase difference φ also induces the Josephson current flowing through the dot. 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, however, most of the calculations have been carried out assuming a highly symmetric condition as |∆ L | = |∆ R | and Γ L = Γ R . 17,18,19,20,21,22,24 Here, ∆ ν ≡ |∆ ν | e iθν is the SC gap of the leads on the left and right (ν = L, R), and Γ ν is the bare level width due to the hybridization. Therefore, the feature of the QPT far from the symmet-ric condition is one of the theoretical issues to be clarified with some accurate approaches.
In a previous work, we have studied the ground-state properties in the case of |∆ L | = |∆ R |. In real quantum dots, it corresponds to the situation that the two superconductors connected to the dot are not identical. 25 Specifically, we have considered the limit where one of the two SC gaps, ∆ L , is much larger than the other one ∆ R . In the limit of |∆ L | → ∞, the left lead has been shown to be separated from the rest of the system, which consists of the dot and right lead. It just leaves a static SC pair potential, which is given by ∆ d = Γ L e iθL , in the dot. The pair potential appears because a SC proximity from the left lead is allowed even in the large gap limit, while the quasi-particle tunneling into the left lead is prohibited. This situation is described by a single-channel Anderson model with an extra SC pair potential ∆ d in the impurity site. The simplified model possesses essential ingredients of the Kondo and Josephson effects, and the phase difference φ ≡ θ R − θ L can be defined between ∆ d and ∆ R . We have studied the features of QPT applying the NRG to this model, and have confirmed, for example, that the Andreev bound state emerging close to the Fermi level causes a re-entrant QPT near half-filling for φ ≃ π and Γ L ≃ Γ R . 25 Another aspect that seems to be important is how the asymmetry in the couplings Γ L and Γ R affects QPT. That is the main subject of the present work. The SC proximity effect depends strongly on the environment surrounding the dot, and the competition between the Kondo and proximity effects determine the low-energy properties. In this paper we present the results of the QPT phase diagram obtained for the asymmetric couplings Γ L = Γ R . The NRG calculations have been carried out mainly using the two-channel model, assuming the two gaps are equal in size |∆ L | = |∆ R |. Our results show that the asymmetry in the couplings tends to enhance the screening of the local moment.
The model and some symmetry properties of the Anderson impurity in the Josephson junctions are described in Sec. II, based on a rotational symmetry in the pseudospin space. The NRG approach and the results of the phase diagram for QPT obtained in some typical parameter regions are discussed in Sec. III. We also discuss in Sec. IV the roles of the asymmetry in two solvable cases: noninteracting case U = 0, and a superconducting version of the atomic limit for |∆ L | → ∞ and |∆ R | → ∞ which can also be considered as one of the fixed points of NRG. A summary is given in Sec. V. We also describe some general properties deduced from the current conservation in the appendix.

II. MODEL FOR A SC-DOT-SC JUNCTION
We describe in this section the model and some of the symmetry properties. Specifically, at half-filling for the Josephson junctions with φ = 0 or π, the system has a global U(1) symmetry in a pseudo-spin space. It enables us to map the SC leads onto the normal conductors with a staggered potential, as shown by Satori et al. 6

A. General formulation
We start with a single Anderson impurity connected to two SC leads on the left (L) and right (R), as illustrated in Fig. 1. The Hamiltonian is given by Here, ξ d ≡ ǫ d + U/2, with ǫ d the impurity level, and U is the onsite Coulomb interaction. The operator d † σ creates an electron with spin σ at the impurity site, and The other operator c ν,0σ annihilates a conduction electron that couples directly to the dot via the tunneling matrix element v ν . The electrons in the leads can be described by the Hamiltonian of a tightbinding form without loss of generalities. 29 The hopping matrix element t ν,n determines a continuous oneparticle spectrum ǫ kν and eigenfunction ϕ kν (n), by which the hybridization strength can be written in the form The symmetry properties of the system can be clearly seen, in the Nambu representation using the spinors Then, eqs.
(2)-(5) can be rewritten in the form Here, ∆ ν and i d are defined by and τ k for k = 1, 2, 3 is the Pauli matrix. The Coulomb interaction H U d is isotropic in the pseudo-spin space, and the expression given in eq. (7), which is identical to that in eq. (2), explicitly shows the rotational symmetry. Furthermore, the current through the dot can be expressed in the form, The current is conserved such that ∂n d /∂t+J R −J L = 0, as described also in the appendix A. The rotational symmetry in the pseudo-spin space are described by the operator The z component corresponds to the half of the total charge Q = 2I z , and I x (I y ) represents the real (imaginary) part of the SC pair potentials. These three components satisfy the same commutation relations as that of the angular momentums, In the pseudo-spin space, the onsite potential ξ d can be interpreted as a local external field along the z axis, and the SC gaps ∆ ν can be regarded as a staggered field in the x-y plane. Therefore, if ξ d = 0 and ∆ ν = 0, which corresponds to normal leads in the particle-hole symmetric case, the system has a full rotational symmetry in the pseudo-spin space as well as the rotational symmetry in the real spin space, because [I k , H] = 0 for k = x, y, z.
Another typical case is that of the Josephson junction ∆ ν = 0 with the particle-hole symmetry ξ d = 0. In this case the z component of the external field vanishes. Therefore, if we take the quantization axis to be in the y-direction by carrying out a spinor rotation of π/2 with respect to the x axis, the Hamiltonian H takes a real symmetric form. Then, with the new axis, the current operators are described by a Pauli matrix τ 2 instead of τ 3 in eqs. (12) and (13).

B. Particle-Hole symmetric case with real gaps
The two SC gaps ∆ L and ∆ R are essentially real for the Josephson junctions with φ = 0 or π. In this case, if the system also has the particle-hole symmetry ξ d = 0, the model can be mapped onto a normal asymmetric Anderson model with a staggered potential. 6 It is owing to a conservation associated with [I x , H] = 0, caused by a U(1) symmetry along the x axis. It enables us to transform I x into the total number of the quasi-particles as shown in eq. (21). This feature can be seen explicitly if we take the quantization axis to be in the x direction, 6 carrying out a spinor rotation of π/2 with respect to the y axis by a unitary transform e i π 4 τ2 as  Then, the Hamiltonian can be expressed in the form Note that H 0 d = 0, because of the particle-hole symmetry, ξ d = 0. Therefore, the energy spectrum is determined by the quasi-particles, the total number of which is conserved Particularly, the local charge q −1 corresponds to the SC pair correlation penetrated into the impurity site. The profile of the potential that the quasi-particles feel is illustrated in Fig. 2. For the Josephson junction with φ = 0 the potential still shows a regular staggered period around the dot, while for φ = π the steps of the potential changes like that in a defect, or dislocation. The potential profile suggests that the SC proximity effect, which causes a finite anomalous average q −1 , is larger for φ = 0 than that in the case of π. The penetration of the SC correlations into the impurity site depends on the tunneling strength Γ ν and the size of the SC gaps ∆ ν . Specifically, for φ = π and Γ R = Γ L , the impurity level evolves to the Andreev bound state emerging just on the Fermi level situated in the middle of the energy gap. In this case, if also the size of the two gaps are equal ∆ R = ∆ L , the SC correlations cancel out q −1 = 0 in the impurity site. It also holds in the particle-hole asymmetric case ξ d = 0 as shown in the appendix A based on the current conservation.
C. Particle-Hole asymmetric case for φ = π There is a different symmetry for a π-junction φ = π, in the case of Γ R = Γ L and ∆ R = ∆ L . It holds without the particle-hole symmetry, and reveals, for instance, in the Nambu form of the Green's function G given in eq. (A12) in the appendix A. The underlying symmetry links to the conservation of a channel charge Q ch defined by The commutation relation, [Q ch , H] = 0, can be confirmed explicitly if we express eqs. (3)-(5) in terms of bonding and antibonding orbitals for v L = v R , and taking t ν,n to be channel independent. The superconducting part, eq. (5), conserves the difference between the total number of the particles with the even symmetry and that of the odd ones, Here, ∆ is real and defined such that ∆ R = −∆ L = ∆. The Hamiltonian can be simplified carrying out the particle-hole transformation for b nσ , and one of the corners for a 0σ is connected to the dot as The dot part of the Hamiltonian, H 0 d + H U d , keeps the original form given in eq. (2). Thus, it is obvious that the SC correlation vanishes, d † ↑ d † ↓ = 0, at the impurity site in the present situation even for the particle-hole asymmetric case ξ d = 0.
The ladder described by eqs. (26) and (27) gives the density of states that shows a square root divergence at the edges of the gap ǫ = ±∆. Therefore, the properties of the π-junction in the symmetric gaps and couplings must be essentially the same as those of the magnetic impurity in a band insulator, which has been studied using mainly a constant density of states with the gap around the Fermi level. 26,27,28 III. GROUND-STATE PROPERTIES

A. NRG approach
We study the ground-state properties using the NRG. In order to examine the effects caused by the asymmetry in the tunneling matrix elements, we consider the Josephson junction with |∆ L | = |∆ R | (≡ ∆), where the two gaps are equal in size. To be specific, we assume that Γ ν (ǫ) is a constant inside the conduction band −D < ǫ < D, and use the standard logarithmic discretization. 30 Then, the hopping and tunneling matrix elements are expressed in the form The factor A Λ is required for reproducing the original model correctly in the continuum limit Λ → 1. 30, 31 We have carried out the successive diagonalizations of NRG keeping typically the lowest 500 energy states for constructing the Hilbert space with extra orbitals from next step. The low-energy properties are determined by the ratios Γ ν /∆, U/∆, ǫ d /∆ and the phase difference φ. In the calculations we have taken the gap to be ∆/D = 1.0 × 10 −5 : this ratio itself does not affect the low-energy properties, provided ∆ ≪ D. We have used the eigenvalue of the pseudo spin I x as a quantum number to classify the Hilbert space in the particle-hole symmetric case for φ = 0 and π. These two limits of φ provide the upper and lower bounds in the ground-state phase diagrams for intermediate φ.
The model for the Josephson junction contains two conduction bands, and the accuracy of the NRG results becomes somewhat worse than that for a single-channel model. For this reason, we have mainly used a rather large value Λ = 6.0 for the discretization parameter to get an early convergence in the NRG steps. To check the validity of it, we have examined the phase diagram of the  Fig. 3. We see that the results obtained at Λ = 6.0 already agree well with those obtained for small Λ's. It indicates that the level crossing of the ground and first excited states is not so sensitive to Λ, and it supports the validity of our results presented in the following. The figure 3 itself shows that the ground state is a nonmagnetic singlet for large Γ or small U , and at the critical point it changes discontinuously to a magnetic doublet for small Γ or large U . 8,25 In order to show an example of the behavior of observables at the critical point, we have also calculated the average number of electrons n d in the dot. The result is shown in Fig. 4 as a function of ǫ d for U = 5.0∆, φ = 0, and Γ L = Γ R . In this parameter set, the ground state for Γ = 0.65∆ is a singlet for all values of ǫ d , and thus n d decreases monotonically with increasing ǫ d . In contrast, for Γ = 0.35∆, the ground state is a doublet for −4.2 ǫ d −0.8, and n d jumps at the critical points. Other observable quantities, such as the Josephson current, will also show similar discontinuous changes, because the QPT between the singlet and double ground states is the first order transition occurring as a result of a level crossing of the Andreev bound state emerging in the SC gap in the energy spectrum.

B. Asymmetric couplings ΓL = ΓR at half-filling
How does the asymmetry in the couplings Γ L and Γ R affect the ground-state properties? We first of all consider the particle-hole symmetric case ǫ d = −U/2. In this case, the dot is occupied by a single electron n d = 1 on average, and magnetic correlations are enhanced maximally. In Fig. 5, the phase diagram of the ground state for asymmetric couplings Γ L = 1.44Γ R is plotted for the Josephson junctions with (•) φ = 0 and ( ) φ = π. For comparison, the results obtained for the symmetric couplings Γ L = Γ R are shown for (•) φ = 0 and (△) φ = π with the dashed lines. The phase boundary for intermediate values of φ, namely for 0 < φ < π, will appear in between the line for φ = 0 and that for φ = π. In the figure, we see a general trend that the phase difference φ between the SC pair potentials enlarges the region for a doublet ground state. 25 It means that the Josephson effects due to a finite φ disturbs the screening of the local moment, and the QPT will be driven by φ if the other parameters U/∆ and Γ ν /∆ bring the system into the parameter region enclosed by the two lines. Note that for the data obtained for Γ L = 1.44Γ R , the sum of the hybridization energies Γ L + Γ R is larger than that for the symmetric couplings. It also favors the singlet ground state.
The results shown in Fig. 5 suggest, however, that the superconducting proximity effects on the impurity site depends substantially on the asymmetry in the Josephson couplings, particularly for the π-junction. The vertical phase boundary at φ = π for Γ L = Γ R is caused by an Andreev bound state. It emerges just on the Fermi level at half-filling, as we see also in the potential profile shown in Fig. 2 (b). Therefore, along the vertical line the ground state has 4-fold degeneracy. An infinitesimal U lifts the degeneracy when the two SC gaps are identical |∆ L | = |∆ R |, and the ground state becomes a doublet in the whole region of U > 0. We will discuss this point more in detail in Sec. IV. The asymmetry in the couplings Γ L = Γ R breaks this situation. Then, the Andreev bound state emerges away from the Fermi level, and it makes the ground state to be a singlet for small U .

C. Away from half-filling
We also consider the ground state away from half-filling ǫ d = U/2. The phase diagram obtained for a fixed value of the onsite energy ǫ d = −2.5∆ is shown in Fig. 6, where the particle-hole symmetry holds along the dashed line at U = 5.0∆. The calculations have been carried out for asymmetric couplings Γ L = 1.44Γ R for (•) φ = 0 and ( ) φ = π, and also for the symmetric couplings Γ L = Γ R for (•) φ = 0 and (△) φ = π. As mentioned, the boundary for 0 < φ < π will appear between the line for φ = 0 and that for φ = π. In Fig. 6, all the boundaries start from the point U = 2.5∆ in the horizontal axis. This is because in the atomic limit Γ ν = 0 the level crossing between the singlet and doublet states occurs at 2ǫ d + U = ǫ d , when the energy of single-particle state and that of two-particle state coincide. We also see in this figure that the phase boundary for φ = π is changed substantially by the asymmetry in the couplings, and the parameter region corresponding to the singlet ground state spreads in an upper-right region for Γ R 1.4∆ and U 5.0∆. For the symmetric couplings, the phase boundary at φ = π approaches asymptotically to the vertical line at U = 5.0∆. This behavior is caused by the Andreev state approaching to the Fermi energy.
In quantum dots ǫ d is a tunable parameter that can be controlled by the gate voltage. The ǫ d dependence of the phase diagram of the ground state for Γ L = Γ R has been obtained first with the slave-boson mean-field approximation by Rozhkov and Arovas. 19 We have examined the effects of the asymmetry. The NRG results are shown in Fig. 7 for (solid line) the asymmetric couplings Γ L = 1.44Γ R , and for (dashed line) symmetric couplings Γ L = Γ R . Here, the Coulomb interaction is fixed to be U = 5.0∆. The ground state is a singlet for ǫ d + U < 0 and ǫ d > 0, because the impurity site is almost fully filled or almost empty in these two regions. We see again in Fig. 7 that the asymmetry in the couplings Γ L = Γ R enlarges the parameter space of the singlet ground state. The phase boundary for φ = π in the symmetric couplings Γ L = Γ R shows a diverging behavior at ǫ d = −2.5∆, which is owing to the Andreev bound state being situated just on the Fermi level at half-filling.

IV. DISCUSSIONS
The phase transition between the singlet and doublet ground states occurs at a level crossing of the lowest and the first many-body excited states, both of which are discrete and are separated from the continuous excitation spectrum at ǫ > min(|∆ R |, |∆ L |) above the energy scale of the SC gap. Therefore, a few discrete states emerging inside the gap region determine the properties at low energy scales. In this section, to obtain some insights into how the asymmetry around the dot affects the ground states, we consider two simple limits where the model can be solved.
We consider here how the asymmetry in the two SC gaps affects the Andreev state in the noninteracting case H U d = 0. The density of states of each SC lead has the square root divergence at the edge ǫ = ±|∆ ν |, and the formation of the bound state depends strongly on these divergences. The edges are separated from each other for |∆ R | = |∆ L |, and they merge for the symmetric gaps to give a linear divergence for 0 < φ ≤ π.
tion always exists because in this range of the phase F (ǫ) is an increasing function and diverges positively at |ǫ| = min(|∆ R |, |∆ L |) with the square root dependence. Specifically, the bound state appears just on the Fermi level ǫ = 0, when the right hand side of eq. (31) equals zero. It happens for ξ d = 0, φ = π, and Γ R = Γ L , provided |∆ R | and |∆ L | are nonzero. In this case the singlet and doublet ground states are degenerate because the energy does not depend on the way the bound state is occupied. The asymmetry in the couplings Γ R and Γ L affects crucially the ground-state property for the πjunction. The Coulomb interaction lifts the degeneracy, and it can be studied with a simple perturbation theory with respect to an infinitesimal U . 25 The results are shown in Fig. 8, assuming that |∆ R | = min(|∆ R |, |∆ L |). The way how the degeneracy is lifted depends quantitatively on the asymmetry in the SC gaps x = |∆ R |/|∆ L |. Specifically, the curve for the phase boundary reaches 0 at x = 1, which corresponds to the symmetric gaps |∆ L | = |∆ R |. It means that the ground state is a doublet for any values of Γ at x = 1. It explains the behavior of the vertical phase boundary for the π-junction in Fig.  5, and in this case the SC correlation in the impurity site vanishes d † ↑ d † ↓ = 0 also for finite U as shown in the appendix A.
Another example seen in the case of asymmetric gaps |∆ L | = |∆ R | is that the bound state disappears in a finite parameter region. It is caused by the difference in the asymptotic behavior F (ǫ) at |ǫ| → |∆ R | − 0 + , where we assume that |∆ R | ≤ |∆ L |. For instance, in the limit of |∆ L | → ∞, the determinant F (ǫ) does not have a zero point for |∆ R | < Γ L cos φ. 25 This is because the SC correlation of the order Γ L , which penetrates from left lead into the dot, pushed the bound state away towards the continuum outside of the gap region of the width |∆ R |. The bound state can remain inside the gap for |∆ R | > Γ L cos φ. The same situation also happens for finite |∆ L |. The existence of the bound state can be deduced from the asymptotic behavior of the determinant at the edge of the gap region. The bound states appears if F (ǫ) → +∞ for |ǫ| → |∆ R | − 0 + , and this condition can be expressed in the form, for 0 ≤ x < 1 with x = |∆ R |/|∆ L |, as The solution does not exist when the both sides of eq. (32) are equal, since F (ǫ) takes a negative finite value for |ǫ| → |∆ R | − 0 + at the critical condition. The parameter region where the bound sates do not exist shrinks as φ increases as shown in Fig. 9. In contrast, for the symmetric gaps |∆ R | = |∆ L |, the determinant shows a positive linear divergence for 0 < φ ≤ π at the edge of the gap, and the bound state always remains inside the gap region.

B. Large gap limit |∆R|, |∆L| → ∞
When both of the two gaps |∆ R | and |∆ L | are larger than the other parameters, the model can be mapped onto a simple single site problem by taking a limit of |∆ L | → ∞ and |∆ R | → ∞. 32 It can be regarded as a superconducting analogue of an atomic limit, and can be solved analytically as summarized in the appendix B. The properties in the symmetric couplings Γ R = Γ L have been studied by Vecino et al. 21 In the following, we discuss mainly the role of the asymmetry in the couplings Γ R = Γ L .
In the case of a normal transport through a quantum dot the equation of motion method, 33 which is essentially the atomic limit in the Hubbard I approximation, 34 has been applied to the properties at high energy scale. It has a limitation, however, at low temperatures because it does not describe properly the Fermi liquid properties due to the gapless low-energy excitations. Nevertheless, in the case of the dot connected to superconducting reservoirs, the low-energy excitations have the gap, and it gives a different meaning to the atomic limit. The single site degrees of freedom can be identified as the Andreev bound state remaining inside the gap. It does describe the lowest few many-body excited states inside the gap properly, and can also be considered as one the fixed points of NRG. Although the virtual processes to the continuous excited states above the SC energy gap may also be important for the correlation functions, the main features of the contributions of the Andreev state to the low-energy properties, especially the bound-state energy itself, can be captured in the atomic limit of |∆ L |,|∆ R | → ∞.
The effective Hamiltonian for the single site model is given in eq. (B4) in the appendix B. In the large gap limit, the role of the SC leads can be replaced by the SC pair potential at the impurity site ∆ d = Γ L e iθL + Γ R e iθR , which is caused by the proximity effects. The energy of the Andreev state E A = ξ 2 d + |∆ d | 2 given in (B5) can be rewritten in the form, (33) Therefore, the equi-energy surface of E A can be described as an ellipsoid in a three-dimensional space consisting of ξ d and Γ R ±Γ L as shown in Fig. 10. The Josephson phase φ changes the aspect ratio, or eccentricity, of the ellipsoid. Specifically, for φ ≃ π, the gradient along the Γ R − Γ L axis becomes large, and it implies that the asymmetry in the coupling plays an important role on the properties near the π-junction.
The ground state is a doublet for E A < U/2, and it is a singlet for E A > U/2. Therefore, the Coulomb interacion determines the size of an ellipsoidal such that The inside corresponds to a doublet ground state, and the outside is a singlet. The equations (33) and (34) Fig. 7 can be related to a cross section of the elliposoid for a fixed U/2. For instance, for φ = π, the phase diagram in the atomic limit is determined by ξ 2 d + (Γ R − Γ L ) 2 = U 2 /4, and thus the phase boundary becomes two open sheets at ξ d = ±U/2 for the symmetric couplings Γ R = Γ L , while the asymmetry Γ R = Γ L closes the sheets to form an elliptic cylinder. As another simple example, we consider the symmetric couplings Γ R = Γ L (≡ Γ), for which the boundary is given by 4Γ 2 cos 2 φ 2 = U 2 /4 − ξ 2 d . This equation corresponds to the dashed lines shown in Figs. 5 and 6 for small Γ.
In the atomic limit the first and second many-body excitation energies E 1 and E 2 , measured from the groundstate energy, are given by singlet GS: doublet GS: Two discrete many-body excited states can appear in the energy gap even for finite SC gaps, for which the energies can be calculated with NRG. Then, eqs. (35) and (36) could be used to determine the renormalized values of U and E A , which asymptotically coincide with the bare ones in the limit of |∆ L |, |∆ R | → ∞. The renormalization is caused by the contribution of the cotinuous states above the energy gap. The analysis of the fixed point along these lines will be discussed elsewhere.

V. SUMMARY
We have studied effects of the asymmetry in the Josephson couplings Γ L = Γ R on the ground state of a quantum dot embedded between two superconductors. Specifically, we have obtained the phase diagram of the ground state in some parameter regions. The results show that the asymmetry in the couplings enhances the screening of the local moment in the dot, particularly for the π junction. For φ ≃ π, the Andreev bound state emerges near the Fermi level in the symmetric couplings Γ L = Γ R , and it favors the magnetic doublet ground state for U > 0. The asymmetry in the couplings moves the Andreev bound state away from the Fermi level, and enhances the screening of the local moment. The boundary between the singlet and doublet ground states for intermediate phase 0 < φ < π must be in between the curves for φ = 0 and that for φ = π.
We have also described some symmetry properties of the Anderson impurity in the Josephson junction in terms of the spinor rotation in the pseudo-spin and the current conservation. Specifically, the similarity between the π-junction and a magnetic impurity in an insulating host has been dicussed based on a symmetry property. Furthermore, the role of the asymmetry in the couplings and gaps has been discussed in the noninteracting case, and also in the limit of |∆ L |, |∆ R | → ∞. Our results suggest that in real quantum dots the ground-state properties depends in various ways on the local conditions surrounding the dot. and a ⊥ αδ;λρ (iω n , iω m ) : Σ ⊥ (iω n ) αδ = 1 β iωm λ,ρ a ⊥ αδ;λρ (iω n , iω m ) The kernel a ⊥ αδ;λρ (iω n , iω m ) for the off-diagonal part of the self-energy Σ ⊥ (iω n ) can be shown, using a Ward-Takahashi identity, 35 to be expressed in the form, a ⊥ αδ;λρ (iω n , iω m ) = µ4 µ2,µ3 {τ 3 } αµ4 Γ µ4δ;µ2µ3 (iω n , iω n ; iω m , iω m ) Here, Γ µ4µ1;µ2µ3 (iω 4 , iω 1 ; iω 2 , iω 3 ) is the vertex function in the Nambu representation, and µ j (j = 1, 2, 3, 4) represents the component of the pseudo spins. From these properties, it is deduced that, if the both couplings and gaps are symmetric Γ R = Γ L (≡ Γ) and |∆ R | = |∆ L | (≡ ∆) for the π-junction φ = π, the off diagonal part of the the self-energy vanishes at the impurity site. It simplifies the Green's function as {G(iω n )} −1 ⇒ iω n 1 + 2Γ Therefore, in this particular case the SC correlation cancels out in the impurity site d † ↑ d † ↓ = 0, as can be deduced from eq. (A1). Furthermore, if the system also has the particle-hole symmetry ξ d = 0 additionally, the Green's function is simply proportional to the unit matrix 1.

APPENDIX B: LIMIT OF |∆R|, |∆L| → ∞
In the limit of |∆ L | → ∞ and |∆ R | → ∞, the model can be mapped onto a simple single-site problem, 21,25,32 which can be solved analytically. In this limit, the Dyson equation eq. (A2) can be rewritten in the form The SC correlation penetrates into the dot from the SC leads via Γ L and Γ R . Therefore, the amplitude depends on the Josephson phase φ = θ R − θ L , 1/2 for E A < U/2 in the doublet ground state. Therefore, at T = 0, the self-energy can be expressed in the form The Josephson current can be calculated substituting eq. (B6) into eq. (A7) or (A8), At T = 0, the current can be expressed in the form where E A0 = ξ 2 d + (Γ R + Γ L ) 2 is the maximum value of the bound-state energy, and T ≡ 4Γ R Γ L /E 2 A0 corresponds to a normal transmission probability for noninteracting electrons. The contribution of the Coulomb interaction disappears at T = 0 in the case of the singlet ground state as discussed by Glazman and Matveev. 10 The Josephson current vanishes for the magnetic doublet ground state in the limit of |∆ R |, |∆ L | → ∞. A finite current flowing in the opposite direction in the case of the finite SC gaps, 3 is caused by the continuous excited states above the gaps.