Constructive influence of the induced electron pairing on the Kondo state

Superconducting order and magnetic impurities are usually detrimental to each other. We show, however, that in nanoscopic objects the induced electron pairing can have constructive influence on the Kondo effect originating from the effective screening interactions. Such situation is possible at low temperatures in the quantum dots placed between the conducting and superconducting reservoirs, where the proximity induced electron pairing cooperates with the correlations amplifying the spin-exchange potential. The emerging Abrikosov-Suhl resonance, which is observable in the Andreev conductance, can be significantly enhanced by increasing the coupling to superconducting lead. We explain this intriguing tendency within the Anderson impurity model using: the generalized Schrieffer-Wolff canonical transformation, the second order perturbative treatment of the Coulomb repulsion, and the nonperturbative numerical renormalization group calculations. We also provide hints for experimental observability of this phenomenon.


Results
In what follows we address the proximity induced electron pairing and study its feedback on the Kondo state, focusing on the deep subgap regime. First, we introduce the model and discuss its simplified version relevant for the deep subgap states. Next, we discuss the issue of singlet-doublet quantum phase transition in the limit of negligible coupling to the normal lead, Γ N → 0, emphasizing its implications for the Kondo-type correlations. We then determine the effective spin exchange potential, generalizing the Schrieffer-Wolff transformation 1 for the proximized quantum dot, and confront the estimated Kondo temperature with the nonperturbative NRG data (showing excellent quantitative agreement over the region Γ S ≤ 0.9 U d ). We also discuss the results obtained from the second-order perturbation theory (SOPT) with respect to the Coulomb potential, that provide an independent evidence for the Kondo temperature enhancement by increasing Γ S (in the doublet state). Finally, we discuss the experimentally measurable conductance for the subgap regime and give a summary of our results.

Microscopic model in the subgap regime. For the description of the N-QD-S junction we use the
Anderson impurity model 28 , and the isotropic superconductor is described by . Here, σβ † c k ( ) denotes the annihilation (creation) operator of a spin-σ electron with momentum k and energy ξ kβ in the lead β, while Δ denotes the superconducting energ y gap. It is convenient to introduce the characteristic couplings , assuming that they are constant within the subgap energy regime ω ≤ ∆. Since we are interested in a relationship between the Andreev/Shiba quasiparticles and the Kondo state we can simplify the considerations by restricting ourselves to an equivalent Hamiltonian 29 relevant for the subgap regime in a weak coupling limit Γ S < Δ. Effects due to the superconducting electrode are here played by the induced on-dot pairing gap Δ d = Γ S /2 9,11 . This Hamiltonian (2) neglects the high-energy states existing outside the energy gap window ω ≥ ∆ (see Methods) that are irrelevant for the present context. Subgap quasiparticles of the proximized quantum dot. To understand the influence of electron pairing on the Kondo effect, it is useful to recall basic aspects of the singlet-doublet quantum phase transition in the 'superconducting atomic limit' Γ N → 0 9,30 . Exact eigenstates of the proximized QD are then represented either by the spinful configurations σ with eigenenergy ε d , or the spinless (BCS-type) states d d whose eigenvalues are  with the BCS coefficients The single particle excitations, between the doublet and singlet configurations, give rise to the following quasipar- can be regarded as the low-energy excitations, whereas the other ones (shifted from them by U d ) represent the high-energy features. In realistic systems (where U d is typically much larger than Δ) the latter ones usually coincide with a continuum formed outside the subgap regime [14][15][16][17][18]31 .
Diagonal part of the single particle Green's function (for its definition see Methods) is in the subgap regime given by 11 the Boltzmann constant equal to unity, k B ≡ 1. The missing amount of the spectral weight 1 − α belongs to the high-energy states existing outside the superconductor gap. At zero temperature, the subgap weight changes abruptly from α = 0.5 (when E d < U d /2) to α = 1 (when E d > U d /2). At E d = U d /2 the quasiparticle crossing is a signature of the quantum phase transition from the doublet σ to the singlet configuration − 9,11,13 . For infinitesimally small coupling Γ N one can extend the atomic limit solution (7) by imposing the quasipar- Figure 2 shows the normalized spectral function ω with ρ d (ω) ≡ − π −1 ImG 11 (ω), for the half-filled quantum dot, ε d = − U d /2. On top of these curves we have added the Abrikosov-Suhl peak (at ω = 0) whose half width is given by the Kondo temperature, see Eq. (21). Upon increasing the ratio Γ S /U d , the Andreev quasiparticle peaks move closer and they ultimately merge at the critical point Γ S = U d , and simultaneously the Abrikosov-Suhl peak gradually broadens all the way up to the QPT. For Γ S > U d , the Andreev peaks drift away from each other (see the dashed lines in Fig. 2) and the Kondo feature disappears for the reasons discussed in the next subsection.

Spin exchange interactions and Kondo temperature.
Adopting the Schrieffer and Wolff approach 1 to the Hamiltonian (2) of the proximized quantum dot we can design the canonical transformation S S which perturbatively eliminates the hybridization term = ∑ + . To simplify the notation, we skip the subindex N that unambiguously refers to the metallic lead. The terms linear in V kN can be cancelled in the transformed Hamiltonian ∼  H by choosing the operator Ŝ from the following constraint of the half-filled quantum dot obtained from the superconducting atomic limit solution (using the quasiparticle broadening Γ N = 10 −1 Γ S ) superposed with the Abrikosov-Suhl peak whose half width T K is expressed by Eq. (21). The solid/dashed lines correspond to the doublet/singlet ground state configuration and the thick-red curve indicates the quantum phase transition at Γ S = U d .
For the Hamiltonian (2) this can be satisfied with the anti-hermitian operator = − †^Ŝ for , The second term of Eq. (10) explicitly differs from the standard operator used by Schrieffer and Wolff 1 . From the lengthy by straightforward algebra we find that the constraint (9) implies the following coefficients γ ν For Δ d = 0, the coefficients γ α k 2, identically vanish and the other ones, given by Eqs (12 and 14), simplify to the standard expressions γ of the Schrieffer-Wolff transformation 1 .
In the transformed Hamiltonian .
we can recognize: the spin exchange term, the interaction between QD and itinerant electrons, the pair hopping term, and renormalization of the QD energy and the on-dot pairing. Since we focus on the screening effects, we study in detail only the effective spin-exchange term Formal expression for the effective exchange potential is analogous to the standard Schrieffer-Wolff result 1 , but here we have different coefficients γ ± k 1, expressed in Eqs (12 and 14). This important aspect generalizes the Schrieffer-Wolf potential 1 and captures the effects induced by the on-dot pairing.
In particular, near the Fermi momentum the exchange potential (18) simplifies to It is worthwhile to emphasize that this formula (19) precisely reproduces constraint for the quantum phase transition discussed in the previous section. To prove it, we remark that J k k , F F changes discontinuously from the negative (antiferromagnetic) to the positive (ferromagnetic) values at ε ε . Such changeover occurs thus at To estimate the effective Kondo temperature in the case of spinful configuration (for Γ S < U d ), we use the formula 32,33 , is the density of states at the Fermi level, D is the cut-off energy and the auxiliary function is defined as In present case the Kondo temperature is expressed by  (19) is then given by   (21) increases versus Γ S , all the way to the critical point at Γ S = U d . Such tendency, indicated previously by the NRG data 13 , is solely caused by the quantum phase transition. In a vicinity of the QPT the divergent exchange coupling (22) is a typical drawback of the perturbative scheme. Figure 3 demonstrates that the formula (21) is reliable over the broad regime Γ S ≤ 0.9 U d . This straightforward conclusion can be practically used by experimentalists.
Equilibrium transport properties. We now corroborate the analytical results with accurate numerical renormalization group calculations 34,35 . In NRG, the logarithmically-discretized conduction band is mapped onto a tight binding Hamiltonian with exponentially decaying hopping, ξ ∝ Λ − n n/2 , where Λ is the discretization parameter and n site index. This Hamiltonian is diagonalized in an iterative fashion and its eigenspectrum is then used to calculate relevant expectation values and correlation functions. In our calculations, we assumed Λ = 2 and kept N k = 2048 states during iteration exploiting Abelian symmetry for the total spin zth component 36 . Moreover, to increase accuracy of the spectral data we averaged over N z = 4 different discretizations 37,38 . We also assumed flat density of states, ρ = 1/2 W, with W the band half-width used as energy unit W ≡ 1, U d = 0.1, Γ N = 0.01 and zero temperature. In the absence of superconducting correlations, Γ S = 0, this yields the Kondo temperature, ≈ − T 10 K 0 5 , obtained from the half width at half maximum (HWHM) of the dot spectral function ρ d (ω) calculated by NRG. Figure 3(a) presents the energy dependence of the normalized spectral function A(ω) of the correlated quantum dot at half-filling for the model Hamiltonian (2) calculated for different values of Γ S . In the case of Γ S = 0, A(ω) exhibits Hubbard resonance for ω = ± U d /2 and the Abrikosov-Suhl peak at the Fermi energy, ω = 0. It is clearly visible that increasing Γ S leads to the broadening of the Abrikosov-Suhl peak. In Fig. 3(b) we compare the Scientific RepoRts | 6:23336 | DOI: 10.1038/srep23336 relative change of the Kondo temperature obtained from the HWHM of A(ω) calculated by NRG (circles) and from the approximate formula (21) based on the generalized Schrieffer-Wolff canonical transformation (solid line). The numerical constant η was estimated to be η = 0.6. The agreement is indeed very good and small deviations occur only close to Γ S = U d , but then the system is no longer in the local moment regime and the Kondo effect disappears.
The normalized spectral function of the half-filled quantum dot in both the doublet, Γ S < U d , and singlet region, Γ S > U d , is shown in Fig. 4. In the doublet region we clearly observe the zero-energy Abrikosov-Suhl peak, whose width gradually increases upon increasing Γ S . Simultaneously the Andreev peak (whose width is roughly proportional to Γ N ) moves toward the gap center. In the singlet state, on the other hand, the Abrikosov-Suhl peak does no longer exist and the Andreev peaks gradually depart from each other for increasing Γ S . The same evolution of the Andreev and the Abrikosov-Suhl quasiparticle peaks is illustrated in Fig. 2, combining the superconducting atomic limit solution with the perturbative estimation of the Kondo temperature (21).
Broadening of the Abrikosov-Suhl peak upon approaching the doublet-singlet transition can be independently supported by the second-order perturbative treatment of the Coulomb interaction term        22 11 and 12 21 . Figure 5 shows the spectral function A(ω) obtained from the numerical self-consistent solution of Eqs (25)(26)(27)(28). For comparison with the NRG results we focused on the half-filled quantum dot = .
In the weakly correlated case U d ≤ Γ S (corresponding to the spinless BCS-type ground state) the subgap spectrum is characterized by two Andreev states (shown by the dashed-line curves). For U d ~ Γ S , these Andreev states merge, forming a broad structure around the zero energy. In the strongly correlated case U d ≥ Γ S (corresponding to the spinful doublet configuration) we observe appearance of the Kondo feature (at zero energy) that coexists with the Andreev states 44 . We also notice that the width of the zero-energy peak (i.e. 2T K ) depends on the ratio U d /Γ S and such tendency qualitatively agrees with our estimations based on the Schrieffer-Wolff transformation and with the nonperturbative NRG data. Differential Andreev conductance. We now analyze how the observed features reveal in the nonlinear response regime. For possible correspondence with the experimentally measurable quantities we consider the subgap Andreev current    Figure 6 shows the qualitative changeover of the subgap conductance for representative values of U d and Γ S , corresponding to doublet and singlet states. While approaching the QPT from the doublet side, we observe that the zero-bias Abrikosov-Suhl peak is gradually enhanced, and its width significantly broadens. This tendency is caused by the characteristic Kondo temperature, which increases with increasing Γ S /U d . For Γ S > U d , however, the Kondo feature is completely absent (in agreement with NRG and Schrieffer-Wolff estimations). The magnitude of the subgap Andreev conductance approaches then the maximum value 4e 2 /h near the Andreev/Shiba states. We notice the quantitative difference between the subgap transport properties (shown in Fig. 6) and the electronic spectrum (displayed in Figs 4 and 5). Observability of the Kondo enhancement would be thus possible only close to the QPT on the doublet side.

Discussion
We have studied the influence of the electron pairing on the Kondo effect in the strongly correlated quantum dot coupled (by Γ N ) to the metallic and (by Γ S ) to superconducting reservoirs by three independent methods. The proximity induced on-dot pairing and the Coulomb repulsion U d are responsible for the quantum phase transition between the (spinless) BCS-like singlet and the (spinful) doublet configurations, depending on the ratio of Γ S /U d . Upon approaching this quantum critical point from the doublet side, one observes the enhancement of the Kondo temperature with increasing Γ S 13 . We have provided the microscopic arguments supporting this behavior based on the generalized Schrieffer-Wolff canonical transformation. This perturbative treatment of the coupling to metallic lead revealed enhancement of the antiferromagnetic spin-exchange potential, responsible for the Abrikosov-Suhl resonance. We have compared the estimated Kondo temperature with the numerical renormalization group calculations, and found excellent agreement over the broad regime Γ S < 0.9 U d . We have confirmed this tendency (for arbitrary Γ N ) using the second-order perturbative treatment of the Coulomb interaction. Our analytical estimation of the Kondo temperature (21) can be quantitatively verified in experimental measurements of the differential Andreev conductance. We have shown, that the zero-bias enhancement of the subgap conductance (already reported 14,16,22,23 for some fixed values of Γ S ) would be significantly amplified with increasing the ratio Γ S /U d , but only on the doublet side. Such behaviour is in stark contrast with the zero-bias anomaly caused by the Majorana quasiparticles due to the topologically non-trivial superconductivity.

Methods
The deep subgap regime |ω| ≪ ▵. When studying the proximity effect of the Anderson-type Hamiltonian (1) one has to consider the mixed particle and hole degrees of freedom. This can be done, by defining the matrix Here we determine its diagonal and off-diagonal parts in the equilibrium case (which is also useful for description of the transport within the Landauer formalism). The Fourier transform of the Green's function can be expressed by the Dyson equation The self-energy ω Σ ( ) d accounts for the coupling of the quantum dot to external reservoirs and for the correlation effects originating from the Coulomb repulsion U d .
The quantum dot hybridization with the leads can be expressed analytically by where g kβ (ω) are the (Nambu) Greens' functions of itinerant electrons. In the wide-band limit this self-energy is given by the following explicit formula 11 for for (31)  In the subgap regime ω < ∆ the Green's function of uncorrelated quantum dot acquires the BCS-type structure . The resulting spectrum consists of two in-gap peaks, known as the Andreev 11,19 or Yu-Shiba-Rusinov 20,21 quasiparticles. Their splitting is a measure of the pairing gap Δ d induced in the quantum dot. Figure 7 displays the characteristic energy scales of the uncorrelated quantum dot. For infinitesimally weak coupling Γ N = 0 + the in-gap states have a shape of Dirac delta functions (i.e. represent the long-lived quasiparticles). Otherwise, they acquire a finite broadening proportional to Γ N . In the absence of correlations (for U d = 0) the quasiparticle energies E A,± can be determined by solving the following equation 45,46 ε 2 , 2 In the strong coupling limit, Γ ∆  S , we can notice that in-gap quasiparticles appear close to the superconductor gap edges , whereas in the weak coupling limit, Γ ∆  S , they approach the asymptotic val- 2 . For Γ N → 0, the latter case is known as the 'superconducting atomic limit' . The self-energy (31) simplifies then to the static value Influence of the correlation effects. We note that the early studies of the nontrivial relationship between the Coulomb repulsion and the proximity induced electron pairing of the normal metal -quantum dot -superconductor (N-QD-S) junctions have adopted variety of the theoretical methods, such as: slave boson approach 47,48 , equation of motion 49 , noncrossing approximation 50 , iterative perturbation technique 39 , path integral formulation of the dynamical mean field approximation 51 , constrained slave boson method 52 , numerical renormalization group [11][12][13]30 , modified equation of motion 45 , functional renormalization group 53 , expansion around the superconducting atomic limit 54 , cotunneling treatment of the spinful dot 55 , numerical QMC simulations 56 , selfconsistent perturbative treatment of the Coulomb repulsion 40 and other 9,46,57 . Amongst them only the The relationship between the proximity induced on-dot pairing and the screening effects can be better understood by analyzing the superconducting order parameter ↓ ↑d d and the QD magnetization . In Fig. 8 we show their dependence on the coupling Γ S for several Γ N /U d ratios calculated by NRG. For finite superconducting energy gap a sign change of the order parameter signals the quantum phase transition 13 . However, in the case of infinite gap considered here, ↓ ↑d d only drops to zero at the transition point 11,30 . As clearly seen in the figure, the order parameter ↓ ↑d d increases from 0 to 1 2 around Γ S ~ U d [ Fig. 8(a)] corresponding to the QPT. Its enhancement is accompanied by the suppression of the dot magnetization, which vanishes in the singlet phase, Γ S > U d , Fig. 8(b)]. This behavior comes from the well known fact, that the local magnetic susceptibility is inversely proportional to the Kondo temperature 3 . Such variation of the QD magnetization Ŝ d z resembles the analogous tendency for the magnetic ordering in heavy fermion compounds 58,59 , where its suppression is driven by a competition between the local Kondo effect with the non-local RKKY interaction. In our case, the quantitative changeover of ↓ ↑d d and Ŝ