Kondo Screening of Local Moments in a Triangular Triple Quantum Dot Connected to Normal and Superconducting Leads

We study the interplay between the Kondo and superconducting (SC) proximity effects, taking place in a triangular triple quantum dot (TTQD) connected to one normal and two SC leads. This system shows various quantum phases. Without the SC leads, the lowest two states that belong to the different spin sectors, $S=0$ and $S=1/2$, become energetically very close to each other near half filling. The singlet one is a Kondo-screened state by conduction electrons from the normal lead, and the doublet one is a resonating valence bond state with unpaired free spin which remains unscreened. Furthermore, when one additional electron enters the TTQD, the ground state becomes a doublet in which the $S=1$ local moment due to the Nagaoka ferromagnetism is partially screened by conduction electrons.The Cooper pairs penetrating into the TTQD from the SC leads reconstruct the wavefunctions and vary the phase boundaries between these quantum states in the parameter space. We calculate ground-state phase diagrams using the numerical renormalization group, and show that the SC proximity effect induces a reentrant transition in-between the three- and four-electron fillings.

At half-filling N d = 3, the ground state of the isolated TTQD cluster for which all the leads are disconnected has a four-fold degeneracy caused by the C 3v symmetry of an equilateral triangle.This symmetry is broken when a single normal lead is connected to one of the dots in the TTQD, and thus the four states are separated into two spin states.One is the singlet bond (SB) state with an unpaired spin, which is eventually screened by conduction electrons, in the dot connected to the normal lead.The other is the resonating valence bond (RVB) state with unpaired spin that remains unscreened in the dots away from the normal lead.Furthermore, the Kondo effect of an S = 1 high-spin Nagaoka state occurs at N d ≃ 4   6)-( 8), are illustrated for U ≫ t. [4,5].The Cooper pairs that can penetrate into the TTQD from superconducting (SC) leads give further variety to the low-energy states.The SC proximity effects in triple-dot systems have theoretically been examined in some special situations so far [12][13][14][15].However, the interplay between the Kondo and the SC proximity effects have not been fully understood.In this report, we clarify how the ground state evolves as a result of the interplay between the Kondo and SC proximity effects.

Formulation
We consider the TTQD connected to one normal and two SC leads, shown schematically in Fig. 1(a) and can be described by the Hamiltonian of the form H = H dot + H N + H TN + H S + H TS : Here, d † i,σ (d i,σ ) is the creation (annihilation) operator for an electron with energy ε d and spin σ in the quantum dot i (= L, R, N ), n d,i,σ ≡ d † i,σ d i,σ is the local number operator, U is the Coulomb interaction, and t (> 0) is the hopping matrix element between the dots.c † ε,σ and c ε,σ are the operators for conduction electrons in the normal lead, defined such that , with ρ c ≡ 1/(2D) the density of states and D the halfband width.v N is the tunneling matrix element between the dot and the normal lead and it determines the resonant width Γ N ≡ πρ c v 2 N .The operators s † ε,α,σ and s ε,α,σ describe electrons in the SC leads on α = L, R, with ∆ S,α ≡ |∆ S,α | e iθα the SC energy gap.This model takes a simplified form in the large gap limit |∆ S,α | → ∞, where the SC proximity effects can be described by the pair potentials penetrating into the adjacent dots [16], In this report, we investigate the case where the Josephson effect is absent, i.e., θ L = θ R = 0.

Various kinds of local moments induced in the isolated TTQD
We discuss here some notable properties that the TTQD already has in the isolated limit Γ N = Γ S = 0. Figure 1(b) shows the occupation number N d ≡ i N d,i , defined by N d,i ≡ σ n d,i,σ at zero temperature T = 0, as a function of ε d for U = 2πt.In the plateau region of the half-filled case N d = 3, the ground state has a four-fold degeneracy which can be classified into two parts, referred to as the singlet bond (SB) and the resonance valence bond (RVB) states for U ≫ t [3,4], In the SB state, a single unpaired spin σ is localized in the dot adjacent to the normal lead.Therefore, this local moment can easily be screened by conduction electrons to form the Kondo singlet when the tunnel coupling H TN is switched on.In contrast, in the RVB state the unpaired spin situates in the dot, α = L or R, away from the normal lead as illustrated in Fig. 1(d).For this reason, the local moment in the RVB state remains unscreened for small tunnel couplings Γ N , as shown in the next section.
In the plateau region of N d = 4 next to the half-filled one, the ground state becomes a spin triplet for U > 0 and t > 0. This is caused by the Nagaoka ferromagnetic mechanism along the closed loop, which is illustrated in Fig. 1(e) for the spin S z = +1 component, This high-spin S = 1 moment can be fully screened via a two-stage processes if two conducting channels are coupled [4].However, in the configuration shown in Fig. 1(a), only one-half of the moment can be screened by conduction electrons from the single normal lead.Furthermore, the leads attached to the TTQD break the C 3v symmetry and lift the four-fold degeneracy of the ground state at the plateau of N d = 3.We show in the following that at small Γ N a phase transition occurs between the Kondo-screened SB and the unscreened spin-1/2 RVB states in the region of N d ≃ 3.0, and then the RVB state continuously evolves into the under-screened Nagaoka state as N d approaches 4.0.

Kondo screening of the local moments by a single conducting channel
In order to clarify how the different kinds of local moments are screened by conduction electrons, we first of all examine the case where Γ S = 0 and only a single normal lead is connected to the TTQD.This configuration of the TTQD has been studied by Michell et al. at half-filling N d = 3 [3].However, here we explore low-energy properties over a wide filling """ "" " " " " " " "" "" " " " " "" " " "" " " """"" " " " " " " " " " " " " " " " " " " " " " " " """""" " """"" " " " " " " " " " "" " " " " """" " " " " " " " range 3.0 N d 6.0, using the numerical renormalization group (NRG) approach [17].We choose the NRG discretization parameter to be Λ = 6.0 and retain typically N kept = 1000 low-lying excited states.In Fig. 2(a), the ground-state phase diagram is plotted as a function of ε d /U and Γ N /t for relatively large interaction U = 2πt.The dot-lead coupling Γ N lifts the four-fold degeneracy of the lowest energy state of the TTQD cluster near half-filling N d ≃ 3.0.The NRG results show that the ground state becomes a doublet in the boot-shaped shaded region while it becomes a spin singlet on the outside.The spin S = 1/2 ground state at the tip of the boot, which spreads over the region −0.82U ε d −0.58U for small couplings Γ N 1.5t, can be identified as the RVB state with the free spin-1/2 degrees of freedom.Furthermore, the spin S = 0 ground state on the outside of the tip, −0.58U ε d −0.32U , can be identified as the Kondo-screened SB state.These identifications can be deduced from Figs. 2(b) and (c).
We see in Fig. 2(b) that the entropy of the TTQD calculated at the two points, (ε d = −0.70U, Γ N = 0.12t) and (ε d = −0.55U, Γ N = 0.12t), show a similar behavior at high temperatures T /D 10 −8 : both show a plateau of the height log 4 reflecting the four-fold degeneracy of the isolated limit at temperatures of the order of T /D ≃ 10 −3 , and then at 10 −8 T /D 10 −5 they take another wide plateau of the height log 2 corresponding to the spin degrees of freedom of the SB or the RVB state.However, at much lower temperatures T /D 10 −11 the moment vanishes for ε d = −0.55U, whereas the moment remains unscreened for ε d = −0.70Uas Γ N is small at this point in the phase diagram.The entropy at the point (ε d = −0.85U, Γ N = 0.12t) is also plotted in Fig. 2(b).It takes a plateau of the height log 3 corresponding to the free S = 1 moment of Nagaoka state at 10 −6 T /D 10 −3 , and then at lower temperatures T /D 10 −9 it takes another plateau of the height log 2, showing that half of this moment is screened by conduction electrons from the normal lead.
Charge distribution of the ground state is plotted vs ε d /U in Fig. 2(c) for small dotlead coupling Γ N = 0.12t.We can see at ε d ≃ −0.6U , where the phase transition between the singlet and doublet ground states takes place, the occupation number N d,N of dot on the normal-lead side and that of the other two dots N d,L + N d,R show a small but finite discontinuity.At this point the total amount of these occupation numbers is almost unchanged N d ≃ 3.0, which indicates the transition occurs between the Kondo screened SB and the unscreened RVB states.As ε d decreases, the total occupation number N d varies from 3.0 to 4.0 through a gradual step at ε d ≃ −0.82U .This means that the unscreened RVB state emerging at the tip of the boot-shaped region can continuously evolve into the under-screened Nagaoka state spreading widely over the other side.
In contrast, in the leg part of the shaded boot-shaped region in Fig. 2(a), i.e., for large dotlead couplings Γ N 1.5t, the total occupation number N d discontinuously changes its value by an approximate amount of 1.0 at the phase boundary along ε d ≃ −0.9U , as demonstrated in Fig. 2(d) for Γ N = 4t.Therefore, the under-screened Nagaoka state spreads across the leg part of the boot to the boundaries on both sides ε d ≃ −0.9U and ε d ≃ −1.15U .

Ground state of the TTQD connected to one normal and two SC leads
We next consider the configuration with two additional SC leads as illustrated in Fig. 1(a).The ground-state phase diagram is shown in Fig. 3(a) as a function of ε d /U and Γ N /t for several values of Γ S /t, choosing the same value U = 2πt for the interaction as in the above.We can see that the region of the S = 1/2 ground state shrinks as couplings to the SC leads Γ S , defined in Eq. ( 5), increases.In particular for Γ S /t = 2.0, the under-screened S = 1/2 region around ε d ≃ −0.93U becomes very narrow and is enclosed by the boundary line as Cooper pairs penetrating from the SC leads dominate the other magnetic correlations.We can also see that a reentrant transition occurs for small Γ N across the S = 1/2 region at −0.9U ε d −0.6U .This is caused by the fact that an infinitesimal Γ S lifts the four-fold degeneracy of the lowest energy state in the half-filled case N d = 3 and makes the SB state the ground state for very small normal tunnel couplings Γ N ≪ t.The singlet region which spreads below the doublet region, for instance the one emerges at Γ N /t 0.4 in Fig. 3(b), can be identified as the Kondo-screened SB state.The other side of the boundary can be identified as the unscreened RVB state for which effects of Γ N dominates that of Γ S .These identifications can be verified from the behaviors of the entropy of the TTQD, calculated for Γ S = 1.0t at five different points on the dashed-vertical and dashed-horizontal lines in the phase diagram Fig. 3(b).
The entropy of the TTQD obtained at the three on line for Γ N = 0.4t are plotted in Fig. 3(c).The entropy at the point ε d = −0.87U, where N d ≃ 4.0, shows a plateau of the height log 3 at high temperatures 10 −4 T /D 10 −3 , and then at low temperatures it converges to log 2, which indicates that the ground state at this point is the under-screened Nagaoka state.In contrast, at the point of ε d = −0.80Uwhere N d ≃ 3.0, the entropy directly approaches the value for the free S = 1/2 moment as T decreases without showing a step at log 3, and thus the ground state at this point can be identified as the unscreened RVB state.We can also see that the entropy at the point (ε d /U = −0.80,Γ N /t = 0.12) plotted in Fig. 3(d) clearly shows the behavior of the fully-screened SB state: " " " " " " " " " " " " " " "" "" " " " " " " " " " " " " " " " We can also see in Fig. 3(a) that the reentrant behavior is suppressed as tunnel coupling to the SC lead Γ S increases.The unpaired spin of the RVB state, illustrated in Fig. 1(d), can be replaced by local Cooper pairs penetrating into the adjacent dots from the SC leads for large Γ S , and it makes the ground state a singlet near the reentrant region.

Summary
We have studied the quantum phase transition and crossover occurring in the TTQD over a wide range of electron fillings.In the case where only a single normal lead is connected, the unscreened RVB state and the under-screened Nagaoka state merge into the doublet phase, which appears as a boot-shaped region in the phase diagram.The RVB state spreading over the tip of the boot can continuously evolve into the under-screened Nagaoka state as ε d decreases.When two additional SC leads are connected, a reentrant transition takes place around the tip part of the doublet phase as the SC proximity makes the Kondo-screened SB state the ground state for small Γ N ≪ t.The magnetic doublet phase shrinks as Cooper pairs penetrating into the TTQD increase with Γ S .

Fig. 2 .
Fig. 2. NRG results for TTQD connected to a single normal lead (Γ S = 0) for U = 2πt.(a): Groundstate phase diagram, classified according to the total spin, S = 1/2 (shaded region) or S = 0 (the outside).(b): Temperature dependence of the entropy of TTQD for Γ N = 0.12t at ε d = −0.85U( ), −0.70(•), and −0.55U ( ).The occupation number N d,N of the dot adjacent to the normal lead, that for the other dots N d,L + N d,R , and the total one N d are plotted vs ε d for Γ N = 0.12t (c), and 4t (d).The dash line represents N d for Γ N = 0.