Fano-Kondo resonance versus Kondo plateau in an Aharonov-Bohm ring with an embedded quantum dot

We theoretically examine the transport through an Aharonov-Bohm ring with an embedded quantum dot (QD), the so-called QD interferometer, to address two controversial issues regarding the shape of the Coulomb peaks and measurement of the transmission phase shift through a QD. We extend a previous model [B. R. Bulka and P. Stefanski, Phys. Rev. Lett. 86, 5128 (2001); W. Hofstetter, J. Konig, and H. Schoeller, ibid. 87, 156803 (2001)] to consider multiple conduction channels in two external leads, L and R. We introduce a parameter p_{\alpha} (|p_{\alpha}| \le 1) to characterize a connection between the two arms of the ring through lead \alpha (=L, R), which is the overlap integral between the conduction modes coupled to the two arms. First, we study the shape of a conductance peak as a function of energy level in the QD, in the absence of electron-electron interaction U. We show an asymmetric Fano resonance for |p_{L,R}| = 1 in the case of single conduction channel in the leads and an almost symmetric Breit-Wigner resonance for |p_{L,R}|<0.5 in the case of multiple channels. Second, the Kondo effect is taken into account by the Bethe ansatz exact solution in the presence of U. We precisely evaluate the conductance at temperature T=0 and show a crossover from an asymmetric Fano-Kondo resonance to the Kondo plateau with changing p_{L,R}. Our model is also applicable to the multi-terminal geometry of the QD interferometer. We discuss the measurement of the transmission phase shift through the QD in a three-terminal geometry by a"double-slit experiment."We derive an analytical expression for the relation between the measured value and the intrinsic value of the phase shift.


I. INTRODUCTION
In the mesoscopic physics, an Aharonov-Bohm (AB) ring with an embedded quantum dot (QD), the so-called QD interferometer, has been intensively studied to elucidate the coherent transport through a QD with discrete energy levels and strong Coulomb interaction [1][2][3][4].
Controversial issues still remain regarding the transport through the interferometer despite long-term experimental and theoretical studies. We theoretically revisit these issues by generalizing a previous model to consider multiple conduction channels in external leads and a multi-terminal geometry.
We first discuss the shape of Coulomb peaks, i.e., conductance G as a function of gate voltage attached to the QD to control the energy levels electrostatically. Kobayashi et al.
observed an asymmetric shape of the Coulomb peaks, which has a peak and dip in accordance with the Fano resonance, using a QD interferometer [5]. The Fano resonance stems from the interference between a discrete energy level in the QD and continuum energy states in the ring [6,7]. Remarkably the resonant shape of the Coulomb peaks changes with a magnetic flux penetrating the ring. However, the other groups observed symmetric Coulomb peaks, which can be fitted to the Lorentzian function of Breit-Wigner resonance [8]. No criteria has been elucidated regarding the Fano or Breit-Wigner resonance in the QD interferometer.
The second issue concerns the measurement of the transmission phase shift through a QD using the QD interferometer as a double-slit experiment. It is well known that the phase shift cannot be observed by the interferometer in the two-terminal geometry [1]. This is due to the restriction by the Onsager's reciprocity theorem: Conductance G satisfies G(B) = G(−B) for magnetic field B, or G(φ) = G(−φ) for the AB phase φ = 2πΦ/(h/e) with magnetic flux Φ penetrating the ring [3,4]. The phase measurement was first reported using the interferometer in a four-terminal geometry [2]. In the Kondo regime, the phase shift through the QD should be locked at π/2 [9][10][11]. This phase locking was also investigated experimentally using four-or three-terminal devices [8,[12][13][14][15][16][17]. It is nontrivial, however, how precisely the phase shift is measured using the multi-terminal interferometer.
Theoretically, Bulka and Stefański studied Fano and Kondo resonances using a model for the two-terminal QD interferometer, in which a QD is coupled to leads L and R and the leads are directly coupled to each other [18]. Hofstetter et al. found an asymmetric Fano-Kondo resonance by applying the numerical renormalization group calculation to an equivalent model [19]. Their works were followed by many theoretical studies, e.g., to elucidate various aspects of the Kondo effect [20][21][22][23][24][25][26][27][28][29], fluctuation theorem [30], and dynamics of electronic states [31]. Recently, the Fano resonance was proposed to detect the Majorana bound states [32,33].
Although the model in Refs. [18,19] was widely used, it is insufficient to describe experimental situations with multiple conduction channels in the leads. In the present paper, we propose an extended model for the QD interferometer to resolve the above-mentioned problems. As shown in Fig. 1(a), our model is the same as the previous model except the tunnel couplings, V L , V R , and W , depend on the states in leads L and R. We show that the state-dependence can be disregarded only in the case of single conduction channel in the leads.
Our model yields a parameter p α (|p α | ≤ 1) to characterize a connection between the two arms of the ring through lead α (= L, R), which is the overlap integral between the conduction modes coupled to the upper and lower arms of the ring. First, we examine the shape of a conductance peak in the two-terminal geometry, in the absence of electronelectron interaction U in the QD. We show an asymmetric Fano resonance for |p L,R | = 1 in the case of single conduction channel in the leads and an almost symmetric Breit-Wigner resonance at |p L,R | < 0.5 in the case of multiple channels. Hence our model could explain the experimental results of both the asymmetric Fano resonance [5] and almost symmetric Breit-Wigner resonance [8], with fitting parameters p L,R to their data.
Second, the transport in the Kondo regime is examined by exploiting the Bethe ansatz exact solution. This method precisely gives us the conductance at temperature T = 0 in the presence of U . We show a crossover from an asymmetric Fano-Kondo resonance [19] to the Kondo plateau with changing p L,R .
Our model is also applicable to the multi-terminal geometry, where state k [k ] belongs to lead L(1) or L(2) [R(1) or R(2)], as depicted in Fig. 1(b). We discuss the measurement of the transmission phase shift through the QD by a "double-slit experiment" using a three-terminal interferometer. We derive an analytical relation between the observed phase shift and intrinsic phase shift in the absence of U . Using a simple model to represent the experiment by Takada et al. [8,16,17], we evaluate the measured phase shift in both the absence and presence of U . For U = 0, we show that the phase locking at π/2 is observable in the Kondo regime although it is slightly different from the behavior of the intrinsic phase shift that satisfies the Friedel sum rule.
The organization of the present paper is as follows. In section II, we present our model and calculation method. The parameters p L and p R are introduced, which are relevant to the shape of a conductance peak. We explain the calculation method of the conductance at T = 0, taking into account the Kondo effect exactly. In section III, the calculated results are given for the shape of the conductance peak in a two-terminal geometry. We discuss the asymmetric Fano resonance versus symmetric Breit-Wigner resonance in the absence of U , by changing p L,R . We also study the conductance in the Kondo regime in the presence of U and show a crossover from an asymmetric Fano-Kondo resonance to the Kondo plateau. In section IV, we examine the phase measurement in a three-terminal geometry by a double-slit interference experiment. We derive an analytical relation between the measured value and R , and µ R , respectively, in the formulation in Appendix A. We fix µ intrinsic value for the transmission phase shift through the QD in the absence of U . Two specific models are studied to see a crossover from two-to three-terminal measurement and to simulate the experimental situation using two quantum wires to form the QD interferometer [8,16,17]. Section V is devoted to the discussion regarding the justification and generality of our model. The conclusions are given in section VI.

A. Model
Let us consider a model for the QD interferometer in a two-terminal geometry, depicted in Fig. 1(a). The Hamiltonian is given by where Here, n σ = d † σ d σ is the number operator for electrons with spin σ in the QD with energy level ε d , where d † σ and d σ are creation and annihilation operators, respectively. a † α,kσ and a α,kσ are those for conduction electrons in lead α (= L, R) with state k and spin σ, whose energy is denoted by ε k . U is the electron-electron interaction in the QD. The tunnel Hamiltonian, H T , connects the QD and state k in lead α by V α,k through the lower arm of the ring, whereas it connects state k in lead L and state k in lead R by W k ,k through the upper arm of the ring. The AB phase is defined by φ = 2πΦ/(h/e) for a magnetic flux Φ penetrating the ring. To make the calculation simple, we decompose W k ,k into the contributions from state k in lead L and state k in lead R as This separable form is justified for the tight-binding models, as discussed in section V.
For lead α, we introduce the following three parameters to describe the contribution to the transport: We assume that the ε-dependence of these parameters is weak around the Fermi level and simply express Γ α , x α , and p α for ε ≈ E F . Γ α (x α ) characterizes the strength of tunnel coupling to the QD (coupling through the upper arm of the ring). Using x = x L x R , the transmission probability through the upper arm of the ring is given by Concerning x L and x R , the physical quantities are always written in terms of x = x L x R in our model [34].
The parameter p α (|p α | ≤ 1) defined by Eq. (8) characterizes a connection between the two arms of the ring through lead α (= L, R). Namely, p α (ε) is an overlap integral between the conduction mode coupled to the QD and that coupled to the upper arm of the ring in lead α at a given energy ε. The tunnel Hamiltonian H T in Eq. (4) indicates that these modes are given by |ψ QD The interference by the AB effect is maximal when |p L | = |p R | = 1, whereas it completely disappears when p L = 0 or p R = 0. In the previous model [18,19], |ψ QD α = |ψ upper α and thus p α = 1 since V α,k and √ w α,k are constant, irrespective of state k. As seen in the following sections, p L and p R play a crucial role in determining the shape of conductance peaks. Although p L and p R should be given by the details of experimental systems, we treat them as parameters as well as Γ L , Γ R , and x.
As an example, let us consider quasi-one-dimensional leads, or leads of a quantum wire.
The state in lead α is specified by k = q in the case of single conduction channel and by k = (q, i) in the presence of multiple channels, where q is the momentum along the wire and i is the index of the subbands. In the former, V α,k = V α (ε k ) and w α,k = w α (ε k ), which yield Γ α (ε) = πρ α (ε)[V α (ε)] 2 with density of states ρ α in the lead, x α (ε) = πρ α (ε)w α (ε), and |p α | = 1 from Eqs. (6)- (8). In the case of multiple channels, |p α | < 1, as shown in section V.
Note that a similar parameter to p α was introduced in the study on a double quantum dot in parallel and was evaluated for three-or two-dimensional leads with a flat surface [35].
B. Formulation of electric current We formulate the electric current using the Keldysh Green's functions along the lines of Ref. [19] (see Appendix A). For example, the current from lead L(1) in Fig. 1(b) is given by where N (1) is the number operator for electrons in the lead. In the stationary state, I L is expressed in terms of the retarded Green's function G r d,d (ε) and lesser Green's function G < d,d (ε) of the QD, in Eq. (A22) in Appendix A.
Next, we eliminate G < d,d (ε) from the expression and write the current using G r d,d (ε) only. We restrict ourselves to the case of with µ L − µ R = eV , to simplify the current expression. Then the current conservation is written as follows in the stationary state: where f α (ε) = [(ε − µ α )/(k B T ) + 1] −1 is the Fermi distribution function in lead α(1) or α(2) [Γ will be given in Eq. (24)]. Using Eq. (18), we eliminate G < d,d (ε) from the current expression, e.g., Eq. (A22) for I with the transmission probability Here, the coefficient C 1 is given by Note that (i) for p L = p R = 1, where a single conduction channel is effective in each lead, Eq. (20) coincides with the current expression derived in Ref. [19]. (ii) For p L = p R = 0, the transmission probability is given by This is the summation of the transmission probability through the upper arm, T upper in Eq. (9), and that through the QD, indicating no interference effect between the two paths in the QD interferometer.
For multi-terminal systems, the current is expressed in terms of the retarded Green's function G r d,d (ε) in a similar way. The expression is given in Eqs. (A29) and (A30) in Appendix A.

D. Exact calculation for Kondo effect
In the absence of Coulomb interaction, U = 0, the retarded Green's function of the QD is given by where the self-energy by the tunnel couplings is with an effective linewidth This expression is common to two-and multi-terminal systems.
In the presence of U , G r d,d (ε) is evaluated exactly in the following way. The Green's function at U = 0 indicates that our models are equivalent to the situation in which a QD with an energy levelε is connected to a lead with linewidthΓ, as shown in Appendix B. In the Fermi liquid theory, the Green's function is written as (25),Γ * = zΓ is that ofΓ in Eq. (24), and z is a factor of wavefunction renormalization by the electron-electron interaction U [36][37][38]. Since the phase shift θ QD at the QD is given by tan θ QD =Γ * /ε * d , the Green's function is determined by θ QD as θ QD is related to the electron number per spin in the QD through the Friedel sum rule, θ QD = π n σ . n σ is evaluated at temperature T = 0 using the Bethe ansatz exact solution [39,40]. Hence we can precisely calculate G r d,d (0) and thus the conductance It is worth mentioning that the effective energy levelε d (φ) in the QD gives rise to the φ-dependent Kondo temperature [20]. It is written as with D being the bandwidth [36,41] although T K (φ) is irrelevant to our study on the transport properties at T = 0.

III. CALCULATED RESULTS IN TWO-TERMINAL GEOMETRY
In this section, we present the calculated results for the two-terminal system, paying attention to the shape of a conductance peak as a function of energy level ε d in the QD. We find that parameters p L and p R are relevant in both the cases of U = 0 and U = 0.

A. Fano versus Breit-Wigner resonance
We begin with the case of no electron-electron interaction in the QD, U = 0. Figure   2 shows the conductance G at T = 0 as a function of energy level ε d in the QD for (a) and π (dotted line). G(φ) = G(−φ) holds by the Onsager's reciprocal theorem.
In panel (a) with p L = p R = 1, the conductance G shows an asymmetric resonant shape with dip and peak in the absence of magnetic field (φ = 0). This is known as the Fano resonance which is ascribable to the interference between the tunneling through a discrete level and that through continuous states [6,7]. A magnetic field changes the resonant shape to be symmetric at φ = ±π/2 and asymmetric with peak and dip at φ = π. This Fano resonance is characterized by a complex Fano factor [5]. Indeed the conductance can be analytically expressed [42] in the form of [Eqs. (25) and (24) for p L = p R = 1]. The complex Fano factor is given by With a decrease in p L and p R , the conductance peak becomes more symmetric and its φ-dependence is less prominent, as shown in panels (b) and (c). The shape of conductance peak is closer to that of the Lorentzian function of Breit-Wigner resonance as p L and p R go to zero.
Note that the conductance G can exceed unity in units of 2e 2 /h when p L , p R < 1,  The AB phase for the magnetic flux penetrating the ring is φ = 0 (solid line), φ = ±π/2 (broken line), and φ = π (dotted line).

B. Fano-Kondo resonance versus Kondo plateau
In the presence of U , the Kondo effect is exactly taken into account in the evaluation of the conductance at T = 0, as described in the previous section. In Fig. 3 For p L = p R = 1, G behaves as a "Fano-Kondo resonance" proposed by Hofstetter et al. [19], which stems from an interplay between the Kondo resonance (G ∼ 2e 2 /h at −U < ε d < 0) and the Fano resonance. When φ = 0 (π), G shows a dip and peak (peak and dip) with a gradual slope around the center of the Kondo valley, i.e., Coulomb blockade regime with a spin 1/2 in the QD. When φ = π/2, G is almost constant at 2e 2 /h in the Kondo valley and symmetric with respect to the valley center.
With decreasing p L and p R , the asymmetric shape of the Fano-Kondo resonance changes to a conductance plateau, the so-called Kondo plateau, in the Kondo valley:

IV. CALCULATED RESULTS IN THREE-TERMINAL GEOMETRY
In this section, we examine a three-terminal system to discuss the measurement of transmission phase shift through the QD by a "double-slit interference experiment." We assume two leads R(1) and R(2) on the right side and a single lead L on the left side in Fig. 1(b).
We evaluate the conductance from lead L to R(1) or to R(2), , as a function of AB phase φ. We define the measured phase shift by the AB phase φ max at which the conductance G (1) (φ) is maximal.
As an intrinsic transmission phase shift through the QD, we introduce θ (0) QD and θ QD by tan θ respectively, in the absence of U . θ QD is the phase shift through the QD without the upper arm of the ring, whereas θ QD satisfies the Friedel sum rule θ QD = π n σ for the QD embedded in the ring. The latter depends on the AB phase φ for the magnetic flux penetrating the ring. In the next subsection, we derive an analytical relation between the measured phase φ max and θ R , to investigate a crossover from two-to three-terminal phase measurement. In Fig. 4(b), we model the experimental situation by Takada et al., in which leads R(1) and R(2) are partly-coupled quantum wires [8,16,17]. For the three-terminal model in Fig. 1(b) with leads L, R(1), and R(2), we introduce the following dimensionless parameters: for j = 1 and 2. They are the ratios of contribution from lead R(j) to Γ R , x R , and √ Γ R x R p R , respectively, and satisfy the relations of γ In the absence of U , Eqs. (A29) and (A30) yield the conductance in the form of where If we neglect the φ-dependence inε d (φ) in the denominator in Eq. (35), the measured phase φ max is given by where θ QD is defined in Eq. (32). This is an approximate formula for the relation between the measured value and intrinsic value of the transmission phase shift through the QD.
In the two-terminal geometry, lead R(2) is absent and thus γ (1) R , respectively. Equation (37) yields an approximate relation of which indicates that the measured phase shift φ max approaches the intrinsic phase shift θ (0) QD with an increase in Γ R .  R /Γ R = 0.2, φ max changes almost abruptly from zero to π around ε d = E F = 0, which is close to the behavior in the two-terminal system. For larger Γ To illustrate the crossover from the two-to three-terminal phase measurement, we replot φ max for three values of Γ (2) R /Γ R in a graph in Fig. 6. Fig. 4

(b)
Now we study the model shown in Fig. 4(b) to examine the experimental situation using partly-coupled quantum wires to form a mesoscopic ring [8,16,17]. We assume that leads R(1) and R(2) consist of two equivalent wires a and b of single conduction channel. They are tunnel-coupled to each other in the vicinity of their edges, which mixes states |a, k in lead a and |b, k in lead b. As a result, the edge states in leads R(1) and R(2) are given by respectively, with real coefficients α R and β R (α 2 R + β 2 R = 1). Far from the edges, |ψ Rk → |a, k in lead R(1) and |ψ (2) Rk → |b, k in lead R(2) in an asymptotic way. As shown in Fig. 4(b), |ψ R = −1) as explained in Appendix C and in consequencẽ ε d (φ) = ε d in Eq. (25). For U = 0, Eq. (37) exactly holds, which yields   It should be mentioned that the sum of the currents to leads R(1) and R(2), I R , does not depend on the AB phase φ, reflecting p R = 0 in this model (see Appendix C).  Therefore, the AB oscillation of G (1) (φ) is out-of-phase to that of G (2) (φ), as indicated in the insets in Fig. 7. φ max evaluated from G (1) behaves similarly to θ (0) QD , while that from G (2) similarly to −θ experimental observation by Takada et al. [8,16].
Finally, the measured phase is discussed in the Kondo regime with U = 0. In Fig. 8, we plot φ max that is numerically evaluated from G (1) , as a function of energy level ε d in the QD.
(a) U/Γ = 8 and (b) 16 with Γ L = Γ R = Γ/2. In the Kondo valley (−U < ε d < 0), the phase locking at π/2 is observable by a "double-slit experiment" using the QD interferometer. We calculate the intrinsic phase shift θ QD using the Friedel sum rule θ QD = π n σ , where n σ is given by the Bethe ansatz exact solution (dotted line). φ max and θ QD are related to each other by Eq. (42). The phase locking seems smeared in the curve of the measured phase shift φ max , in comparison with the intrinsic phase shift θ QD .

V. DISCUSSION
In our models shown in Figs. 1(a) and (b), we assume a separable form for the tunnel coupling between the leads in Eq. (5). Here, we discuss the justification of this form using a tight-binding model. We also show that |p α | < 1 in the presence of multiple conduction channels in lead α.
As a simple example, let us consider the model depicted in Fig. 9(a). The leads consist of two sites in width and N sites in length (N 1). The eigenvalues of the Hamiltonian for leads L and R form two subbands ε ± (q), where q is the wavenumber in the x direction (0 < q < π/a) with a being the lattice constant [ Fig. 9(b)]. The corresponding states are where |j, is the Wannier function at site (j, ). The tunnel coupling between |L; q, γ and |R; q , γ (γ, γ = ±) is expressed as W q ,γ ;q,γ = ψ R;q ,γ (1, 2)W ψ L;q,γ (−1, 2) using the wavefunctions at the edge of the leads, ψ L;q,± (−1, 2) = −1, 2|L; q, ± and ψ R;q ,± (1, 2) = 1, 2|R; q , ± . In consequence W q ,γ ;q,γ has a separable form in Eq. (5) with √ w L;q,γ = √ W ψ L;q,γ (−1, 2), When the Fermi level intersects both the subbands, there are two conduction channels, labeled by k = (q, ±), as indicated in Fig. 9(b). Then where q ± are the intersections between the subband ± and Fermi level, as derived in Appendix D. Thus |p L,R | < 1. On the other hand, p L,R = ±1, in the case of single conduction channel when E F crosses one of the subbands.
Although we have considered a specific model in Fig. 9(a), the separable form of W k ,k in Eq. (5) should be justified when the system is described by a tight-binding model in general. Then √ w L,k ( √ w R,k ) is proportional to the wavefunction ψ L,k (ψ R,k ) at the edge of the lead, as in Eqs. (45) and (46). We could also claim that p L,R < 1 for the leads of multiple conduction channels and p L,R = 1 for the leads of single channel in usual cases.
Precisely speaking, the presence of multiple channels is a necessary condition for p L,R < 1: p α is determined by the detailed shape of the system around a junction between the ring and lead α through Eq. (10).
We comment on the generality of our models. In this section, we have examined a model in which the subbands (±) are well defined in the leads. Then the state in the leads is labeled by k = (q, ±) in the presence of two conduction channels. This is not the case in experimental systems of various shape. We believe that Γ α , x α , and p α can be defined in Eqs. (6)-(8) using state-dependent tunnel couplings without loss of generality. In our models in Figs. 1(a) and (b), we assume a single conduction channel in the upper arm of the ring.
The multiple channels in the arm should be beyond the scope of our study.

VI. CONCLUSIONS
We have theoretically examined the transport through an Aharonov-Bohm ring with an embedded quantum dot (QD), the so-called QD interferometer, to address two controversial issues, one concerns the shape of the conductance peak as a function of energy level ε d in the QD and the other is about the phase measurement in the multi-terminal geometry as a double-slit experiment. For the purpose, we have generalized a previous model in Refs. [18,19] to consider multiple conduction channels in leads L and R. In our model, the tunnel couplings between the QD and leads and that between the leads depend on the states in the leads, as shown in Figs. 1(a) and (b). This gives rise to a parameter p α (|p α | ≤ 1) to characterize a connection between the two arms of the ring through lead α (= L, R), which is equal to the overlap integral between the conduction modes coupled to the upper and lower arms of the ring.
First, we have examined the shape of the conductance peak in the two-terminal geometry, in the absence of electron-electron interaction U in the QD. We have shown an asymmetric subbands in the leads, ε ± (q) = ∓t 1 − 2t cos qa, as a function of wavenumber q in the x direction (0 < q < π/a). There are two conduction channels when the Fermi level E F intersects both the subbands at q = q ± .
Bethe ansatz exact solution, and precisely evaluated the conductance at temperature T = 0.
We have shown a crossover from an asymmetric Fano-Kondo resonance [19] to the Kondo plateau with changing p L,R .
Our model is also applicable to the multi-terminal geometry to address the second issue on the measurement of the transmission phase shift through the QD by a double-slit experiment.
We have studied the measured phase φ max , the AB phase at which the conductance G (1) (φ) to lead R(1) is maximal in Fig. 1(b). In the absence of U , Eq. (37) indicates the relation of φ max to an intrinsic phase shift θ QD that is the phase shift through the QD without the upper arm of the ring. We have examined two specific models in the three-terminal geometry, depicted in Fig. 4. We have discussed a crossover from two-to three-terminal phase measurement in the former and simulated the experimental system consisting of two quantum wires [8,16,17] in the latter. Using the latter model, we have shown how precisely the phase locking at π/2 is measured in the Kondo regime.

ACKNOWLEDGMENTS
We appreciate fruitful discussions with Dr. Akira Oguri. This work was partially sup- The current is formulated for the multi-terminal model depicted in Fig. 1(b), using the Keldysh Green's functions [43][44][45]. The chemical potential in lead L(j) [R(j)] is denoted by The spin index σ is omitted in this appendix.

Keldysh Green's functions
The retarded, advanced, and lesser Green's functions are defined by We also introduce the Green's functions in isolate leads L and R, in the absence of tunnel coupling, H T in Eq. (4). For example, where f (j) L )/(k B T ) + 1] −1 is the Fermi distribution function in lead L(j) that state k belongs to (j = 1 or 2). The Fourier transformation leads to In the following calculations, the real part (principal value) of g r αk (ω) and g a αk (ω) = [g r αk (ω)] * is disregarded in the summation over k, assuming a wide band limit.
In the next subsection, G < d,Lk is replaced by G r d,d and G < d,d . For this purpose, their relation is derived in the following. In the Baym-Kadanoff-Keldysh nonequilibrium techniques, a complex-time contour is considered from t = −∞ to t = t 0 just above the real axis and from t = t 0 to t = −∞ just below the real axis. For the contour-ordered Green's function, the equation-of-motion method yields [44,45] According to the Langreth's theorem [45,46], this results in and We have added a factor of two by the summation over spin index σ. This equation is rewritten as Hence we need to calculate two terms in the integral, Let us consider X 0 . Using the Fourier transformation of Eq. (A11), we obtain Then we need For X 1 , we use an equation for G r d,Rk corresponding to Eq. (A10) for G r d,Lk , which leads to Using the Fourier transformation of Eq. (A10), we obtain From Eqs. (A19) and (A21), we express X 1 in terms of G r d,d (ω). In the same way, X 2 can be written using G r d,d (ω) and G < d,d (ω). A similar procedure is adopted for Y 0 . The final result is so lengthy that we show the current expression in the case of Eq. (17), i.e., µ After the variable conversion ofhω → ε, where The current I   For the two-terminal model in Fig. 1(a), the current from lead L is I L = I As a three-terminal model, we examine the model in Fig. 1(b) where with Regarding the φ-dependence of the conductance at T = 0, Eqs. (A29) and (A30) yield Eqs. (35) and (36) in the absence of U . In the presence of U , however, we cannot obtain such a simple form in general.
Appendix B: Green's function in the presence of U For our models shown in Figs. 1(a) and (b), the Green's function of the QD is solvable in the case of U = 0. As discussed in section II.D, the retarded Green's function is given by with the effective energy levelε d (φ) in Eq. (25) and effective linewidthΓ in Eq. (24). The renormalization due to the direct tunneling between the leads and the Aharonov-Bohm effect by the magnetic flux is included in these effective parameters.
In the presence of U , we formulate the perturbation with respect to the electron-electron interaction in the QD, H U = U n ↑ n ↓ . The Hamiltonian in Eq. (1) is divided into the noninteracting part H 0 and H U ; H = H 0 + H U . The contour-ordered Green's function of the , is written as where ρ 0 is the density matrix for U = 0 and index I indicates the operator in the interaction picture, O I (τ ) = e iH 0 τ /h Oe −iH 0 τ /h . In the perturbative expansion, the unperturbed Green's function is given by Eq. (B1). This problem is equivalent to that of the conventional Anderson impurity model, in which an impurity with energy levelε d (φ) and Coulomb interaction U is connected to an energy-band of conduction electrons via the effective hybridizationΓ: whereΓ = πρ|v| 2 , with the density of states ρ for the conduction electrons.
In the equilibrium with eV = 0, the physical quantities of electrons in our model can be evaluated by exploiting the established methods for the Anderson impurity model [19]. The retarded Green's function is given by with use of the self-energy Σ U (ε) due to the electron-electron interaction in the QD. Note (26). G r d,d (0) is expressed in Eq. (27) using the phase shift θ QD . The Friedel sum rule connects the phase shift to the electron occupation per spin in the QD, θ QD = π n σ , where We use the Bethe ansatz exact solution to evaluate n σ [39,40].
As mentioned in section IV.C, V R,k = V R α R and √ w R,k = √ w R β R when state k belongs to lead R(1) while V R,k = V R β R and √ w R,k = − √ w R α R when state k belongs to lead R (2) in the tunnel Hamiltonian H T . This results in Γ (1) We also find that p (1) R = −1, and hence p R = p (1) R = 0. From p R = 0,ε d (φ) = ε d in Eq. (25), which is independent of the AB phase φ for the magnetic flux. The Green's function in the absence of U becomes The substitution of γ (1) Sinceε d (φ) = ε d in this model, Eq. (37) exactly holds in the absence of U , which leads to Eq. (41). Besides, even in the presence of U , a relation between φ max and θ QD is derived in the following. The substitution of Eq. (26) into Eq. (C3) yields For φ = φ max at which F 1 (φ) is maximal, where tan θ QD =Γ * /ε * d . θ QD satisfies the Friedel sum rule in the presence of U .
The current to lead R(2), −I (2) R , is given by replacing (1) → (2) in Eq. (A29). T (2) R is obtained from T (1) R in Eq. (C3), replacing α R → β R and β R → −α R . In T (1) R and T (2) R , coefficients of cos φ and sin φ are the same in magnitude and opposite in sign. As a result, the total current to leads R(1) and R(2) does not depend on the AB phase φ for the magnetic flux: where T (ε) = T (1) with This coincides with Eq. (20) for the current in the two-terminal system with p R = 0.
Appendix D: Tight-binding model in Fig. 9 In the tight-binding model in Fig. 9(a), leads L and R consist of two sites in width and N sites in length (N 1). There are two subbands in the leads, as depicted in Fig. 9(b), ε ± (q) = ∓t 1 − 2t cos qa, where t (t 1 ) is the transfer integral in x (y) direction and a is the lattice constant. q is the wavenumber in the x direction, q = πn/[(N + 1)a] with n = 1, 2, · · · , N . The corresponding states are given by Eqs. (43) and (44).
We calculate Γ α , x α , and p α in Eqs. (6)-(8) at ε = E F . We focus on lead L because lead R is identical to lead L. The density of states for subband ± is given by where q ± is defined by ε ± (q ± ) = E F , as depicted in Fig. 9(b), x L = W 2t (sin q + a + sin q − a), and in consequence we obtain p L in Eq. (47).