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We propose a numerical method to simulate a transport experiment using a quantum dot interferometer made of two quantum wires in parallel [S. Takada et al., Phys. Rev. Lett. 113, 126601 (2014)]. The wires are partly tunnelcoupled to each other to form a mesoscopic ring with an embedded quantum dot. Our method consists of two stages. In the first stage, we represent the experimental system with a tightbinding model by discretizing the space. The conductance around a Coulomb peak is evaluated as a function of magnetic field in fourterminal geometry, where the Coulomb interaction is irrelevant. We show clear Aharonov–Bohm (AB) oscillations despite the multiple conduction channels and magnetic field inside the wires. In the second stage, we adopt a model of double quantum dot (DQD) in parallel. The model parameters are chosen to reproduce the Coulomb peak and AB oscillations obtained in the first stage in the absence of the Coulomb interaction U. Finally, we calculate the conductance in the Kondo valley using the DQD model in the presence of U. We observe phase locking at π/2, which is consistent with experimental results.
Transport through an Aharonov–Bohm (AB) ring with an embedded quantum dot (QD), the socalled QD interferometer, has been studied extensively since the pioneering works by Yacoby et al.^{1}^{)} and Schuster et al.^{2}^{)} Despite its similarity to a doubleslit experiment, the transmission phase shift through the QD cannot be measured using the interferometer in the twoterminal geometry.^{1}^{)} This is due to the restriction by Onsager's reciprocity theorem: The conductance G satisfies
There is another issue regarding the QD interferometer, the shape of the Coulomb peak. Kobayashi et al. observed an asymmetric shape with a peak and a dip, i.e., the Fano resonance.^{16}^{)} It stems from the interference between a discrete energy level in the QD and continuum energy states in the ring.^{17}^{,}^{18}^{)} The peak shape changes with magnetic flux, which can be expressed using a complex Fano factor.^{16}^{,}^{19}^{)} However, other research groups observed symmetric Coulomb peaks, which can be fitted to the Lorentzian function of the Breit–Wigner resonance.^{6}^{,}^{7}^{)} Regarding the resonant shape in the Kondo regime, an asymmetric Fano–Kondo resonance was theoretically proposed by Bułka and Stefański^{20}^{)} and Hofstetter et al.^{21}^{)} Although their works were followed by many theoretical studies,^{22}^{–}^{31}^{)} the Fano–Kondo effect has not been observed experimentally.
To examine the resonant shape in the QD interferometer, two of the present authors extended the model proposed by Hofstetter et al. to include the multiple conduction channels in the leads.^{32}^{)} Their model involves the parameters
In this paper, we propose a numerical method to simulate the experimental systems of a QD interferometer in the multiterminal geometry and in the presence of the Kondo effect. Although our method is generally applicable to realistic systems, we focus on the interferometer made of two quantum wires in parallel used by Takada and coworkers.^{6}^{,}^{7}^{)} In their system, two wires are partly tunnelcoupled to each other to form a mesoscopic ring with an embedded QD. Each quantum wire includes several conduction channels.
The purposes of our present work are as follows. (i) We demonstrate a method to derive the transport properties from a given device structure. This would be useful for designing a suitable device to examine the fundamental physics of the QD interferometer. (ii) We take into account multiple conduction channels in the arms of the ring and magnetic field inside the arms and leads, which were disregarded in previous works.^{32}^{,}^{33}^{)} We elucidate the reasons why clear AB oscillations are observed experimentally despite these effects.^{6}^{,}^{7}^{)} (iii) The electron–electron interaction U in the QD is considered in the device structure. The conductance and measured phase shift are evaluated at temperature
Our method consists of two stages. In the first stage, we represent the experimental system with a tightbinding model by discretizing the space. We consider two conduction channels in each wire for simplicity. Although the number of channels is less than that in experimental systems, the effect of multiple channels should be clarified by this calculation. Using Green's function recursion formula,^{34}^{)} we calculate the conductance around a Coulomb peak in a magnetic field, where the Coulomb interaction is irrelevant. Our simulation shows clear AB oscillations despite the multiple conduction channels and magnetic field inside the quantum wires, indicating a dominant contribution from one of the conduction channels and a small orbital magnetization.
In the second stage, we adopt the DQD model for the QD interferometer. We extend the previous work^{33}^{)} to include two energy levels in QD2 to consider two conduction channels in the reference arm. The model parameters are chosen to reproduce the Coulomb peaks and AB oscillations obtained in the first stage. Finally, we calculate the conductance in the Kondo valley of the DQD model in the presence of the Coulomb interaction U in QD1. We observe phase locking at
This paper is organized as follows. We explain our two models and calculation methods in Sect. 2. Section 3 is devoted to the calculated results in the first stage. We begin with the interferometer without an embedded QD to elucidate the AB effect on the conductance. Then, we examine the transport properties of the interferometer with an embedded QD. In Sect. 4, we calculate the conductance of the DQD model in the absence and presence of U. We discuss the phase measurement around the Coulomb peak and in the Kondo valley. We present the conclusions and discussion in Sect. 5.
Our models are depicted in Fig. 1 for the QD interferometer in the fourterminal geometry. In the first stage, we examine the twodimensional tightbinding model to simulate the experimental systems consisting of two quantum wires in parallel (a) without and (b) with an embedded QD.^{6}^{,}^{7}^{)}
Figure 1. (Color online) Models for the QD interferometer in the fourterminal geometry. (a) Tightbinding model for the interferometer made of two quantum wires without an embedded QD. The hopping between the sites is given by −t and the onsite energy is
The tightbinding model is made by discretizing the twodimensional space into a square lattice with the lattice constant a. Let us consider the original Hamiltonian
Figure 1(a) shows two quantum wires of length
To form a ring, the wires are partly connected to each other at “tunnel areas” with
To embed an QD in the interferometer, a region of the model is replaced by a region shown in Fig. 1(b). The QD is formed by the tunnel barriers of height
The conductance is calculated by the Green's function recursion method.^{34}^{)} We consider the injection of electrons from lead
In the second stage, we examine a DQD model in Fig. 1(c) with four external leads. One of the DQDs (QD1) represents an embedded QD with single energy level
The Hamiltonian for the DQD model is given in Appendix Sect. A.1. The strength of the tunnel coupling between the DQD and lead α (
For the offdiagonal elements of
The current is formulated using Keldysh Green's function method, as explained in Appendix Sects. A.2 and A.3. In the absence of U, the current is given by Eq. (A·25), where the retarded and advanced Green's functions are in the form of
In the presence of U, we evaluate the conductance at
We present the calculated results in the first stage using the tightbinding models in Figs. 1(a) and 1(b). We begin with the interferometer without embedding a QD. The amplitude and period are analyzed for the AB oscillations of the conductance. Then, we study the interferometer with an embedded QD. We calculate the conductance around a Coulomb peak, AB oscillations, and transmission phase shifts through the QD, in the absence of the electron–electron interaction U in the QD.
First, we examine the interferometer depicted in Fig. 1(a) without an embedded QD. Figure 2 shows the conductance from lead
Figure 2. Calculated results of AB oscillations in the interferometer without an embedded QD using the tightbinding model depicted in Fig. 1(a). The conductance from lead
The inset in Fig. 2(a) presents the
We comment on the shape and period of AB oscillations. The AB oscillations in Fig. 2(b) look similar to a sinusoidal function of
Note that (i)
A small deviation from the sinusoidal function is seen in Fig. 2(c), which is attributable to a contribution from the second conduction channel. In the figure, the outofphase relationship is broken.
Now we examine the interferometer with an embedded QD shown in Figs. 1(a) and 1(b). The potential is
In Fig. 3, we plot the first peak of the Coulomb oscillation, i.e., the conductance through the lowest bound state in the QD for the current from lead
Figure 3. Calculated results of a Coulomb peak in the QD interferometer using the tightbinding model depicted in Figs. 1(a) and 1(b). The conductance from lead
We plot the AB oscillations in Fig. 4 for the same
Figure 4. Calculated results of the AB oscillations in the QD interferometer using the tightbinding model depicted in Figs. 1(a) and 1(b). The conductance from lead
Figure 5 shows the magnetic field B at the maximum of the AB oscillation as a function of
Figure 5. Phase shift of the AB oscillations in the QD interferometer obtained using the tightbinding model depicted in Figs. 1(a) and 1(b). The magnetic field B at the maximum of the AB oscillation is plotted as a function of the potential
In the second stage, we examine the DQD model depicted in Fig. 1(c). First, we calculate the conductance peak and AB oscillations in the absence of the electron–electron interaction U in QD1. We determine the parameters that reproduce the calculated results in the first stage. Second, in the presence of U, we calculate the conductance and transmission phase shift in the Kondo valley.
In the absence of U, we calculate the conductance peak as a function of the energy level
For the following calculations, we choose the values of
Figure 6. Calculated results of the Coulomb peak in the DQD model depicted in Fig. 1(c) in the absence of U. The conductance from lead

Figure 7 depicts the AB oscillations of the conductance for the current from lead
Figure 7. Calculated results of the AB oscillations in the DQD model depicted in Fig. 1(c) in the absence of U. The conductance from lead
In Fig. 8, we plot the AB phase ϕ at which the conductance is maximal as a function of the energy level
Figure 8. Phase shift of the AB oscillations in the DQD model depicted in Fig. 1(c) in the absence of U. The AB phase ϕ at the maximum of the AB oscillation is plotted as a function of the energy level
Now we introduce the Coulomb interaction U in QD1. We set
In Fig. 9, we plot the conductance at
Figure 9. Calculated results of the Kondo valley in the DQD model depicted in Fig. 1(c) in the presence of U in QD1.
Figure 10 shows the AB phase ϕ at which the conductance is maximal. We clearly observe phase locking at
Figure 10. Phase shift of the AB oscillations in the DQD model depicted in Fig. 1(c) in the presence of U in QD1.
Takada et al. observed phase locking at
We have examined the transport through a QD interferometer made of two quantum wires in parallel, an experimental system used by Takada and coworkers.^{6}^{,}^{7}^{)} Our method consists of two stages. In the first stage, the experimental system is represented by a tightbinding model. Using the recursion method of Green's function, we have calculated the conductance around a Coulomb peak as a function of magnetic field and found clear AB oscillations despite the multiple conduction channels and magnetic field within the wires, indicating a dominant contribution from one of the conduction channels and a small orbital magnetization. The phase shift of the AB oscillations changes gradually with the QD potential in accordance with the Breit–Wigner resonance. In the second stage, we have adopted the model of DQD in parallel, one QD (QD1) for the embedded QD and the other (QD2) for the reference arm. QD2 includes two energy levels to represent two conduction channels. The parameters of the tunnel couplings are chosen to reproduce the Coulomb peaks and AB oscillations obtained in the first stage. Finally, we have calculated the conductance in the Kondo valley in the presence of the Coulomb interaction U in QD1 and shown phase locking at
We comment on the comparison between the experimental system^{6}^{,}^{7}^{)} and our models. Although the tightbinding model in the first stage seems similar to the experimental system, the number of conduction channels is different. Our model includes two channels in each wire, which is less than those in the experimental system, to simply elucidate the effect of multiple channels. As a result, a small Fano resonance (Fano–Kondo effect) appears in our calculations for the Coulomb peak (Kondo valley), whereas a symmetric shape of the Lorentzian function (symmetric Kondo plateau) is found experimentally.^{6}^{,}^{7}^{)} We speculate that our models with more conduction channels yield a more symmetric shape of the Coulomb peak and Kondo plateau, whereas the visibility of the AB oscillations (the ratio of the amplitude of the oscillations to the total current) is reduced. Regarding the strength of the Coulomb interaction, we assume that U is much larger than the level broadening
In general, the DQD model can represent N conduction channels in the reference arm using N levels in QD2. In this study, we have tried several values of tunnel couplings
Acknowledgments
This work was partially supported by JSPS KAKENHI Grant Numbers JP20K03807 and JP21K03415, and JSTCREST Grant Number JPMJCR1876.
Three of the present authors previously examined the DQD model for the QD interferometer with a single energy level in each QD.^{33}^{)} We extend the model to include two energy levels
The Hamiltonian for the DQD model is given by
The tunnel couplings between the DQD and lead α give rise to the linewidth function in the form of
Keldysh Green's functions are used to formulate the electric current. The retarded, advanced, lesser, and greater Green's functions of the DQD are denoted by
In the absence of the electron–electron interaction U in QD1, the equationofmotion method yields the retarded Green's function
In the presence of U in QD1,
Equations (A·14)–(A·18) can be used even in the presence of U in QD1. As a result, we obtain all the elements of
The current from lead α to the DQD is given by
In the presence of U, the current is expressed as
Although we speculate the irrelevance of
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