For the last several years, systems of coupled multi-quantum dots have gained significant interests both experimentally^[1–6] and theoretically, due to rapid progress in spintronics^[7–9] and quantum information.^[10,11] An important example of a multi-dot system is the triple quantum dots (TQD). Such structure exhibits many interesting quantum phenomena in the strong correlation limit. For instance, the Aharanov–Bohm (AB) effect and magnetic frustration are studied in systems with triangular geometry,^[12–15] and the multi-channel Kondo effect related to a number of Fermi-liquid and non-Fermi-liquid behaviors are investigated in various TQD models.^[16–20] Furthermore, TQD systems are also considered as ideal models to show the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction,^[21,22] the quantum interference,^[23–26] and various kinds of quantum phase transition (QPT).^{[13,15,27–32]} Basically, these behaviors depend closely on the organizations of the quantum dots and the conduction leads, as well as the coupling/interaction elements which have been taken into account.

Within the above phenomena, the level difference plays an important role as suggested in some double quantum dot (DQD) structures. For instance, a local spin triplet–singlet transition of Kosterlitz–Thouless (KT) type is found as the difference increases.^[33] For a parallel DQD system, it is revealed that the triplet–doublet transition could be KT type or first order depends on the breaking of the spin-rotation SU(2) symmetry,^[34] and the linear conductance is shown to have an asymmetric line shape of the Fano resonance when the interdot tunneling coupling is taken into account.^[35] In a spinless two-level dot system, charge oscillation occurs in the presence of level spacing as the gate voltage sweeps.^[36] These works are limited in the double dot structures, but less is studied in the TQD system. Compared to the double dot system, TQD structure exhibits more attractive behaviors, owing to its complex geometry and much more Feynman paths for the electron transmission.^[25] Furthermore, experimentally speaking, scaling up the number of spin-1/2 qubits is an inevitable step towards the realization of quantum computation, not only for extending the qubit system, but also for performing practical quantum algorithms.^[37] Very recently, plenty of experimental works have been carried out on this topic in TQD systems,^{[5,6,38–41]} therefore, to illustrate the electronic transport, phase transition and relevant quantum phenomena in such systems also becomes an emergent task.

In this paper, we consider a parallel TQD system with level spacings between neighboring dot sites. With the help of the numerical renormalization group (NRG) method, and fixing dot 2 at the half-filled level, we demonstrated that the level spacings play important roles in the linear conductance and the QPT. Our main findings include the following: by tuning the level differences, the ground state of the system transits from local spin quadruplet to triplet, then doublet sequentially, and three kinds of Kondo peaks at the Fermi surface could be found, corresponding to spin-3/2, spin-1, and spin-1/2 Kondo effect, respectively. Two Kosterlitz–Thouless (KT) type transitions are clarified, resulting from asymmetric Kondo couplings between conduction leads and different dots, as well as the unequal charge occupation of three dots. To handle these problems, some important physical quantities are shown, and necessary physical arguments are given.

The outline of the paper is as follows. In Section 2, we define the model Hamiltonian of the TQD system, and present the calculation algorithms and formulations. In Section 3, we show the quantum phase transitions and transport properties with respect to the increasing level spacings. Finally, a summary is given in Section 4.

We illustrate the second-quantized form of the model Hamiltonian for the triple dot device in Fig. 1,

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. Schematic view of the parallel TQD device connected to the L and R leads.

The second part is for the non-interacting electrons in the conduction lead, , where create a spin-σ electron of wave vector k and energy ε_k in lead ν (ν = R or L). By contrast, the last term is for the coupling interaction energy between the conduction lead and the dots, , where V_k is the tunnel matrix element, and is assumed to be σ-independent, identical for three dots, and also symmetric to the right and left leads.

In the following discussion, we concentrate on the quantum behaviors in the strong correlation limit, hence a sophisticated theoretical technique must be adopted. Therefore, we use the celebrated NRG method^[42–44] to solve Eq. (1). To simplify the problem into a more convenient form, we first assume the density of state for the conduction lead ρ₀ = 1/(2D) and the hybridization function (dot–lead coupling) Γ = 2πρ₀|V_k|² are constant. Here, D is the half width of the conduction band, and is chosen for the energy unit throughout this paper. The strong correlation limit is defined as U ≫ Γ, and the number of low-lying states kept at each iteration is about 3000. Furthermore, the discretization parameter Λ, which characterizes the logarithmic discretization of the conduction band, is set to be 1.8–2.2.

The total charge number N_tot and the local spin S_dot² are defined as

The linear conductance through the device G is determined by the Landauer formula^[45]

Finally, we define the temperature-dependent magnetic moment of the dots μ²(T) at temperature T as

In this section, we study the phase transition and the electronic transport of the TQD device in the strongly correlated regime, focusing on the phenomena when the level differences sweep upwards. The on-site Coulomb repulsion, the dot-lead coupling, and the charge energy of dot 2 are fixed at U = 0.1, Γ = 0.01, and ε₂ = −U/2, respectively, throughout this paper. For convenience, we choose Δ₂ = Δ₁/2. However, it is worth noting that our conclusions are robust for general cases, and do not really require Δ₁ and Δ₂ to satisfy the above relation.

Let us start our discussion from the quantum phase transition for the TQD system. In Figs. 2(a)–2(c), we depict the charge occupation 〈n_i〉 of each dot, the spin correlation 〈S_iS_j〉 between dots i and j, and the local spin on the dots as functions of Δ₁. We can see when the level differences are absent, three dots are singly occupied with 〈n_i〉 = 1.0, since the intra-dot repulsion U is strong enough. In this case, 〈S_iS_j〉 ≈ 0.20 and , implying the local spins on three dots are arranged parallelly and a local spin quadruplet is generated, owing to the ferromagnetic RKKY interaction mediated by the Kondo exchange coupling between electrons on the leads and those on the dots. It is noted that 〈S_iS_j〉 and could not reach the ideal values of 1/4 and 15/4 for spin quadruplet, as a finite U/Γ is adopted. With increasing Δ₁, the level ε₁ decreases, while ε₃ increases but with a lower speed. Therefore, 〈 n₁ 〉 grows to 2.0 gradually, while 〈n₂〉 and 〈n₃〉 retain nearly at 1.0. In this process, changes continuously to about 1.75, describing that the ground state is dominated by the spin triplet. In this triplet, dot 1 is almost doubly occupied, hence 〈S₁S₂〉 and 〈S₁S₃〉 decrease to about 0, while 〈S₂S₃〉 has nearly no change. As Δ₁ increases continuously, 〈n₃〉 decreases to the regime n₃ ≈ 0. In this case, the system maps to the single impurity case and the ground state is a local spin doublet with 〈S_iS_j〉 ≈ 0 and .

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) (a) Charge occupation 〈ni〉 on each dot, (b) spin correlation 〈SiSj〉 between dots i and j, and (c) local spin of the dots at zero temperature as functions of Δ1. Here, Γ = 0.01, U = 0.1, ε2 = −U/2 and Δ2 = Δ1/2.

Figures 3(a)–3(c) show the transmission coefficient T(ω) at zero temperature for various Δ₁. It is seen that the spectral weight is symmetric to the Fermi level ω = 0 when Δ₁ = 0, since Eq. (2) satisfies particle–hole (p–h) symmetry. Two Coulomb peaks located at ω = ± U/2 are found, referring to the process of annihilating (creating) an additional electron (hole) on the bonding orbital . Besides, a Kondo peak could also be seen at the Fermi level, which reaches the unitary limit of 2e²/h corresponding to full conductance. It could be attributed to the process where the itinerant electron screens a local spin-1/2 degree of freedom and results in a partially screened spin-3/2 Kondo effect.^[15,21] As Δ₁ increases, the spectral weight moves to the left and is not symmetric to ω = 0, for the p–h symmetry is broken and N_tot is away from a triply occupied state. In this process, the amplitude of the Kondo resonance peak decreases, implying that the linear conductance is reduced slightly (see Δ₁ = 0.043 in panel (a)). As Δ₁ exceeds the first critical point , a sharp Kondo peak appears at ω = 0 (see Δ₁ = 0.05 in Fig. 3(b)), which is related to the partially screened spin-1 Kondo effect, and originates from the screening of the local spin triplet between dot 2 and 3 by the conduction electrons. With increasing Δ₁, the spectral weight moves away from ω = 0 and the Kondo peak is broadened, indicating a decrease of the conductance (e.g., Δ₁ = 0.07). When Δ₁ grows larger than the second critical point , another sharp Kondo peak is observed at the Fermi surface (see Δ₁ = 0.10 in panel (c)), which results from the antiferromagnetic Kondo coupling between the conduction leads and dot 2, since dot 1 is almost doubly occupied and dot 3 is empty. As Δ₁ grows continuously, the Kondo peak keeps at ω = 0, for the TQD is stable in the spin doublet (see Δ₁ = 0.16).

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) (a)–(c) Transmission coefficient T(ω) at zero temperature for various Δ1. The remaining parameters are the same as in Fig. 2.

To get more information about the phase transition, we show T(ω) on the triplet side near in Fig. 4(a). One finds that the width of the spin-1 Kondo peak W₁ enlarges with increasing Δ₁. Here, the width of the spin-1 Kondo peak is defined as the half width at half maximum of the Kondo peak. Figure 4(b) suggests that W₁ depends exponentially on the distance to the critical point (Δ₁ − Δ₁^c1) and can be adequately described using an exponential function , where the fitting parameters are given by P₁ = −0.0192, P₂ = 3.9409, and P₃ = 0.0552. It is seen that the fitting function (solid line) agrees very well with our NRG results (scatter dots). This behavior describes that the quadruplet–triplet transition at is a KT transition. Figure 4(c) gives T(ω) for various Δ₁ on the doublet side near . One may also find that the width of the spin-1/2 Kondo peak W₂ approaches zero exponentially as Δ₁ is close to , and can be described by an exponential function . Here, the fitting parameters are P₄ = −0.0019, P₅ = 6.4015, P₆ = 0.0431. Figure 4(d) indicates that the fitting function (solid line) agrees very well with the NRG results (scatter dots), illustrating the triplet–doublet transition is also of the KT type.

Fig. 4.

Figure Option ViewDownloadNew Window

Fig. 4. (color online) (a) T(ω) on the triplet side near the first critical points . The curves from top to bottom on the left side are for Δ1 = 0.051 to 0.060 in steps of 0.001. (b) Width of the spin-1 Kondo peak W1 (scatter dots) and its fitting exponential functions (solid lines). (c) T(ω) on the doublet side near the second critical point . The curves from top to bottom on the right side are for Δ1 = 0.091 to 0.099 in steps of 0.001. (d) Width of the spin-1/2 Kondo peak W2 (scatter dots) and its stimulant functions (solid lines). The remaining parameters are the same as in Fig. 2.

To exhibit more information about the low-temperature scenario, we present μ²(T) for different Δ₁ in Fig. 5. The upmost curve is for the case of Δ₁ = 0. It is seen that TQD goes through four different regimes as T decreases. For example, when T is high enough (e.g., T > U), the electrons on three dots are independent, then each dot contributes 1/8 to μ²(T). As T decreases to T < U (e.g., T ~ 0.01), the electrons are then in the local moment regime, thus the full- and zero-occupied states are strongly suppressed. Therefore, each of them contributes 1/4 to μ²(T). When T decreases to the order of the RKKY interaction (e.g., T ~ 0.001), the electrons on three dots form a spin quadruplet and contribute S_dot(S_dot + 1)/3 = 5/4 to μ²(T). As a result, μ²(T) grows to a higher spin stage. Here, μ²(T) in the local moment regime (μ²(T) ~ 0.62) and the RKKY regime (μ²(T) ~ 0.80) are smaller than the ideal values, for the TQD is not in a pure ground state in these temperature regimes. When T reduces to the Kondo temperature scale (e.g., T ~ 10⁻⁷), the itinerant electron screens a local spin-1/2 degree of freedom, thus one finds an obvious drop in μ²(T) and μ²(T) ~ 0.67 at low temperature. As Δ₁ increases, the height of μ²(T) in the RKKY regime is reduced, since the possibilities of generating the RKKY interaction decreases due to increasing double occupancy on dot 1. On the other hand, μ²(T) at low temperature μ²(T = 0) decreases exponentially. When Δ₁ is large enough (e.g., Δ₁ = 0.052), μ²(T = 0) ≈ 0.25, since in this case the spin triplet is partially screened by the conduction leads, and the remaining spin-1/2 degree of freedom contribute to μ²(T). As Δ₁ grows continuously, μ²(T = 0) reduces exponentially again. When , μ²(T) → 0, μ²(T) → 0 at low temperature, for in this regime the local spin singlet is totally screened by the conduction electrons.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. Total magnetic moment of the TQD μ2(T) as a function of temperature T according to different Δ1, the curves from top to bottom are for Δ1 = 0, 0.04, 0.046 to 0.050 in steps of 0.001, 0.052 (dash line), and Δ1 = 0.08, 0.087 to 0.09 in steps of 0.001, 0.094, 0.1, respectively. The remaining parameters are the same as in Fig. 2.

To explore the physical origination of the KT transitions, we change Eq. (1) to a three impurities spin-1/2 Kondo model by adopting the Schrieffer–Wolff transformation.^{[?, 21]} The effective Hamiltonian then could be written as

With increasing Δ₁, J_ki are asymmetric. On the other hand, the charge occupation in three dots also become different. The symmetric broken by these features are the origination of the KT transitions, similar to a two-level dot system with level difference^[33] and inter-dot Coulomb repulsion.^[34] Furthermore, it is worth noting that one can estimate the critical points by considering the energy levels of an isolated TQD model. For instance, the energy level for the spin quadruplet can be written as E_Q = ε₁ + ε₂ + ε₃= −3U/2. While that for the triplet is E_T = 2ε₁ + ε₂ + ε₃ + U = −U − 3Δ₁/2, and the level for the doublet is given by E_D = 2ε₁ + ε₂ + U = −U/2 − 2Δ₁. As Δ₁ increases, the energy levels for the triplet and the singlet are pulled down, and eventually degenerate with the quadruplet at Δ₁ = U/3 and the triplet at U respectively. As a result, two QPTs of the KT type can be found at and .

In conclusion, we have studied the phase transition and Kondo behavior in a triple quantum dot device with parallel organization in the strongly correlated regime. We concentrate on the effect of the level differences Δ₁ = ε₂ − ε₁ and Δ₂ = ε₃ − ε₂. By keeping dot 2 at the half-filled level and tuning the level differences, it is shown that the ground state of the system transits from local spin quadruplet to triplet, then doublet sequentially, three kinds of Kondo peaks at the Fermi surface could be found in the transmission coefficient, which correspond to spin-3/2, spin-1, and spin-1/2 Kondo effect, respectively. Two KT type phase transitions are clarified, resulting from asymmetric Kondo couplings between the conduction leads and different dots, as well as unequal electron occupation of three dots. We believe our work not only clarifies the effect of the level difference on the phase transition and the electronic transport in a triple dot structure, but it may also afford useful guidance for spintronics and molecular electronics devices.