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Quantum phase transition in trigonal triple quantum dots: The case of quantum dots deviated from particle–hole symmetric point
Physica B 465 (2015) 55–59
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Physica B journal homepage: www.elsevier.com/locate/physb
Quantum phase transition in trigonal triple quantum dots: The case of quantum dots deviated from particle–hole symmetric point Song-Hyok Kim a, Chol-Jin Kang a, Yon-Il Kim a, Kwang-Hyon Kim b,n a b
Department of Theoretical Physics, Institute of Physics, Unjong district, Pyongyang, DPR Korea Institute of Laser, Unjong district, Pyongyang, DPR Korea
art ic l e i nf o
a b s t r a c t
Article history: Received 2 November 2014 Received in revised form 25 January 2015 Accepted 5 February 2015 Available online 7 February 2015
We consider a triple quantum dot system in a triangular geometry with one of the dots connected to metallic leads. We investigate quantum phase transition between local moment phase and Kondo screened strong coupling phase in triple quantum dots where energy levels of dots are deviated from the particle–hole symmetric point. The effect of on-site energy of dots on quantum phase transition between local moment phase and Kondo screened strong coupling phase in triple quantum dots is studied based on the analytical arguments and the numerical renormalization group method. The results show that the critical value of tunnel coupling between side dots decreases when the energy level of embedded dot rises up from the symmetric point to the Fermi level and the critical value increases when the energy levels of two side dots rise up. The study of the influence of on-site-energy changes on the quantum phase transitions in triple quantum dots has the importance for clarifying the mechanism of Kondo screening in triple quantum dots where energy levels of dots are deviated from the particle–hole symmetric point. & 2015 Elsevier B.V. All rights reserved.
Keywords: Triple quantum dots Quantum phase transition Numerical renormalization group method RKKY interaction
1. Introduction The triangle is the simplest polygon that has a closed loop which plays an important role on various fascinating phenomena in the condensed matter physics. The closed path in metal and semiconductor allows the electron to move around the loop, and causes quantum interference effects, such as an Aharonov–Bohm effect [1,2]. The closed path consisting of the odd-number of links also causes frustration, leading to resonating valence bonds for some anti-ferromagnetic systems [3]. Single triangle is also a fundamental unite of the triangular and Kagome lattices. In these systems, the geometrical frustration affects significantly the magnetic properties and the behavior at the Mott–Hubbard metal–insulator transition [4,5]. Another interesting example is the triangular trimer of Cr atoms placed upon an Au surface [6–8], expected to show a non-Fermi-liquid behavior due to the multi-channel Kondo effect [9,10]. The quantum phase transition between Kondo screened strong coupling (SC) phase and local moment (LM) phase in triple quantum dot system has attracted much interest and has been studied theoretically [11,12]. In triple quantum dots with week inter-dot coupling, the quantum phase transition between those n
Corresponding author. E-mail address: [email protected] (K.-H. Kim).
http://dx.doi.org/10.1016/j.physb.2015.02.005 0921-4526/& 2015 Elsevier B.V. All rights reserved.
phases is the consequence of competition between direct exchange interaction and the indirect interaction between two side dots. There are two types of interactions between two side dots coupled to the embedded dot: the direct exchange interaction between two side dots and the indirect RKKY-type interaction mediated by the Kondo singlet formed between the embedded dot and the leads. The signs of those interactions are different from each other. The sign of the actual effective exchange interaction energy between two side dots depends therefore on which of the two interactions prevails. The quantum phase of the system depends on the sign of the effective exchange interaction energy between two side dots. In the preceding report [11], quantum phase transitions between the Kondo screened SC phase and the LM phase have been investigated only for the systems consisting quantum dots whose energy levels are fixed at the particle–hole symmetric point. In real triple quantum dot systems, it is, however, difficult to produce dots with identical properties. In particular, it is not easy to achieve equal on-site energies of three dots at the particle–hole symmetric point and the energy levels of dots are generally deviated from the particle–hole symmetric point. In this work, we study how the phase diagram, representing the quantum phase transition between the Kondo screened SC phase and the LM phase in triple quantum dots with week interdot coupling, is modified as the levels of dots are deviated from the particle–hole symmetric point. In the analytical argument we
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assume that RKKY interaction between side dots can also be mediated by virtual ‘conductance electrons’ at the Fermi level in the density of states, representing the Kondo singlet state formed between embedded dot and leads. In Ref. [12], it was shown that Kondo effect induces entanglement between the spins in the side dots and the quantum phase transition between unentangled and entangled states is analyzed by an effective spin model where the triple quantum dot is anti-ferromagnetically coupled to an additional singly occupied quantum dot, representing the conduction band using a single spin. Kondo coupling between the embedded dot and leads is, however, many-body effect like the screening of a local moment by conduction electrons, which produces Kondo peak at the Fermi level in the density of states. It is, therefore, reasonable to consider the influence of Kondo coupling between embedded dot and leads on the quantum phase transition by using the density of states including Kondo peak. It also allows us to obtain the analytical expression for the critical value of tunnel coupling between side dots as the function of on-site energies of dots. The analytical results qualitatively agree well with the result of the numerical renormalization group method. The results show that the critical value of tunnel coupling between side dots decreases when the energy level of embedded dot rises up from the symmetric point to the Fermi level and the critical value increases when the energy levels of two side dots rise up. The study of the influence of on-site-energy changes on the quantum phase transitions in triple quantum dots is important for clarifying the mechanism of Kondo screening in triple quantum dots where energy levels of dots are deviated from the particle–hole symmetric point. Our study also allows wide applications of such systems, enabling the control of quantum phase transition properties in triple quantum dots by changing their energy levels.
electrons on dot site with i = 1, 2, 3. HTQD describes the isolated three quantum dots and Hleads the leads, essentially being non-interacting but macroscopic metal. The hybridization term Hhyb shows that dot1 is coupled to leads with tunnel-coupling Vαk . 2.1. Analytical argument When a small number of magnetic atoms are added to the nonmagnetic metals, Kondo effect, the minimum phenomenon in the resistance of the metal, appears. In such systems, magnetic impurities also interact with one another via the RKKY interaction. Such an indirect spin–spin interaction is mediated by conduction electrons at the Fermi level in the density of states with bandwidth D. In this work, we assume that RKKY interaction between dots 2 and 3 (side dots) is mediated by virtual ‘conductance electrons’ at the Fermi level in the density of states with Kondo peak (its width Tk and its height 1/Tk ) which describes the Kondo singlet state formed between dot 1 and leads. We neglect much narrower gap inside Kondo peak, since its contribution to the RKKY interaction is significantly small when the coupling between the embedded dot and the side dots is week. The weaker the coupling between the embedded dot and the side dot is, the narrower the gap inside Kondo peak becomes [13]. Under this assumption, we can rewrite the Hamiltonian (1) as following:
H = H0 + HD + H′, H0 =
∑ Ek f k+σ fkσ , k, σ
∑
HD =
σ , i = 2,3
⎡ + ⎤ 1 ⎢⎣εi diσ diσ + Ui niσ niσ¯ ⎥⎦ , 2
H′ = t12 ∑ ⎡⎣d2+σ fkσ + h. c ⎤⎦ + t13 ∑ ⎡⎣d3+σ fkσ + h. c ⎤⎦ ,
2. Model and methods
k, σ
We consider the triple quantum dots (TQD) device illustrated in Fig. 1. It consists of locally correlated single-level sites, with level energy εi and on-site coulomb repulsion Ui . Dots i and j are tunnelcoupled by a matrix element tij to form a triangular arrangement. Dot 1 (embedded dot) is also coupled to source and drain leads. We consider the case of zero-bias when the system is in equilibrium. The Hamiltonian that we study reads Hleads =
∑ ∑ ∑ ε k cα+kσcαkσ , d = s, d σ
HTQD =
∑ i = 1,2,3
Hhyb =
∑ α, k, σ
k, σ
(2)
where H0 describes virtual ‘conductance electron’ system that has the density of states with Kondo peak (its width Tk and its height 1/Tk ), and H′ describes that the dots 1 and 2 are coupled to virtual ‘conductance electron’ system. The RKKY interaction Hamiltonian can be derived from the fourth-order perturbation expansion of H′ [14]
⇀ ⇀ HRKKY = JRKKY S2 ⋅ S3,
(3)
k
⎡ε i n i ↑ + n i ↓ + U i n i ↑ n i ↓ ⎤ + ) ⎣ ( ⎦
∑ i < j, σ
(tij di+σd jσ + h. c),
JRKKY =
k, k ′
⎡V d + c ⎤ ⎣ αk 1σ αkσ + h . c ⎦ ,
(1)
where niσ = di+σ diσ is the number operator for spin σ = ↑ /↓
t23
t12 s
(4)
⎞ 1 2⎛ 1 1 1 1 t1i ⎜ + − − ⎟, 2 ⎝ Ek − εi Ek ′ − εi Ek − εi − Ui Ek ′ − εi − Ui ⎠
(5)
where Ek and Ek′ are energies of virtual ‘conductance electrons’ and they are approximately equal to zero. At zero temperature, RKKY coupling is described by
t13 1
1 , Ek − Ek ′
where JRKKY describes RKKY coupling between dots 2 and 3 and Jikk′ denotes Kondo-type coupling between dot i and ‘conductance electron’ system. Kondo-type coupling is given by
Jikk′ =
3
2
∑ fk (1 − fk ′ ) J2kk′ J3kk′
d
Fig. 1. Geometrical structure of the triple quantum dot system. Electrons are transported between source (s) and drain (d) leads.
JRKKY = J2 J3
0
∫−∞ dε ρ (ε) ∫0
∞
dε′ ρ (ε′)
1 , ε − ε′
(6)
where ρ (ε) is the density of states of virtual ‘conductance electron’ system:
ρ (ε) =
1 Tk . π ε2 + Tk2
(7)
S.-H. Kim et al. / Physica B 465 (2015) 55–59
57
Consequently,
JRKKY ≈ − 0.637
J2 J3 Tk
,
(8)
where
⎛ 1 1 ⎞ Ji = ti2 ⎜− + ⎟ (i = 2, 3) εi + Ui ⎠ ⎝ εi
(9)
and Kondo temperature is [15]
⎛ ⎞1/2 ⎛ π ε1 ε1 + U1 ⎞ ΓD ⎟⎟ exp ⎜⎜− Tk ∼ D ⎜⎜ ⎟⎟. π ε ε + U 2ΓD ⎝ 1 1 ⎝ ⎠ 1 ⎠
(10)
Considering both the direct exchange coupling and the indirect RKKY coupling, the total effective exchange coupling between dots 2 and 3 is described by
Jeff = J′ + JRKKY , where the direct exchange coupling is
J′ =
4t′2 U
(11)
for U = U2 = U3, ε = ε2 = ε3 and t′ = t23. When Jeff > 0 , singlet lies lowest in energy-level scheme and decouples from the rest of the system. At the temperature of
T ≪ Jeff , the entire system is therefore in a singlet state that is characteristic of the strong coupling phase. When Jeff < 0, the triplet state formed between dots 2 and 3 lies lowest in energylevel scheme and is characteristic of the local moment phase. In consequence, there is a quantum phase transition between a Kondo screened strong coupling phase and a local moment phase. The transition is expected to occur at
Jeff = J′ + JRKKY = 0, 4tc′2 = − JRKKY , U
(12)
where tc′ is the critical value of tunnel coupling between dots 2 and 3 when the transition occurs between two phases. From Eq. (12), we can study how the phase diagram obtained in the previous paper [11] is modified when each dot is deviated from the particle–hole symmetric point. First, we consider the phase diagram when the energy level of dot1 rises up from the particle–hole symmetric point (δ1 = 0) to U
Fermi level (δ1 = 2 ), while the dots 2 and 3 are at the particle–hole symmetric point (δ = δ2 = δ 3 = 0). From Eq. (10), In this case, Kondo temperature Tk increases from Eq. (10) and the critical value of tc′ decreases. Such behavior of critical value tc′ is shown in Fig. 2. Fig. 2 shows critical values t′c /πΓ of tunnel coupling between side dots as the function of tunnel coupling t/πΓ between side dot and embedded dot for the embedded dot unfixed at the particle– hole symmetric point and fixed side dots. The results presented in Fig. 2 have been obtained by using the analytical Eq. (12) for U/πΓ = 7, D/Γ = 100 and on-site energy of the embedded dot δ1 = 0, 0.8, 1.6, 2.4, 3.5, and show that critical values decrease with rising the energy level of the embedded dot. Second, we study the phase diagram when the energy levels of dots 2 and 3 rise up from the particle–hole symmetric point to Fermi level and the energy level of dot 1 is at the particle–hole symmetric point. From Eq. (9), we can see that Kondo couplings J2 and J3 increase and critical value tc′ therefore increases, in this case. Such behavior of critical value tc′ is shown in Fig. 3. Fig. 3 shows critical values t′c /πΓ of tunnel coupling between
Fig. 2. Critical values t′c /πΓ of tunnel coupling between side dots as the function of tunnel coupling t/πΓ between side dot and embedded dot for the embedded dot unfixed at the particle–hole symmetric point and side dots fixed. These curves have been obtained by using the analytical Eq. (12) forU/πΓ = 7, D/Γ = 100 and on-site energy of the embedded dot δ1 = 0, 0.8, 1.6, 2.4, 3.5, and show that the higher the energy level of the embedded dot rises up, the smaller the critical value.
side dots as the function of tunnel coupling t/πΓ between side dot and embedded dot for side dots unfixed at the particle–hole symmetric point and the embedded dot fixed. The results in Fig. 3 have been obtained by using the analytical Eq. (12) for U/πΓ = 7, D/Γ = 100 and on-site energy of the side dots δ ( = δ2 = δ 3 ) = 0, 1, 2, 3, and show that the higher the energy level of the side dots rises up, the larger the critical value becomes. Such behaviors of critical value tc′ shown in Figs. 2 and 3 are confirmed by the exact numerical results obtained by using the numerical renormalization group method. 2.2. Numerical renormalization group method The numerical renormalization group (NRG) technique consists of logarithmic discretization of the conduction band, mapping onto one-dimensional chain with exponentially decreasing hopping constants, and iterative diagonalization of the resulting Hamiltonian [16–18]. Only low-energy part of the spectrum is kept after each iteration step; in our calculations we kept 1500 states, using discretization parameter Λ = 3. Using the NRG method, we explore the effect of the change in on-site energies of dots deviated from the particle–hole symmetric point on the quantum phase transition between the Kondo
Physica B
Fig. 3. Critical values t′c /πΓ of tunnel coupling between side dots as the function of tunnel coupling t/πΓ between side dot and embedded dot for side dots unfixed at the particle–hole symmetric point and the embedded dot fixed. These curves have been obtained by using the analytical Eq. (12) forU/πΓ = 7, D/Γ = 100 and on-site energy of the side dots δ ( = δ2 = δ 3 ) = 0, 1, 2, 3, and show that higher the energy level of the side dots rises up, the larger the critical value.
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screened strong coupling phase and the local moment phase. We now consider the case of δ1 ≠ 0, δ = δ2 = δ 3 = 0 and then the other case of δ1 = 0, δ = δ2 = δ 3 ≠ 0 . Typically, we consider only the parameter range of t/πΓ = 3 × 10−3–6 × 10−3. This study will be conducted by computing the impurity susceptibility
χimp (T ) =
(gμ B ) kB T
(S
2 z
− S z2
0
),
where the first expectation value refers to the system with the triple quantum dot, while the second refers to the system without the dots. Fig. 4 shows the plots of Tχimp vs. ln (T /πΓ), illustrating the transition from the local moment phase to the strong coupling and phase for U/πΓ = 7, t/πΓ = 3 × 10−3, D/Γ = 100 t′/πΓ = 4.1 × 10−4 , 4.22 × 10−4 , 4.24 × 10−4 , 4.2 × 10−4 , 4.26 × 10−4 , 4.28 × 10−4 , 4.3 × 10−4 , 4.4 × 10−4 . The dashed lines correspond to the local moment phase and the solid ones to the Kondo screened strong coupling phase. We find the critical value t′c by comparing temperature-dependent susceptibilities calculated for a range of parameters t′ expected to be the critical value for fixed values of t and δ ( ≠ 0). In Fig. 4, the critical value of t′c /πΓ exists in the range between 4.26 × 10−4 and 4.28 × 10−4 , and can be determined to be about 4.26 × 10−4 . In the same way as we found the critical values t′c in Fig. 4, we can obtain the critical values t′c for the various values of t and δ1, and δ ( = δ2 = δ 3 ) to draw the phase diagrams. Fig. 5 shows the critical values t′c /πΓ of tunnel coupling between side dots as a function of tunnel coupling t/πΓ between side dot and embedded dot, where the embedded dot is unfixed at the particle–hole symmetric point and side dots are fixed. The curves have been obtained by using the numerical renormalization group method for U/πΓ = 7, D/Γ = 100, and on-site energy of the embedded dot δ1 = 0, 0.8, 1.6, 2.4, 3.5. They show that the higher the energy level of the embedded dot goes up, the smaller the critical value becomes. Fig. 6 shows the critical values t′c /πΓ of tunnel coupling between side dots as the function of tunnel coupling t/πΓ between side dot and embedded dot for side dots unfixed at the particle– hole symmetric point and the fixed embedded dot. The curves in the figure have been obtained by using the numerical renormalization group method for U/πΓ = 7, D/Γ = 100, and on-
Fig. 4. Plots of Tχimp vs. ln (T /πΓ) which illustrate the transition from the local moment phase to the strong coupling phase for U/πΓ = 7,t/πΓ = 3 × 10−3 , andt′/πΓ = 4.1 × 10−4 , 4.2 × 10−4 , 4.22 × 10−4 , 4.24 × 10−4 , D/Γ = 100 4.26 × 10−4 , 4.28 × 10−4 , 4.3 × 10−4 , 4.4 × 10−4 . The dashed curves correspond to the local moment phase and the solid curves to the strong coupling phase. The critical value t′c /πΓ exists in the range between 4.26 × 10−4 and 4.28 × 10−4 , and approximates to 4.26 × 10−4 .
Fig. 5. Critical values t′c /πΓ of tunnel coupling between side dots as a function of tunnel coupling t/πΓ between side dot and embedded dot for the embedded dot unfixed at the particle–hole symmetric point and side dots fixed. These curves have been obtained by the numerical renormalization group method for U/πΓ = 7, D/Γ = 100 and on-site energy of the embedded dot δ1 = 0, 0.8, 1.6, 2.4, 3.5, and show that higher the energy level of the embedded dot goes up, the smaller the critical value. In the figure, we have shown the results for the different on-site energies of embedded dot, as in Fig. 2. In comparison of this figure with Fig. 2, we find that the analytical result qualitatively agrees well with the numerical one.
site energy of the side dots δ ( = δ2 = δ 3 ) = 0, 1, 2, 3. They show that the higher the energy level of the side dots goes up, the larger the critical value becomes. Comparing Figs. 5 and 6 with Figs. 2 and 3, we can see that the analytical results qualitatively agree with numerical results.
3. Discussions The RKKY interaction appears, mediated by conduction electrons between impurities. In this paper, we present the assumption that RKKY interaction between side dots is mediated by virtual ‘conductance electrons’ at the Fermi level in the density of states with Kondo peak (its width Tk and its height 1/Tk ), representing the Kondo singlet state formed between embedded dot and leads. This allows us to obtain the analytical expression of the
Fig. 6. Critical values t′c /πΓ of tunnel coupling between side dots as a function of tunnel coupling t/πΓ between side dot and embedded dot for side dots unfixed at the particle–hole symmetric point and the embedded dot fixed. These curves have been obtained by the numerical renormalization group method for U/πΓ = 7, D/Γ = 100 and on-site energy of the side dots δ ( = δ2 = δ 3 ) = 0, 1, 2, 3, and show that higher the energy level of the side dots goes up, the larger the critical value. In the figure, we have shown the results for the different on-site energies of side dots, as in Fig. 3. In comparison of this figure with Fig. 3, we find that the analytical result qualitatively agrees with the numerical one. Note, however, that the result for δ = 3 does not quantitatively agree with the analytic one due to the invalidity of Eq. (12) for large deviation of energy levels of dots from the particle– hole symmetric point.
S.-H. Kim et al. / Physica B 465 (2015) 55–59
Physica B
59
clarifying the mechanism of Kondo screening in triple quantum dots where energy levels of dots are deviated from the particle– hole symmetric point.
4. Conclusion
Fig. 7. Schematic phase diagrams for the TQD with energy levels of dots deviated from the particle–hole symmetric point. In the figure, solid line corresponds to energy levels of all the quantum dots at the particle–hole symmetric point. The dashed lines (a)–(d) show that the critical value t′c decreases with raising the energy level of embedded dot, while (e)–(g) demonstrate that t′c increases with raising the energy levels of side dots. For the embedded dot, its on-site energy has been changed such that δa < δb < δc < δd . For the side dots, their on-site energies have been equally changed such that δe < δ f < δ g .
critical value t′c as the function of on-site energies of dots. The behaviors of critical value expected by analytical expression for t′c qualitatively agree with numerical results obtained by the numerical renormalization group method. Nevertheless, the quantitative differences between the analytical and numerical results become larger when the energy levels in quantum dots are further deviated from the particle–hole symmetric point, since the analytic Eq. (12) is invalid for large deviation of energy levels of dots from the particle–hole symmetric point. For significant deviation of energy levels of dots from the particle–hole symmetric point, the numerical results obtained by using Anderson Hamiltonian cannot be strictly explained by the analytic expression based on the Kondo Hamiltonian. In this work, we focused on the general behaviors of critical value for the changes of on-site energies of dots, rather than quantitative treatment for its value. In analytical argument, the proportional coefficient in the expression (10) for Kondo temperature is determined by the fact that TK ≈ 10−4 for the systems considered in this paper. πΓ Fig. 7 shows schematic phase diagrams for the TQD with energy levels of dots deviated from the particle–hole symmetric point. In the figure, the solid line corresponds to energy levels of all the quantum dots at the particle–hole symmetric point. The dashed lines (a)–(d) show that the critical value t′c decreases with raising the energy level of embedded dot, while (e)–(g) demonstrate that t′c increases with raising the energy levels of side dots. For the embedded dot, its on-site energy has been changed such thatδa < δb < δc < δ d . For the side dots, their on-site energies have been equally changed such that δe < δ f < δ g . The behaviors of quantum phase transition shown in Fig. 7 have the importance for
In this work, we have investigated quantum phase transition between the local moment and the Kondo screened strong coupling phase for the case of weak inter-dot coupling in a triple quantum dot system with a triangular geometry, in which one of the dots is connected to metallic leads. The dependence of the critical value of tunnel coupling between side dots on the on-site energies of dots has been studied by using the numerical renormalization group method in combination with analytical arguments. The results show that the critical value decreases when the energy level of embedded dot rises up from the particle–hole symmetric point to the Fermi level and energy levels of side dots are fixed in the particle–hole symmetric point. They also demonstrate that the critical value increases when the energy level of embedded dot is fixed and the energy levels of two side dots rise up. The results are valid for the week inter-dot coupling, corresponding to the maxima of t and t′ around UTK . The results presented in this paper have the importance to clarify the mechanism of Kondo screening in triple quantum dots where energy levels of dots are deviated from the particle–hole symmetric point and to extend practical applications of TQD.
References [1] Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485. [2] R.A. Webb, S. Washburn, C.P. Umbach, R.B. Laibowitz, Phys. Rev. Lett. 54 (1985) 2696. [3] P.W. Anderson, Mater. Res. Bull. 8 (1973) 153. [4] N. Bulut, W. Koshibae, S. Maekawa, Phys. Rev. Lett. 95 (2005) 037001. [5] Y. Furukawa, T. Ohashi, Y. Koyama, N. Kawakami, Phys. Rev. B 82 (2010) 161101. [6] T. Jamneala, V. Madhavan, M.F. Crommie, Phys. Rev. Lett. 87 (2001) 256804. [7] K. Ingersent, A.W. Ludwig, I. Affleck, Phys. Rev. Lett. 95 (2005) 257204. [8] B. Lazarovits, P. Simon, G. Zarand, L. Szunyogh, Phys. Rev. Lett. 95 (2005) 077202. [9] P. Nozieres, A. Blandin, J. Phys. 41 (1980) 193. [10] D.L. Cox, A. Zawadowski, Adv. Phys. 47 (1998) 599. [11] A.K. Mitchell, T.F. Jarrold, M.G. Galpin, P.E. Logan, J. Phys. Chem. B 117 (2013) 12777. [12] S.B. Tooski, A. Ramsak, R. Zitko, B.R. Bulka, Eur. Phys. J. B 87 (2014) 145. [13] R. Zitko, J. Bonca, J. Phys.: Condens. Matter 19 (2007) 255205. [14] M. Yang, S.S. Li, Phys. Rev. B 74 (2006) 073402. [15] A.C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, Cambridge, 1993. [16] K.G. Wilson, Rev. Mod. Phys. 47 (1975) 773. [17] H.R. Krishnamurthy, J.W. Wilkins, K.G. Wilson, Phys. Rev. B 21 (1003) (1980) 1044. [18] R. Bulla, T. Costi, T. Pruscke, Rev. Mod. Phys. 80 (2008) 395.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Quantum phase transition in trigonal triple quantum dots: The case of quantum dots deviated from particle–hole symmetric point Song-Hyok Kim a, Chol-Jin Kang a, Yon-Il Kim a, Kwang-Hyon Kim b,n a b
Department of Theoretical Physics, Institute of Physics, Unjong district, Pyongyang, DPR Korea Institute of Laser, Unjong district, Pyongyang, DPR Korea
art ic l e i nf o
a b s t r a c t
Article history: Received 2 November 2014 Received in revised form 25 January 2015 Accepted 5 February 2015 Available online 7 February 2015
We consider a triple quantum dot system in a triangular geometry with one of the dots connected to metallic leads. We investigate quantum phase transition between local moment phase and Kondo screened strong coupling phase in triple quantum dots where energy levels of dots are deviated from the particle–hole symmetric point. The effect of on-site energy of dots on quantum phase transition between local moment phase and Kondo screened strong coupling phase in triple quantum dots is studied based on the analytical arguments and the numerical renormalization group method. The results show that the critical value of tunnel coupling between side dots decreases when the energy level of embedded dot rises up from the symmetric point to the Fermi level and the critical value increases when the energy levels of two side dots rise up. The study of the influence of on-site-energy changes on the quantum phase transitions in triple quantum dots has the importance for clarifying the mechanism of Kondo screening in triple quantum dots where energy levels of dots are deviated from the particle–hole symmetric point. & 2015 Elsevier B.V. All rights reserved.
Keywords: Triple quantum dots Quantum phase transition Numerical renormalization group method RKKY interaction
1. Introduction The triangle is the simplest polygon that has a closed loop which plays an important role on various fascinating phenomena in the condensed matter physics. The closed path in metal and semiconductor allows the electron to move around the loop, and causes quantum interference effects, such as an Aharonov–Bohm effect [1,2]. The closed path consisting of the odd-number of links also causes frustration, leading to resonating valence bonds for some anti-ferromagnetic systems [3]. Single triangle is also a fundamental unite of the triangular and Kagome lattices. In these systems, the geometrical frustration affects significantly the magnetic properties and the behavior at the Mott–Hubbard metal–insulator transition [4,5]. Another interesting example is the triangular trimer of Cr atoms placed upon an Au surface [6–8], expected to show a non-Fermi-liquid behavior due to the multi-channel Kondo effect [9,10]. The quantum phase transition between Kondo screened strong coupling (SC) phase and local moment (LM) phase in triple quantum dot system has attracted much interest and has been studied theoretically [11,12]. In triple quantum dots with week inter-dot coupling, the quantum phase transition between those n
Corresponding author. E-mail address: [email protected] (K.-H. Kim).
http://dx.doi.org/10.1016/j.physb.2015.02.005 0921-4526/& 2015 Elsevier B.V. All rights reserved.
phases is the consequence of competition between direct exchange interaction and the indirect interaction between two side dots. There are two types of interactions between two side dots coupled to the embedded dot: the direct exchange interaction between two side dots and the indirect RKKY-type interaction mediated by the Kondo singlet formed between the embedded dot and the leads. The signs of those interactions are different from each other. The sign of the actual effective exchange interaction energy between two side dots depends therefore on which of the two interactions prevails. The quantum phase of the system depends on the sign of the effective exchange interaction energy between two side dots. In the preceding report [11], quantum phase transitions between the Kondo screened SC phase and the LM phase have been investigated only for the systems consisting quantum dots whose energy levels are fixed at the particle–hole symmetric point. In real triple quantum dot systems, it is, however, difficult to produce dots with identical properties. In particular, it is not easy to achieve equal on-site energies of three dots at the particle–hole symmetric point and the energy levels of dots are generally deviated from the particle–hole symmetric point. In this work, we study how the phase diagram, representing the quantum phase transition between the Kondo screened SC phase and the LM phase in triple quantum dots with week interdot coupling, is modified as the levels of dots are deviated from the particle–hole symmetric point. In the analytical argument we
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assume that RKKY interaction between side dots can also be mediated by virtual ‘conductance electrons’ at the Fermi level in the density of states, representing the Kondo singlet state formed between embedded dot and leads. In Ref. [12], it was shown that Kondo effect induces entanglement between the spins in the side dots and the quantum phase transition between unentangled and entangled states is analyzed by an effective spin model where the triple quantum dot is anti-ferromagnetically coupled to an additional singly occupied quantum dot, representing the conduction band using a single spin. Kondo coupling between the embedded dot and leads is, however, many-body effect like the screening of a local moment by conduction electrons, which produces Kondo peak at the Fermi level in the density of states. It is, therefore, reasonable to consider the influence of Kondo coupling between embedded dot and leads on the quantum phase transition by using the density of states including Kondo peak. It also allows us to obtain the analytical expression for the critical value of tunnel coupling between side dots as the function of on-site energies of dots. The analytical results qualitatively agree well with the result of the numerical renormalization group method. The results show that the critical value of tunnel coupling between side dots decreases when the energy level of embedded dot rises up from the symmetric point to the Fermi level and the critical value increases when the energy levels of two side dots rise up. The study of the influence of on-site-energy changes on the quantum phase transitions in triple quantum dots is important for clarifying the mechanism of Kondo screening in triple quantum dots where energy levels of dots are deviated from the particle–hole symmetric point. Our study also allows wide applications of such systems, enabling the control of quantum phase transition properties in triple quantum dots by changing their energy levels.
electrons on dot site with i = 1, 2, 3. HTQD describes the isolated three quantum dots and Hleads the leads, essentially being non-interacting but macroscopic metal. The hybridization term Hhyb shows that dot1 is coupled to leads with tunnel-coupling Vαk . 2.1. Analytical argument When a small number of magnetic atoms are added to the nonmagnetic metals, Kondo effect, the minimum phenomenon in the resistance of the metal, appears. In such systems, magnetic impurities also interact with one another via the RKKY interaction. Such an indirect spin–spin interaction is mediated by conduction electrons at the Fermi level in the density of states with bandwidth D. In this work, we assume that RKKY interaction between dots 2 and 3 (side dots) is mediated by virtual ‘conductance electrons’ at the Fermi level in the density of states with Kondo peak (its width Tk and its height 1/Tk ) which describes the Kondo singlet state formed between dot 1 and leads. We neglect much narrower gap inside Kondo peak, since its contribution to the RKKY interaction is significantly small when the coupling between the embedded dot and the side dots is week. The weaker the coupling between the embedded dot and the side dot is, the narrower the gap inside Kondo peak becomes [13]. Under this assumption, we can rewrite the Hamiltonian (1) as following:
H = H0 + HD + H′, H0 =
∑ Ek f k+σ fkσ , k, σ
∑
HD =
σ , i = 2,3
⎡ + ⎤ 1 ⎢⎣εi diσ diσ + Ui niσ niσ¯ ⎥⎦ , 2
H′ = t12 ∑ ⎡⎣d2+σ fkσ + h. c ⎤⎦ + t13 ∑ ⎡⎣d3+σ fkσ + h. c ⎤⎦ ,
2. Model and methods
k, σ
We consider the triple quantum dots (TQD) device illustrated in Fig. 1. It consists of locally correlated single-level sites, with level energy εi and on-site coulomb repulsion Ui . Dots i and j are tunnelcoupled by a matrix element tij to form a triangular arrangement. Dot 1 (embedded dot) is also coupled to source and drain leads. We consider the case of zero-bias when the system is in equilibrium. The Hamiltonian that we study reads Hleads =
∑ ∑ ∑ ε k cα+kσcαkσ , d = s, d σ
HTQD =
∑ i = 1,2,3
Hhyb =
∑ α, k, σ
k, σ
(2)
where H0 describes virtual ‘conductance electron’ system that has the density of states with Kondo peak (its width Tk and its height 1/Tk ), and H′ describes that the dots 1 and 2 are coupled to virtual ‘conductance electron’ system. The RKKY interaction Hamiltonian can be derived from the fourth-order perturbation expansion of H′ [14]
⇀ ⇀ HRKKY = JRKKY S2 ⋅ S3,
(3)
k
⎡ε i n i ↑ + n i ↓ + U i n i ↑ n i ↓ ⎤ + ) ⎣ ( ⎦
∑ i < j, σ
(tij di+σd jσ + h. c),
JRKKY =
k, k ′
⎡V d + c ⎤ ⎣ αk 1σ αkσ + h . c ⎦ ,
(1)
where niσ = di+σ diσ is the number operator for spin σ = ↑ /↓
t23
t12 s
(4)
⎞ 1 2⎛ 1 1 1 1 t1i ⎜ + − − ⎟, 2 ⎝ Ek − εi Ek ′ − εi Ek − εi − Ui Ek ′ − εi − Ui ⎠
(5)
where Ek and Ek′ are energies of virtual ‘conductance electrons’ and they are approximately equal to zero. At zero temperature, RKKY coupling is described by
t13 1
1 , Ek − Ek ′
where JRKKY describes RKKY coupling between dots 2 and 3 and Jikk′ denotes Kondo-type coupling between dot i and ‘conductance electron’ system. Kondo-type coupling is given by
Jikk′ =
3
2
∑ fk (1 − fk ′ ) J2kk′ J3kk′
d
Fig. 1. Geometrical structure of the triple quantum dot system. Electrons are transported between source (s) and drain (d) leads.
JRKKY = J2 J3
0
∫−∞ dε ρ (ε) ∫0
∞
dε′ ρ (ε′)
1 , ε − ε′
(6)
where ρ (ε) is the density of states of virtual ‘conductance electron’ system:
ρ (ε) =
1 Tk . π ε2 + Tk2
(7)
S.-H. Kim et al. / Physica B 465 (2015) 55–59
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Consequently,
JRKKY ≈ − 0.637
J2 J3 Tk
,
(8)
where
⎛ 1 1 ⎞ Ji = ti2 ⎜− + ⎟ (i = 2, 3) εi + Ui ⎠ ⎝ εi
(9)
and Kondo temperature is [15]
⎛ ⎞1/2 ⎛ π ε1 ε1 + U1 ⎞ ΓD ⎟⎟ exp ⎜⎜− Tk ∼ D ⎜⎜ ⎟⎟. π ε ε + U 2ΓD ⎝ 1 1 ⎝ ⎠ 1 ⎠
(10)
Considering both the direct exchange coupling and the indirect RKKY coupling, the total effective exchange coupling between dots 2 and 3 is described by
Jeff = J′ + JRKKY , where the direct exchange coupling is
J′ =
4t′2 U
(11)
for U = U2 = U3, ε = ε2 = ε3 and t′ = t23. When Jeff > 0 , singlet lies lowest in energy-level scheme and decouples from the rest of the system. At the temperature of
T ≪ Jeff , the entire system is therefore in a singlet state that is characteristic of the strong coupling phase. When Jeff < 0, the triplet state formed between dots 2 and 3 lies lowest in energylevel scheme and is characteristic of the local moment phase. In consequence, there is a quantum phase transition between a Kondo screened strong coupling phase and a local moment phase. The transition is expected to occur at
Jeff = J′ + JRKKY = 0, 4tc′2 = − JRKKY , U
(12)
where tc′ is the critical value of tunnel coupling between dots 2 and 3 when the transition occurs between two phases. From Eq. (12), we can study how the phase diagram obtained in the previous paper [11] is modified when each dot is deviated from the particle–hole symmetric point. First, we consider the phase diagram when the energy level of dot1 rises up from the particle–hole symmetric point (δ1 = 0) to U
Fermi level (δ1 = 2 ), while the dots 2 and 3 are at the particle–hole symmetric point (δ = δ2 = δ 3 = 0). From Eq. (10), In this case, Kondo temperature Tk increases from Eq. (10) and the critical value of tc′ decreases. Such behavior of critical value tc′ is shown in Fig. 2. Fig. 2 shows critical values t′c /πΓ of tunnel coupling between side dots as the function of tunnel coupling t/πΓ between side dot and embedded dot for the embedded dot unfixed at the particle– hole symmetric point and fixed side dots. The results presented in Fig. 2 have been obtained by using the analytical Eq. (12) for U/πΓ = 7, D/Γ = 100 and on-site energy of the embedded dot δ1 = 0, 0.8, 1.6, 2.4, 3.5, and show that critical values decrease with rising the energy level of the embedded dot. Second, we study the phase diagram when the energy levels of dots 2 and 3 rise up from the particle–hole symmetric point to Fermi level and the energy level of dot 1 is at the particle–hole symmetric point. From Eq. (9), we can see that Kondo couplings J2 and J3 increase and critical value tc′ therefore increases, in this case. Such behavior of critical value tc′ is shown in Fig. 3. Fig. 3 shows critical values t′c /πΓ of tunnel coupling between
Fig. 2. Critical values t′c /πΓ of tunnel coupling between side dots as the function of tunnel coupling t/πΓ between side dot and embedded dot for the embedded dot unfixed at the particle–hole symmetric point and side dots fixed. These curves have been obtained by using the analytical Eq. (12) forU/πΓ = 7, D/Γ = 100 and on-site energy of the embedded dot δ1 = 0, 0.8, 1.6, 2.4, 3.5, and show that the higher the energy level of the embedded dot rises up, the smaller the critical value.
side dots as the function of tunnel coupling t/πΓ between side dot and embedded dot for side dots unfixed at the particle–hole symmetric point and the embedded dot fixed. The results in Fig. 3 have been obtained by using the analytical Eq. (12) for U/πΓ = 7, D/Γ = 100 and on-site energy of the side dots δ ( = δ2 = δ 3 ) = 0, 1, 2, 3, and show that the higher the energy level of the side dots rises up, the larger the critical value becomes. Such behaviors of critical value tc′ shown in Figs. 2 and 3 are confirmed by the exact numerical results obtained by using the numerical renormalization group method. 2.2. Numerical renormalization group method The numerical renormalization group (NRG) technique consists of logarithmic discretization of the conduction band, mapping onto one-dimensional chain with exponentially decreasing hopping constants, and iterative diagonalization of the resulting Hamiltonian [16–18]. Only low-energy part of the spectrum is kept after each iteration step; in our calculations we kept 1500 states, using discretization parameter Λ = 3. Using the NRG method, we explore the effect of the change in on-site energies of dots deviated from the particle–hole symmetric point on the quantum phase transition between the Kondo
Physica B
Fig. 3. Critical values t′c /πΓ of tunnel coupling between side dots as the function of tunnel coupling t/πΓ between side dot and embedded dot for side dots unfixed at the particle–hole symmetric point and the embedded dot fixed. These curves have been obtained by using the analytical Eq. (12) forU/πΓ = 7, D/Γ = 100 and on-site energy of the side dots δ ( = δ2 = δ 3 ) = 0, 1, 2, 3, and show that higher the energy level of the side dots rises up, the larger the critical value.
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screened strong coupling phase and the local moment phase. We now consider the case of δ1 ≠ 0, δ = δ2 = δ 3 = 0 and then the other case of δ1 = 0, δ = δ2 = δ 3 ≠ 0 . Typically, we consider only the parameter range of t/πΓ = 3 × 10−3–6 × 10−3. This study will be conducted by computing the impurity susceptibility
χimp (T ) =
(gμ B ) kB T
(S
2 z
− S z2
0
),
where the first expectation value refers to the system with the triple quantum dot, while the second refers to the system without the dots. Fig. 4 shows the plots of Tχimp vs. ln (T /πΓ), illustrating the transition from the local moment phase to the strong coupling and phase for U/πΓ = 7, t/πΓ = 3 × 10−3, D/Γ = 100 t′/πΓ = 4.1 × 10−4 , 4.22 × 10−4 , 4.24 × 10−4 , 4.2 × 10−4 , 4.26 × 10−4 , 4.28 × 10−4 , 4.3 × 10−4 , 4.4 × 10−4 . The dashed lines correspond to the local moment phase and the solid ones to the Kondo screened strong coupling phase. We find the critical value t′c by comparing temperature-dependent susceptibilities calculated for a range of parameters t′ expected to be the critical value for fixed values of t and δ ( ≠ 0). In Fig. 4, the critical value of t′c /πΓ exists in the range between 4.26 × 10−4 and 4.28 × 10−4 , and can be determined to be about 4.26 × 10−4 . In the same way as we found the critical values t′c in Fig. 4, we can obtain the critical values t′c for the various values of t and δ1, and δ ( = δ2 = δ 3 ) to draw the phase diagrams. Fig. 5 shows the critical values t′c /πΓ of tunnel coupling between side dots as a function of tunnel coupling t/πΓ between side dot and embedded dot, where the embedded dot is unfixed at the particle–hole symmetric point and side dots are fixed. The curves have been obtained by using the numerical renormalization group method for U/πΓ = 7, D/Γ = 100, and on-site energy of the embedded dot δ1 = 0, 0.8, 1.6, 2.4, 3.5. They show that the higher the energy level of the embedded dot goes up, the smaller the critical value becomes. Fig. 6 shows the critical values t′c /πΓ of tunnel coupling between side dots as the function of tunnel coupling t/πΓ between side dot and embedded dot for side dots unfixed at the particle– hole symmetric point and the fixed embedded dot. The curves in the figure have been obtained by using the numerical renormalization group method for U/πΓ = 7, D/Γ = 100, and on-
Fig. 4. Plots of Tχimp vs. ln (T /πΓ) which illustrate the transition from the local moment phase to the strong coupling phase for U/πΓ = 7,t/πΓ = 3 × 10−3 , andt′/πΓ = 4.1 × 10−4 , 4.2 × 10−4 , 4.22 × 10−4 , 4.24 × 10−4 , D/Γ = 100 4.26 × 10−4 , 4.28 × 10−4 , 4.3 × 10−4 , 4.4 × 10−4 . The dashed curves correspond to the local moment phase and the solid curves to the strong coupling phase. The critical value t′c /πΓ exists in the range between 4.26 × 10−4 and 4.28 × 10−4 , and approximates to 4.26 × 10−4 .
Fig. 5. Critical values t′c /πΓ of tunnel coupling between side dots as a function of tunnel coupling t/πΓ between side dot and embedded dot for the embedded dot unfixed at the particle–hole symmetric point and side dots fixed. These curves have been obtained by the numerical renormalization group method for U/πΓ = 7, D/Γ = 100 and on-site energy of the embedded dot δ1 = 0, 0.8, 1.6, 2.4, 3.5, and show that higher the energy level of the embedded dot goes up, the smaller the critical value. In the figure, we have shown the results for the different on-site energies of embedded dot, as in Fig. 2. In comparison of this figure with Fig. 2, we find that the analytical result qualitatively agrees well with the numerical one.
site energy of the side dots δ ( = δ2 = δ 3 ) = 0, 1, 2, 3. They show that the higher the energy level of the side dots goes up, the larger the critical value becomes. Comparing Figs. 5 and 6 with Figs. 2 and 3, we can see that the analytical results qualitatively agree with numerical results.
3. Discussions The RKKY interaction appears, mediated by conduction electrons between impurities. In this paper, we present the assumption that RKKY interaction between side dots is mediated by virtual ‘conductance electrons’ at the Fermi level in the density of states with Kondo peak (its width Tk and its height 1/Tk ), representing the Kondo singlet state formed between embedded dot and leads. This allows us to obtain the analytical expression of the
Fig. 6. Critical values t′c /πΓ of tunnel coupling between side dots as a function of tunnel coupling t/πΓ between side dot and embedded dot for side dots unfixed at the particle–hole symmetric point and the embedded dot fixed. These curves have been obtained by the numerical renormalization group method for U/πΓ = 7, D/Γ = 100 and on-site energy of the side dots δ ( = δ2 = δ 3 ) = 0, 1, 2, 3, and show that higher the energy level of the side dots goes up, the larger the critical value. In the figure, we have shown the results for the different on-site energies of side dots, as in Fig. 3. In comparison of this figure with Fig. 3, we find that the analytical result qualitatively agrees with the numerical one. Note, however, that the result for δ = 3 does not quantitatively agree with the analytic one due to the invalidity of Eq. (12) for large deviation of energy levels of dots from the particle– hole symmetric point.
S.-H. Kim et al. / Physica B 465 (2015) 55–59
Physica B
59
clarifying the mechanism of Kondo screening in triple quantum dots where energy levels of dots are deviated from the particle– hole symmetric point.
4. Conclusion
Fig. 7. Schematic phase diagrams for the TQD with energy levels of dots deviated from the particle–hole symmetric point. In the figure, solid line corresponds to energy levels of all the quantum dots at the particle–hole symmetric point. The dashed lines (a)–(d) show that the critical value t′c decreases with raising the energy level of embedded dot, while (e)–(g) demonstrate that t′c increases with raising the energy levels of side dots. For the embedded dot, its on-site energy has been changed such that δa < δb < δc < δd . For the side dots, their on-site energies have been equally changed such that δe < δ f < δ g .
critical value t′c as the function of on-site energies of dots. The behaviors of critical value expected by analytical expression for t′c qualitatively agree with numerical results obtained by the numerical renormalization group method. Nevertheless, the quantitative differences between the analytical and numerical results become larger when the energy levels in quantum dots are further deviated from the particle–hole symmetric point, since the analytic Eq. (12) is invalid for large deviation of energy levels of dots from the particle–hole symmetric point. For significant deviation of energy levels of dots from the particle–hole symmetric point, the numerical results obtained by using Anderson Hamiltonian cannot be strictly explained by the analytic expression based on the Kondo Hamiltonian. In this work, we focused on the general behaviors of critical value for the changes of on-site energies of dots, rather than quantitative treatment for its value. In analytical argument, the proportional coefficient in the expression (10) for Kondo temperature is determined by the fact that TK ≈ 10−4 for the systems considered in this paper. πΓ Fig. 7 shows schematic phase diagrams for the TQD with energy levels of dots deviated from the particle–hole symmetric point. In the figure, the solid line corresponds to energy levels of all the quantum dots at the particle–hole symmetric point. The dashed lines (a)–(d) show that the critical value t′c decreases with raising the energy level of embedded dot, while (e)–(g) demonstrate that t′c increases with raising the energy levels of side dots. For the embedded dot, its on-site energy has been changed such thatδa < δb < δc < δ d . For the side dots, their on-site energies have been equally changed such that δe < δ f < δ g . The behaviors of quantum phase transition shown in Fig. 7 have the importance for
In this work, we have investigated quantum phase transition between the local moment and the Kondo screened strong coupling phase for the case of weak inter-dot coupling in a triple quantum dot system with a triangular geometry, in which one of the dots is connected to metallic leads. The dependence of the critical value of tunnel coupling between side dots on the on-site energies of dots has been studied by using the numerical renormalization group method in combination with analytical arguments. The results show that the critical value decreases when the energy level of embedded dot rises up from the particle–hole symmetric point to the Fermi level and energy levels of side dots are fixed in the particle–hole symmetric point. They also demonstrate that the critical value increases when the energy level of embedded dot is fixed and the energy levels of two side dots rise up. The results are valid for the week inter-dot coupling, corresponding to the maxima of t and t′ around UTK . The results presented in this paper have the importance to clarify the mechanism of Kondo screening in triple quantum dots where energy levels of dots are deviated from the particle–hole symmetric point and to extend practical applications of TQD.
References [1] Y. Aharonov, D. Bohm, Phys. Rev. 115 (1959) 485. [2] R.A. Webb, S. Washburn, C.P. Umbach, R.B. Laibowitz, Phys. Rev. Lett. 54 (1985) 2696. [3] P.W. Anderson, Mater. Res. Bull. 8 (1973) 153. [4] N. Bulut, W. Koshibae, S. Maekawa, Phys. Rev. Lett. 95 (2005) 037001. [5] Y. Furukawa, T. Ohashi, Y. Koyama, N. Kawakami, Phys. Rev. B 82 (2010) 161101. [6] T. Jamneala, V. Madhavan, M.F. Crommie, Phys. Rev. Lett. 87 (2001) 256804. [7] K. Ingersent, A.W. Ludwig, I. Affleck, Phys. Rev. Lett. 95 (2005) 257204. [8] B. Lazarovits, P. Simon, G. Zarand, L. Szunyogh, Phys. Rev. Lett. 95 (2005) 077202. [9] P. Nozieres, A. Blandin, J. Phys. 41 (1980) 193. [10] D.L. Cox, A. Zawadowski, Adv. Phys. 47 (1998) 599. [11] A.K. Mitchell, T.F. Jarrold, M.G. Galpin, P.E. Logan, J. Phys. Chem. B 117 (2013) 12777. [12] S.B. Tooski, A. Ramsak, R. Zitko, B.R. Bulka, Eur. Phys. J. B 87 (2014) 145. [13] R. Zitko, J. Bonca, J. Phys.: Condens. Matter 19 (2007) 255205. [14] M. Yang, S.S. Li, Phys. Rev. B 74 (2006) 073402. [15] A.C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, Cambridge, 1993. [16] K.G. Wilson, Rev. Mod. Phys. 47 (1975) 773. [17] H.R. Krishnamurthy, J.W. Wilkins, K.G. Wilson, Phys. Rev. B 21 (1003) (1980) 1044. [18] R. Bulla, T. Costi, T. Pruscke, Rev. Mod. Phys. 80 (2008) 395.