Thesis Title

Institute of Molecular Physics Polish Academy of Sciences ´ , Poland Poznan

Doctoral Thesis

Quantum entanglement, Kondo effect, and electronic transport in quantum dots system Author: Sahib Babaee Tooski

Supervisor: Professor Bogdan R. Bulka

A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in Physics

September 2014

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Abstract This thesis presents numerical renormalization group studies of quantum entanglement, Kondo effect and electronic transport through a system of quantum dots. The results are first presented for a triangular molecule built of coherently coupled quantum dots. Then, a single quantum dot with an assisted hopping is considered. The first part of thesis presents studies of three electrons confined in a triple quantum dot with one of the dots connected to metallic electrodes which is modeled by a three-impurity Anderson Hamiltonian. It is focused on the pairwise quantum entanglement of a three-spin system and its relation to the thermodynamic and transport properties. It is shown that two many-body phenomena compete with each others, the Kondo effect and the inter-dot exchange interactions. In fact, coupling triple quantum dots to the electrodes results a formation of the Kondo singlet which can switch the entanglement due to the interplay between the interdot spin-spin correlations and various Kondo-like ground states. The quantum phase transition between unentangled and entangled states is studied quantitatively and the corresponding phase diagram is explained by exactly solvable four spin model. Although the work concentrates on the system of quantum dots, the model is more general and can be applied to the Kondo physics in molecules with a triangular symmetry. The second part of thesis extends the transport properties of the triple quantum dot system for whole range of electron fillings. The study shows many-body feature of the ground states, which manifests itself in the conductance. Transport properties are also explained by an underlying Friedel-Luttinger sum rule which is applicable to both the regular- and singular-Fermi liquid ground states. The Friedel-Luttinger sum rule relates the conductance to the impurity charge and the Luttinger-integral. It is shown that the Luttinger integral is zero for the regularFermi liquid ground state and π/2 for the singular-Fermi liquid phase ground state. The main attention is to electronic correlations, formation of many-body states and their role in electronic transport. Detail investigations of correlation

ii functions and conductance present the ground state characteristics specially their local magnetic moment formation, and a corresponding quantum phase transition which separates the regular- and singular-Fermi liquid ground states. More interestingly, it has been shown that one can obtain the underscreened Kondo effect related to partial screening of spin S = 1. The last part of thesis studies conductance and thermopower of a single quantum dot coupled to the electrodes. The system is described by an extended single impurity Anderson model which takes into account the assisted hopping processes, i.e., the occupancy-dependence of the tunneling rates. The gate-voltage and temperature dependencies are discussed in various regimes to show that the thermopower and the conductance are very sensitive probes of assisted hopping and Kondo correlations. It is found that the assisted hopping modifies the width and position of the levels, breaks the electron-hole symmetry, shifts the Kondo temperature, and strongly affects the conductance and the thermopower of a quantum dot. The assisted hopping can lead to anomalies in the mixed-valence regime, in particular with a very high Seebeck coefficient.

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Abstract This thesis presents numerical renormalization group studies of quantum entanglement, Kondo effect and electronic transport through a system of quantum dots. The results are first presented for a triangular molecule built of coherently coupled quantum dots. Then, a single quantum dot with an assisted hopping is considered. The first part of thesis presents studies of three electrons confined in a triple quantum dot with one of the dots connected to metallic electrodes which is modeled by a three-impurity Anderson Hamiltonian. It is focused on the pairwise quantum entanglement of a three-spin system and its relation to the thermodynamic and transport properties. It is shown that two many-body phenomena compete with each others, the Kondo effect and the inter-dot exchange interactions. In fact, coupling triple quantum dots to the electrodes results a formation of the Kondo singlet which can switch the entanglement due to the interplay between the interdot spin-spin correlations and various Kondo-like ground states. The quantum phase transition between unentangled and entangled states is studied quantitatively and the corresponding phase diagram is explained by exactly solvable four spin model. Although the work concentrates on the system of quantum dots, the model is more general and can be applied to the Kondo physics in molecules with a triangular symmetry. The second part of thesis extends the transport properties of the triple quantum dot system for whole range of electron fillings. The study shows many-body feature of the ground states, which manifests itself in the conductance. Transport properties are also explained by an underlying Friedel-Luttinger sum rule which is applicable to both the regular- and singular-Fermi liquid ground states. The Friedel-Luttinger sum rule relates the conductance to the impurity charge and the Luttinger-integral. It is shown that the Luttinger integral is zero for the regularFermi liquid ground state and π/2 for the singular-Fermi liquid phase ground state. The main attention is to electronic correlations, formation of many-body states and their role in electronic transport. Detail investigations of correlation

iv functions and conductance present the ground state characteristics specially their local magnetic moment formation, and a corresponding quantum phase transition which separates the regular- and singular-Fermi liquid ground states. More interestingly, it has been shown that one can obtain the underscreened Kondo effect related to partial screening of spin S = 1. The last part of thesis studies conductance and thermopower of a single quantum dot coupled to the electrodes. The system is described by an extended single impurity Anderson model which takes into account the assisted hopping processes, i.e., the occupancy-dependence of the tunneling rates. The gate-voltage and temperature dependencies are discussed in various regimes to show that the thermopower and the conductance are very sensitive probes of assisted hopping and Kondo correlations. It is found that the assisted hopping modifies the width and position of the levels, breaks the electron-hole symmetry, shifts the Kondo temperature, and strongly affects the conductance and the thermopower of a quantum dot. The assisted hopping can lead to anomalies in the mixed-valence regime, in particular with a very high Seebeck coefficient.

Acknowledgements In the first place, I thank my supervisor professor Bogdan R. Bulka for his guidance, teaching, and support. His intuitive approach and inspiring suggestions deeply influenced me and this work. It is him who is responsible also for having me infected with his passion in the mesoscopic physics. In fact, his comments and advices were crucial. Next, I thank professor Anton Ramˇsak for his support during my accommodations in Ljubljana. I thank him for his scientific advice and knowledge and many insightful discussions and suggestions. I praise the enormous ˇ amount of help by him throughout these years. I also want to thank Rok Zitko for his advice, for sharing his knowledge with me, and particularity for his NRG code. ˇ I acknowledge enlightening discussions with Rok Zitko, Jukub Luczak, Marcin ˇ Urbaniak, Tomaz Rejec, Tilen Cadez and Piotr Stefanski. They generously shared their expertise in strongly correlated electron systems and computing science. I thank all the colleagues from the department of solid state theory at the Institute of Molecular Physics in Poznan and department of theoretical physics at Jozef Stefan Institute in Ljubljana. Institutes have been during past few years a second home to me, and my coworkers have all contributed to making it a really pleasant one. Computing facilities at Jozef Stefan Institute and Institute of Molecular Physics are also gratefully acknowledged. This thesis was funded by an Early Stage Training Marie Curie Doctoral Fellowship within the EU FP7 project Marie Curie ITN NanoCTM and in part by National Science Centre (Poland) under the contract DEC-2012/05/B/ST3/03208.

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Contents Abstract

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Abstract

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Acknowledgements

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Contents

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1 Aim of the study 2 General introduction 2.1 Quantum entanglement . . 2.2 Kondo effect . . . . . . . . 2.3 Anderson impurity model 2.4 Conductance . . . . . . . 2.5 Thermopower . . . . . . . 2.6 Quantum phase transition

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3 Numerical renormalization group method 3.1 Numerical renormalization group method . . . . . . . . . 3.1.1 Reduction to a one-dimensional problem . . . . . 3.1.2 Logarithmic discretization of the conduction band 3.1.3 Mapping on a semi-infinite chain . . . . . . . . . 3.1.4 RG transformation . . . . . . . . . . . . . . . . . 3.1.5 Iterative diagonalization . . . . . . . . . . . . . . 3.2 Calculation of physical properties . . . . . . . . . . . . . 3.2.1 Thermodynamic and static properties . . . . . . . 3.2.2 Dynamic properties . . . . . . . . . . . . . . . . . 3.3 NRG Ljubljana code . . . . . . . . . . . . . . . . . . . .

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4 Entanglement, the Kondo effect, and quantum phase transition in a triple quantum dot 29 4.1 Spins in triple quantum dots . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Modeling of triple quantum dot system . . . . . . . . . . . . . . . . 31 vii

Contents 4.3 4.4 4.5

4.6

States for three spins . . . . . . . . . . . . . . . . . . Effective four-spin model . . . . . . . . . . . . . . . . Numerical results . . . . . . . . . . . . . . . . . . . . 4.5.1 Concurrence . . . . . . . . . . . . . . . . . . . 4.5.2 Spin-spin correlation . . . . . . . . . . . . . . 4.5.3 Charge fluctuation . . . . . . . . . . . . . . . 4.5.4 Entropy and susceptibility . . . . . . . . . . . 4.5.5 Conductance . . . . . . . . . . . . . . . . . . 4.5.6 Phase diagram . . . . . . . . . . . . . . . . . 4.5.7 Molecular structure and the Kondo effect . . . 4.5.8 Kosterlitz-Thouless quantum phase transition Summary . . . . . . . . . . . . . . . . . . . . . . . .

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5 Friedel-Luttinger sum rule and Kondo effect in a triple quantum dot 5.1 Isolated triple dot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 One and five electrons . . . . . . . . . . . . . . . . . . . . . 5.1.2 Two and four electrons . . . . . . . . . . . . . . . . . . . . . 5.1.3 Correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 NRG results of correlators . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Ferromagnetic case . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Antiferromagnetic case . . . . . . . . . . . . . . . . . . . . . 5.3 General derivation of Friedel sum rule . . . . . . . . . . . . . . . . . 5.4 NRG results of conductance and Friedel Luttinger sum rule . . . . . 5.4.1 Ferromagnetic case . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Antiferromagnetic case . . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 54 54 56 58 62 62 64 66 70 71 73 75

6 Effect of assisted hopping on transport in a Kondo-correlated quantum dot 6.1 Anderson model with assisted-hopping and perturbative analysis . . 6.1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Perturbative analysis . . . . . . . . . . . . . . . . . . . . . . 6.2 Correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Spectral densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Tranport properties: thermopower and conductance . . . . . . . . . 6.4.1 Gate-voltage dependence . . . . . . . . . . . . . . . . . . . . 6.4.1.1 High-temperature regime . . . . . . . . . . . . . . 6.4.1.2 Low-temperature regime . . . . . . . . . . . . . . . 6.4.2 Temperature dependence . . . . . . . . . . . . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 78 78 79 83 86 86 87 87 91 95 98

7 Conclusion

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A List of the publications

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B Conferences

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Bibliography

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Chapter 1 Aim of the study The main goal of thesis is to study quantum entanglement, Kondo correlations and electronic transport in system of quantum dot by means of the numerical renormalization group approach. Entanglement is recognized to be a key resource in quantum information processing tasks [1, 2]. Since entanglement expresses as an exclusive quantum correlations, the concept was applied for studies phenomena in strongly correlated many-fermion systems in order to gain insight into the nature of quantum phase transitions [3], which appear as a result of competing between different ground state phases [4]. Electron transport through a quantum dot (QD) reveals that the Kondo effect is a fully entangled state in which a localized electron spin is entangled perfectly with its surrounding of conduction electron spins [5]. The situation is richer for a system with several dots where ground states with different spin arrangements are constructed according the Hund’s rules [6]. A spin of an electron in a QD and coupled spins in double quantum dots (DQD) have appeared as potential building blocks for quantum information [5]. Moreover, it has been shown that triple quantum dots (TQD) can provide additional tools and functionalities [7–10]. DiVincenzo et al. [11] proposed a scheme to measure and control states in a TQD with three spins, in which the doublet ground

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states can be controlled using purely electrically gates [7, 12]. Moreover, manipulation of spins and ground states within the scheme of Di Vincenzo are much less sensitive to decoherence processes. In turn, it was also shown that in a subspace of doublet states one can obtain an explicit relation between entanglement and spin correlation functions [13, 14]. The interaction of such a spin system with the environment is in general a complicated many-body process [15, 16]. In a case of multiple spins, there are two regimes of interest that compete with each other, one is the entanglement between localized electrons, and another one is the entanglement between electrons with their environment, i.e. the Kondo effect. The first part of the thesis presents the entanglement between three electrons confined in a triangular molecule built of coherently coupled quantum dots where one of them is connected with electrodes. The main purpose is to show how the Kondo entanglement competes with the internal entanglement for the spins inside the triangle. In fact, the coupling of TQD to the leads can induce the Kondo cloud which can switch the inter-dot spin-spin correlations due to the interplay between various Kondo-like ground states. Quantum phase transitions and different kinds of the Kondo screening are expected due to two distinct ground states, which can affect spin correlations, entanglement and charge transport measurement. The quantum phase transition should be manifested itself in a sudden jump of the entropy and the spin susceptibility. However the simplest way to detect the quantum phase transition seems to be a measurement of electrical conductance which should be different for the unentangled and entangled states. The results are expected to be different from those in the DQD configurations in which the Kondo interaction reduces the entanglement between spins in DQD [17], because for TQD one has the doublet ground states with S = 1/2 whereas in DQD the ground state is a singlet. The isolated TQD in the whole region of the electron filling is an interesting system, which can demonstrate different ground states. The electronic properties of TQD as a function of the number of electrons has been studied theoretically in

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Ref. [6]. It has shown that depending on the TQD topology and the electron filling, the isolated TQD exhibits different ground states with the total spin S = 1/2 and 1. In fact, various types of the Kondo effects and quantum phase transitions are expected to take place which are related with different spin configurations in the ground state. The question is whether the quantum phase transitions and different types of the Kondo effects can be understood in terms of correlators and conductance. The main purpose of the second part of the thesis is to consider a problem of electronic correlations and a role of many-particle states in coherent transport through the TQD system in whole range of electron fillings. In turn, a natural question arises: is there also a general formula to describe the zero-bias conductance in terms of the electron filling? The Friedel sum rule relates electron occupancy to the phase shift of scattered electrons [18]. Since the Friedel phase is connected to the density of states [19], it is also related to conductance. One of the principal aim of the second part of the thesis is to study the behavior of the phase and its corresponding conductance. In particular, I would like to understand whether the Friedel sum rule fullfills for the underscreened Kondo effect with a singular Fermi liquid ground state. It was shown that the zero-bias conductance can be expressed only in terms of the dot occupancy according to a Friedel-Luttinger sum rule, which is applicable to both screened and underscreened Kondo effects [20, 21]. In turn, I want to show how the Friedel-Luttinger sum rule captures the essential physics of the screened and underscreened Kondo phases in the TQD system. The inclusion of the assisted hopping in the single-impurity Anderson model can also lead to new physical properties in quantum impurity systems. The assisted hopping is always present in real QD systems, however it is usually ignored in theoretical modelling despite the fact that it can be relatively large [22, 23]. Recently, there has also been growing interest in the thermopower of Kondo correlated quantum dots, which has been measured [24, 25] and theoretically analyzed [26–29]. However, until now, the effect of the assisted-hopping on the thermopower for a Kondo-correlated dot has not been examined.

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The aim of the last part of thesis is to study conductance and thermopower of a QD coupled to the leads which is described by an extended single impurity Anderson model which takes into account the assisted hopping processes. More interesting, the inclusion of this term in the initial Anderson Hamiltonian breaks the particle-hole symmetry. In turn, the thermopower is very sensitive to the particle-hole asymmetry. Therefore, the thermopower can be a good tool to clarify details of the assisted hopping processes. According to the Mott formula, the thermopower is related to the slope of the local density of states at the Fermi energy. As a result, any sign change or special behavior in the thermopower provides additional information about the Abrikosov-Suhl peak which is formed in the Kondo regime. Therefore, I would like also to study the effect of the assisted hopping on the spectral function and to see changes in the the Abrikosov-Suhl peak.

Chapter 2 General introduction This chapter aims to cover the basic ideas of quantum impurity systems that are frequently used in the main body of the thesis (Chapter 4, 5 and 6). The Chapter is organized as follows. Sec. 2.1 presents quantum entanglement and its relation to spin-spin correlation functions. Sec. 2.2 explains the Kondo effect and why it is more interesting in quantum dots rather than the bulk materials. The Anderson and Kondo Hamiltonians and their relation to each other are discussed in Sec. 2.3. Sec. 2.4 and 2.5 show how the Kondo effect exhibits itself in conductance and thermopower of a quantum dot attached to metallic electrodes. The impurity quantum phase transitions are presented and discussed in Sec. 2.6.

2.1

Quantum entanglement Einstein, Podolski and Rosen (EPR) [30] argued that quantum mechanics

can not provide a complete description of the physical systems. They claimed that a pair of quantum systems, i.e. two spins A and B in the singlet state, can be described independent from their distance. EPR thought that quantum entanglement violates the speed limit of light on the transmission of information. The main problem of EPR was that two quantum spins A and B should be considered 5

Chapter 2. General introduction

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as a wave-function that describes two ”entangled” particles, instead of considering each one separately. Therefore, the distance between them do not play any role and do not violate the speed limit of light as he was concerned. John Stewart Bell [31] showed that two quantum objects can communicate faster than the speed of light. He showed that two disconnected observers are permitted to make two different measurements on their own systems. In fact, he derived an inequality that relates the expectation value of the observables. He showed that entangled states violate the inequality. This means that either quantum mechanics is not complete or the EPR assumptions are not accurate. The only way to understand which one of them is true is to experimentally test Bell’s inequality. Aspect [32] finally showed the violation of Bell’s inequality for an entangled pairs of photons, as expected for entangled particles . Quantum entanglement is a quantum correlation that comes to play because of the interaction between two or more quantum objects. It does not have any classical counterpart. The quantum states are entangled when they are not separable. States of two electron in quantum dots are separable when one can write them as a simple tensor product of two states | ↓> and | ↑>, i.e. | ↓↑>= | ↓> | ↑>. However, if it cannot be written as a tensor product of two states | ↓> and | ↑>, the state are called entangled states. One of the examples of entangled states are Bell states [33–36] 1 √ (| ↑> | ↓> ±| ↓> | ↑>) , 2 1 √ (| ↑> | ↑> ±| ↓> | ↓>) . 2

(2.1) (2.2)

The quantum entanglement between two electrons in quantum dots can be obtained analytically. In fact, the quantum entanglement of the mixed state is related to the quantity known as the concurrence [33]. The concurrence C = 0 for a separable state and C = 1 for fully entangled states, such as Bell states. It can


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be calculated from the Wootters formula [33],

C = max 0, 2λmax −

X

j=1−4

λj

!

,

(2.3)

where λj are positive square roots of the eigenvalues of the non-Hermitian matrix ρ˜ ρ in which λmax is the largest one, and ρ˜ = (σy ⊗ σy ) ρ∗ (σy ⊗ σy ) is the timereversed density matrix ρ. As an example, lets me consider a pure state of two electrons in quantum dots |Ψ >=

X µ

αµ |µ >,

(2.4)

which has been expressed in the standard basis {| ↑↑>, | ↑↓>, | ↓↑>, | ↓↓>}. The concurrence can be calculated easily from C = 2|α1 α4 − α2 α3 | and in reality it is zero for separable states ´ ↑> +´ γ | ↓> . |Ψ >= (β| ↑> +γ| ↓>) β|

(2.5)

In the remainder part of thesis only bipartite entanglement which one can quantify it by the concurrence is considered and discussed. Lets me now to study the entanglement of electron spins in quantum dots. It is possible to write the density matrix ρ of electron spins in terms of spin-spin correlation functions without taking into account information of the wave functions. In some special case, it is also possible to calculate analytically concurrence of electron spins. For example, if the density operator and the square of the total spin projection operator, S z = SAz + SBz , commutes with each other [ρ, (S z )2 ] = 0, the matrix ρ is a block form such that ρ12 = ρ13 = ρ24 = ρ34 = 0 or, equivalently, E D D E x,y x,y z SA(B) = 0 and SA(B) SB(A) = 0 when all of the operators has been expressed

in the standard basis. In this situation, the reduced density matrix in the standard


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basis is expressed as [37–39] D E

− −  ↑ ↑ 0 0 SA SB   PA PB D E

  ↑ ↓ − +   0 P P S S 0 A B A B   D E ρ= .

+ − ↓ ↑   0 SA SB PA PB 0   E D 

SA+ SB+ 0 0 PA↓ PB↓

(2.6)

where Si+ = (Si− )† is the electron spin raising operator for dot i = A, B. Piσ = |σi ihσi | = niσ (1 − niσ ) is the projection operator onto the subspace where dot i is singly occupied with one electron and with the spin σ. Applying the Wootters formula (2.3) for the above density matrix (2.6), the concurrence of two electron spins in quantum dots can then be written as [38] max(0, C↑↓ , C|| ) , P↑↓ + P|| rD ED E

+ − = 2| SA SB | − 2 PA↑ PB↑ PA↓ PB↓ , rD ED E

+ + PA↑ PB↓ = 2| SA SB | − 2 PA↓ PB↑ ,

CAB = C↑↓ C||

(2.7) (2.8) (2.9)

D E D E P|| = PA↑ PB↑ + PA↓ PB↓ and P↑↓ = PA↑ PB↓ + PA↓ PB↑ are probabilities for the spins

to be aligned in the same (parallel) and opposite (anti-parallel) directions, respectively.

For axially symmetric systems, one has both hSAx i = 0 and hSBy i = 0 (conserving the total spin projection). The feature that distinguishes such a system from the general case is that the z component of the total spin Sz is conserved, the conservation of magnetization, and [ρ, Sz ] = 0,

(2.10)

Therefore, for an axially symmetric system hSA+ SB+ i is equal to zero, which makes C|| always negative and irrelevant in concurrence formula. Therefore, the formula


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(2.7) simplifies as

max 0, 2| CAB =

SA+ SB−

rD ED E ↑ ↓ ↓ ↑ PA PB PA PB |−2

P↑↓ + P||

,

(2.11)

where P ↑ = (1 + Sz ) /2, P ↓ = (1 − Sz ) /2, and S ± = (S x ± iS y ) /2. In terms of these operators, the correlators become D D

E 1 hS z − SBz i , PA↓ PB↑ = − hSAz SBz i + A 4 2

(2.12)

hS z − SBz i 1 − hSAz SBz i − A , 4 2

(2.13)

PA↑ PB↓

E

=

SA+ SB− = hSAx SBx i + hSAy SBy i − i (hSAx SBy i − hSAy SBx i) .

(2.14)

For states with translational invariance, i.e., x = y one has hSAz i = hSBz i, and E E D D PA↓ PB↑ = PA↓ PB↑ . Substituting the above correlators into Eq. (2.11), the concurrence to be CAB

−2hSAx SBx i − 2hSAy SBy i − 2 (1/4 − hSAz SBz i) = max 0, , P↑↓ + P||

(2.15)

and since hSA · SB i = hSAx SBx i + hSAy SBy i + hSAz SBz i,

(2.16)

the concurrence simplifies further to CAB

−2hSA · SB i − 1/2 = max 0, P↑↓ + P||

.

(2.17)

There are two different interesting regimes in the case of a pure spin system, where P↑↓ + P|| = 1. On one hand, the concurrence is zero for −2hSA · SB i − 1/2 ≤ 0 (i.e. hSA · SB i ≥ −1/4). On the other hand, the concurrence is non-zero for −2hSA · SB i − 1/2 > 0 (i.e. hSA · SB i ≤ −1/4). The above formula also shows


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that the concurrence CAB = 1 for singlet state hSA · SB i = −3/4, and CAB = 0 for triplet state hSA · SB i = 1/4. In general, the spin-spin correlation function of two spin system is bounded as −3/4 ≤ hSA · SB i ≤ 1/4. Therefore, the concurrence is important for the spin-spin correlation function in the range −3/4 ≤ hSA · SB i ≤ −1/4 and otherwise is zero. In my thesis, I am interested in the entanglement of an electron pair extracted from a many-body state, which can be, for example, an open system of interacting electrons in a solid-state structure of several coupled quantum dots. In the first part of the thesis, I therefore study entanglement of three electrons confined in a triple quantum dot which interact with a metallic environment.

2.2

Kondo effect From the quantum entanglement point of view, the Kondo effect is a fully

entangled state in which a localized electron spin is entangled perfectly with its surrounding of conduction electron spins [40]. I have been intrigued by the Kondo effect in part because understanding how a localized electron can be entangled with its environment could help to overcome problems in quantum information. Magnetic impurities in a bulk metal can considerably change the transport and physical characteristics of an otherwise pure metal at sufficiently low temperature [18]. This phenomenon is known as the Kondo effect. The Kondo effect is due to strong interactions of magnetic impurity spin with the sea of conduction electron spins. At temperature lower than the Kondo temperature, a singlet entangled state forms between a single magnetic impurity spin and the conduction electron spins. It has been theoretically proposed that the Kondo singlet can also form between an unpaired electron in a quantum dot and the surrounding electrodes [41–44]. A decade ago, the Kondo effect was observed in a quantum dot [45], which causes the revival of studying Kondo physics [46]. For transport in a quantum dot, all the electrons have to travel through the dot because there is no any other electrical path to go to the other electrode. In contrast in a bulk metal


11

magnetic impurities scatter the electrons which leads to increase of resistance in low temperatures [46]. In principle, the quantum dot system shows a number of important advantages for studying the Kondo effect: (i) The relevant parameters are easily controllable by gate potentials in contrast to dilute concentration of magnetic impurities in bulk materials [47, 48]; (ii) It is also possible to study the Kondo effect out of equilibrium, i.e. when the electric current is flowing through the dot [49]; (iii) One can study the Kondo effect when the electrodes are ferromagnetic [50–53], superconducting [54], and topological insulator [55]; (iv) It is possible to study the Kondo effect in more complex systems (with many quantum dots) such they can mimic multi-magnetic impurities [35]; (v) Different kinds of the Kondo effects can be studied with various ground states as regular Fermi-liquid, singular Fermiliquid [56] and non-Fermi-liquid [57–59]; (vi) Quantum phase transitions can be studied easily in the Kondo regimes [60]; (vii) Interplay of the Kondo effect with interference processes such as a Fano resonance can be observed [49, 61–63]. In the first part of my thesis, I study a system of three tunnel-coupled semiconductor quantum dots in a ring geometry, one of which is connected to a metallic lead, in the regime where each dot is essentially singly occupied. It is focused on a triple quantum dots with three electrons in the Coulomb blockade valley and study its evolution as a function of the interdot tunnel couplings. In this situation, there is a competition between interdot correlation with the Kondo correlation. In the second part of my thesis, the dependence on the gate-voltage which naturally is more complex and richer has been examined. Various kinds of the Kondo effects take place in such a system depending on the electron filling. It is mainly focused on the possibility of underscreened S = 1 Kondo effect with a singular Fermi liquid ground state. The last part of thesis presents conductance and thermopower of a quantum dot coupled to the electrodes which is described by an extended single impurity Anderson model which takes into account the assisted hopping processes, i.e. the occupancy-dependence of the tunneling rates.


2.3

12

Anderson impurity model The Anderson impurity model [64] was introduce as a microscopic model in

the sixties of the past century to describe the the quantum impurity systems in which certain magnetic ions are embedded in a non-magnetic bulk metal. Recently the model was also applied to describe the Kondo effect in quantum dots [42]. Anderson assumed that the conduction band is a broad band and conduction electrons are noninteracting. The model takes into account just the low energy excitations related to the impurity and ignores interactions which are not directly related to impurity effects. The Anderson impurity Hamiltonian is expressed as [18, 26, 65] H=

X k,σ

ǫk c†kσ ckσ +

X σ

ǫ fσ† fσ + U n↑ n↓ +

X

(Vk c†αkσ fσ + h.c.).

(2.18)

k,σ

The first term in the Hamiltonian shows the kinetic energy of the conduction electrons which are labelled by their momentum and spin. The second term in the Hamiltonian is the quantized energy of electrons in the single magnetic impurity which is a spin-degenerate ground state. All of the other magnetic impurity levels are supposed to be either full of electron when it is below the Fermi energy ǫF or empty of electron when it is above the Fermi energy ǫF . The third term demonstrates the interactions between localized electrons. U is the energy that one has to pay for putting a second electron with the opposite spin direction into a singly occupied impurity state, i.e. once an energy state with energy ǫ is filled with one electron, adding another electron costs ǫ + U . The last term describes the hybridization Vk between the impurity and the conduction band. It shows spin-conserving tunneling on and off the magnetic impurity. Another way of studying the quantum impurity systems is to use the Kondo Hamiltonian. Three years after Anderson [64] proposed his model, Jun Kondo [66] showed that a resonant spin-flip scattering process is responsible for increasing electrical resistivity with lowering temperature in metals containing diluted


13

magnetic impurities. The Kondo Hamiltonian is given by H=

X

ǫk c†kσ ckσ +

k,σ

X

´ σ k,k,σ,´

JK S · c†kσ σσ´σ ck´ ´σ .

(2.19)

The last term describes spin-flip processes of conducting electrons on the localized magnetic impurity spin S with some exchange coupling JK . The Kondo Hamiltonian can be constructed straightforwardly from the Anderson Hamiltonian by means of the Schrieffer-Wolff transformation as long as U is sufficiently large [18, 67]. The Schrieffer-Wolff transformation is a unitary transformation [68] in which one can include relevant low energy excitations of a physical system. A lot of methods have been used to solve the Anderson and Kondo Hamiltonians. Phillip Anderson [64] developed a perturbative renormalization group method which is known as Poor Man’s Scaling. This method includes perturbatively removing excitations to the edges of the noninteracting band, and shows that with decreasing the temperature the effective coupling between the local moment and the conduction band rises without any limitation. Since the method is perturbative, it is not reliable when JK is large. Therefore, the method did not correctly capture the Kondo problem. The problem was finally solved by Kenneth Wilson [69–71] who applied the numerical renormalization group method. He showed numerically for the first time that the resistivity goes to a constant value with lowering the temperature. The next chapter is devoted to explain the numerical renormalization group method in details.

2.4

Conductance This section shows how the Kondo effect manifests itself in conductance of a

quantum dot system. One of the characteristic signature of the Kondo effect in a quantum dot system is an increase of conductance with lowering the temperature at zero-bias voltage limit.


14

It was predicted theoretically in 1988 that a quantum dot embedded between the two metallic leads mimics the Kondo effect in bulk materials [42, 72]. Ten years later the Kondo effect in a quantum dot system was first observed experimentally by Goldhaber et al. [73]. They showed that the conductance is reduced with lowering the temperature for even number of electrons in the dot. This characteristic showed that there is no the Kondo effect when the number of electrons is even. However, for odd number of electrons they observed the Kondo effect in which the conductance increases and reaches its unitary limit G0 = 2e2 /h [74] in low temperatures. I explain here the conductance behavior of a quantum dot in the Kondo regime. The quantum dot originally comprises of two potential barriers and a Coulomb intra-dot repulsion energy, U . The Coulomb repulsion tries to suppress the transport of electrons into or out of the dot. In this situation, the energy level of the dot is out of resonance with the metallic leads and transport through the dot is suppressed. Nevertheless, the higher-order spin-flip processes [66] which lead to the Kondo effect totally change the situation and raise the conductance until it reaches its unitary limit 2e2 /h [74]. Definitely, the conductance reaching its unitary limit shows that electron transmission is perfect [42]. In the linear response limit and when the source-drain voltage goes to zero, the conductance can be written as [74, 75] 2e2 G= h

Z

dωT (ω)[−

∂f ], ∂ω

(2.20)

where T (ω) is the transmission probability for an electron with energy ω, and f is the Fermi distribution function that describes the thermal distribution of electrons in the metallic leads. At low temperatures, the Fermi distribution function is a step function and therefore G = 2e2 T (ǫF )/h. Since in the Kondo regime T (ǫF ) = 1, the conductance reaches its quantum limit G0 = 2e2 /h [42].


2.5

15

Thermopower Additional information about a Kondo-correlated quantum dot can be ob-

tained by the thermopower which is sensitive to the particle–hole asymmetry [26]. I therefore discuss here the concept of thermopower in nanoscale junctions. The configuration is a junction composed of two electrodes which has been connected by a quantum dot. The thermopower (Seebeck coefficient S) measures the induced voltage in response to the temperature differences [76]. If the temperature difference ∆T between the two electrodes is small, the thermopower of the system can be written as follows [76, 76–78] S=−

∆V , ∆T

(2.21)

where ∆V is the thermoelectric voltage has been observed at the electrodes when the total current is zero. In the linear response limit, the thermopower of a quantum dot system in a coherent regime is given by [26, 76] 1 S(ω, T ) = − eT with a Fermi energy µ.

R

dωT (ω)(ω − µ)[−∂f /∂ω] R , dωT (ω)[−∂f /∂ω]

(2.22)

The above formula for the thermopower (at temperature T and Fermi energy ǫF ) can be written by the following Sommerfeld expansion [28, 76] 2 π 2 kB T S(ǫF , T ) = − 3e

2 π 2 kB T ∂ 3 G(ω, T ) 1 ∂G(ω, T ) + + ... G(ω, T ) ∂ω 15G(ω, T ) ∂ω 3

(2.23) , ω=ǫF

where kB is the Boltzmann constant. The well-known semiclassical Mott formula [79] is the expansion of the above formula up to the first order. The formula relates thermopower to the slope (asymmetry) of conductance at the Fermi energy. The conductance is furthermore proportional to the density of states at the Fermi


16

level: limT →0 G (ω, T ) ∝ ρd (ω = ǫF ). [80, 81]. Therefore, the Mott formula gives us information about the structure of density of state and additional information about the electronic transport. The Kondo effect shows up as an additional peak, Abrikosov-Suhl peak, in the spectral function close to the Fermi level ǫF when the temperature decreases in equilibrium situation. Moreover, thermopower is determined by the slope of spectral function at the Fermi level ǫF , according to the Mott formula. Therefore, the variation of thermopower in the Kondo regime is related to changes in the Abrikosov-Suhl peak. In fact, thermopower is a powerful tool to detect the appearance of the Kondo effect when the system is cooled down below the Kondo temperature. The last part of thesis, chapter 6, presents the thermopower of a quantum dot system with assisted hopping, occupancy-dependence of the tunneling.

2.6

Quantum phase transition Quantum phase transition is a transition between different ground states of

a mater at absolute zero temperature as one changes a parameter rather than temperature [60, 82]. Classical phase transitions is usually triggered due to thermal fluctuations. However, by lowering the temperature, thermal fluctuations are reduced and finally they are stop when temperature goes to zero. With reducing the thermal fluctuations, quantum fluctuations start to play an important role. Quantum phase transition can usually be classified as first- or second-order phase transitions [83, 84]. First-order phase transitions display a jump in the first derivative of the free energy versus some thermodynamic quantities. Second-order phase transitions exhibit a continuous change in the first derivative of the free energy across the transition point. However, second-order phase transitions show a discontinuity and jump in a second derivative of the free energy. Another type


17

of quantum phase transition is the Kosterlitz–Thouless transition, which is characterized by logarithmic (exponentially) rather than power-law behavior of thermodynamic quantities near the transition point, i.e., in spin correlations [85, 86]. The Kosterlitz-Thouless transition does not show a singularity in derivative of the thermodynamic quantities at the transition point [87]. Throughout my thesis, I deal with the first-order and the Kosterlitz–Thouless quantum phase transitions. Quantum phase transition in a quantum dot has approximately the same behavior as a bulk material. To study a quantum phase transition in bulk materials, one needs to change precisely the magnetic impurity concentration in the samples, which makes it hard for experimentalist to study [88]. In contrast to bulk materials, the parameters within the quantum dot can be tuned independently from other parameters. It is therefore easy to obtain the different quantum phases, the quantum critical point, and the quantum phase transitions [65]. An electron within a quantum dot is entangled with its surrounding sea of conduction electrons in the Kondo regime. In the multi-dot cases, different ground states (or inter-dot and intra-dot entanglements) compete with each other. Therefore, it is useful to study the quantum entanglement of quantum dots in the presence of its environment. Quantum phase transition which gives information about a change in the ground state of a many-body system provides new view into understanding the Kondo effect and quantum entanglement [16]. Throughout my thesis the relation between quantum entanglement and quantum phase transition as well as quantum transport in a Kondo-correlated quantum dot is studied in details. In this thesis, I study the quantum phase transitions in a triple quantum dot system, in which only one dot is coupled to the metallic leads. For such a system, two distinct phases arise in general: one associated with fully screened S = 1/2 Kondo regime and the other with partially screened S = 1 Kondo regime. The quantum phase transitions for two different cases has been studied: In one case, I have considered the system when there is approximately one electron in each dot. A first-order quantum phase transition between two distinct ground state arises


18

on tuning the interdot tunnel couplings. It is also shown that system undergoes Kosterlitz-Thouless quantum phase transition with broken symmetry of the dot couplings. In another case, I have studied the quantum phase transitions driven by varying the gate-voltage in the whole region of the electron filling. In this case, the system undergoes a first-order quantum phase transition between different ground states. Friedel sum rule, which relates the dot occupancy to the conductance [18], is also used to describe qualitatively the quantum phase transition.

Chapter 3 Numerical renormalization group method The purpose of this chapter is to give a brief introduction to the numerical renormalization group (NRG) method, as well as some guidelines for calculating physical quantities. At the end I also discuss the NRG code established in Ljubljana.

3.1

Numerical renormalization group method Any theoretical method to study a quantum impurity problem encounters

a lot of obstacles [18, 65]. On one hand, one has to take into account a broad range of energies from a high-energy cut-off down to small excitation energies in order to solve a quantum impurity problem, because the environment has usually a continuum degrees of freedom. On the other hand, the coupling of a magnetic impurity to a continuum of excitations with small energies can causes divergences in perturbation studies, because the impurity degrees of freedom ordinarily form an interacting quantum-mechanical system due to an onsite Coulomb repulsion. The problem is thus related on facing to an interacting strongly correlated system with a continuum of excitations containing a wide energy spectrum. Therefore, 19

Chapter 3. Numerical renormalization group method

20

one needs to solve this problem nonperturbatively. NRG [65, 69–71] is a professional method to solve a quantum impurity problem containing such a broad and continuous energy spectrum. NRG is a method in which the idea of poor man’s scaling [89] is implemented into the numerical diagonalization procedure, developed by Kenneth Wilson [69] to solve a quantum impurity problem. It is a non-perturbative method, which was first showed that a ground state of a single impurity is a singlet state. NRG method showed the crossover from the high-temperature local moment regime of a decoupled free spin to the low-temperature strong coupling regime where the spin is screened entirely. Wilson won the Nobel Prize in 1982 for implemention this method in the quantum impurity problem. Later Krishnamurthy and Wilson [70, 71] extended the NRG method to deal with the Anderson model to study Kondo physics not only in the strongly correlated regime but also in the valence fluctuation regime. The NRG method was later used in a lot of quantum impurity problems. An overall strategy and steps performed in the NRG calculation are explained in the following.

3.1.1

Reduction to a one-dimensional problem

I consider the single impurity Anderson Hamiltonian mentioned in the chapter 2. The effect of the bath on the impurity is determined by a hybridization function: ∆ (ω) = π

X k

Vk2 δ (ω − ǫk ) .

(3.1)

One can rewrite the Anderson Hamiltonian such that the involved manipulations do not vary the form of ∆ (ω). ∆ (ω) lies within the interval [−D, D], where D is half of the bandwidth. Later D = 1 is considered as the energy unit. It was shown that after some possible reformulation the Anderson Hamiltonian can be rewritten as [90] H = Himp +

XZ σ

−1 1

dǫg

(ǫ) a†ǫσ aǫσ

+

XZ σ

−1 1

dǫh (ǫ) fσ† aǫσ + a†ǫσ fσ .

(3.2)

Chapter 3. Numerical renormalization group method where Himp =

P

† σ ǫ0 f σ f σ

21

+ U n↑ n↓ describes the isolated impurity. In the above

Hamiltonian, the fermionic operators a†ǫσ correspond to band states with spin σ and a dispersion g (ǫ), and h (ǫ) is hybridization between two subsystems. The g (ǫ) and h (ǫ) are related to the hybridization function ∆ (ω) according to [90] ∆ (ω) = π

dǫ (ω) h [ǫ (ω)]2 , dω

(3.3)

where ǫ (ω) denotes the inverse function to g (ǫ), g [ǫ (ω)] = ω. Krishnamurthy et al. [70, 71] showed that in Eq. (3.2) only the spherical s waves couple to the impurity and the conduction states with other symmetries (p, d, ...) are irrelevant.

3.1.2

Logarithmic discretization of the conduction band

The one-dimensional conduction band in Eq. (3.2) contains a lot of degrees of freedom. For numerical calculations, one should decrease the number of states to a controllable amount. For this reason, the conduction bands should be discretized logarithmically. In this situation, the conduction band is divided into a set of intervals with discretization xn = ±Λ−n , with the discretization parameter Λ, an

integer number n, and the intervals width dn = Λ−n (1 − Λ−1 ). Within each inter√ ± val corresponding orthonormal wave functions are ψnp (ǫ) = exp (±iωn pǫ) / dn . The index p, the Fourier harmonic index, considers all of the integer values from −∞ to +∞, and ωn = 2π/dn is a frequencies associated to each interval. In the next step, one needs to expand the conduction electron operators aǫσ in this basis, i.e., aǫσ =

X np

+ − anpσ ψnp (ǫ) + bnpσ ψnp (ǫ) ,

(3.4)


22

which is related to a Fourier expansion in each of the intervals. The inverse transformation gives anpσ =

bnpσ =

Z

Z

−1 1

−1 1

+ ∗ dǫ ψnp (ǫ) aǫσ ,

(3.5)

− ∗ dǫ ψnp (ǫ) aǫσ .

(3.6)

Eq. (3.2) can now be rewritten in terms of these discrete operators anpσ and bnpσ . The p 6= 0 components of the conduction band states couple only indirectly to the impurity via their coupling to the p = 0 states with a prefactor (1 − Λ−1 ),

which vanishes when Λ goes to 1. In this sense, the impurity couples just to the p = 0 states. One can therefore drop the p 6= 0 terms and relabel the operators an0σ = anσ to obtain a discretized Hamiltonian H =Himp +

X

ξn+ a†nσ anσ + ξn− b†nσ bnσ

nσ

1 X †X + +√ γn anσ + γn− bnσ fσ π σ n ! 1 X X + † +√ γn anσ + γn− b†nσ fσ , π σ n

(3.7)

R ±,n R ±,n R ±,n where γn± = dǫ∆ (ǫ) and ξn± = dǫ∆ (ǫ) ǫ/ dǫ∆ (ǫ), with a definition R −,n R −xn+1 R +,n R xn dǫ = −xn dǫ. written as dǫ = xn+1 dǫ and

3.1.3

Mapping on a semi-infinite chain

The next step is now to transform the discretized Hamiltonian Eq. (3.7) into a form of a semi-infinite chain of a tight-binding Hamiltonian, in which each site couples just to its nearest neighbors with an exponentially reducing matrix elements. In this Hamiltonian, the magnetic impurity couples to an electron with


23

an operator c†0σ expressed as c0σ =

1 X + γn anσ + γn− bnσ , ξ0 n

where the normalization constant is ξ0 =

(3.8)

P + 2 − 2 (γ ) + (γ ) . Constructing new n n n

operators, the Hamiltonian in Eq. (3.7) can be written as follow r

ξ0 X † fσ c0σ + c†0σ fσ π σ i Xh + ǫn c†nσ cnσ + tn c†nσ cn+1σ + c†n+1σ cnσ ,

H =Himp +

(3.9)

σn

where c†nσ denotes a creation operator of a conduction electron at the nth site within the semi-infinite chain. ǫn and tn ≈ Λ−n/2 show the on-site energy and the hopping. The above equation is a special form of the single impurity Anderson model expressed as a semi-infinite chain Hamiltonian such that the hopping tn fall off exponentially.

3.1.4

RG transformation

To obtain the eigenvalues and eigenvectors of the chain Hamiltonian Eq. (3.9), one needs to diagonalize it. In this situation, an iterative renormalization group (RG) procedure can be applied. The RG transformation relates the effective Hamiltonians on energy scales Λ−n/2 and Λ−(n+1)/2 . The chain Hamiltonian can be expressed as H = lim Λ−(N −1)/2 HN ,

(3.10)

N →∞

where HN =Λ

(N −1)/2

+

N X σn

h

Himp +

ǫn c†nσ cnσ

r

+

ξ0 X † † fσ c0σ + c0σ fσ π σ

N −1 X σn

tn

c†nσ cn+1σ

+

c†n+1σ cnσ

i .

(3.11)


24

The factor Λ(N −1)/2 in the above equations are selected to cancel the N-dependence of tN −1 . The Hamiltonians between two intervals are related by X √ HN +1 = ΛHN + ΛN/2 ǫN +1 c†N +1σ cN +1σ σ

+ ΛN/2

X σ

tN c†N σ cN +1σ + c†N +1σ cN σ .

(3.12)

The above equation can now be understood in terms of the RG transformation HN +1 = R (HN ). This helps to construct the eigenvalues and eigenvectors of the Hamiltonian.

3.1.5

Iterative diagonalization

The RG transformation can be applied to diagonalize the Hamiltonian HN iteratively and obtain the energies EN . One first needs to construct a matrix for the Hamiltonian H0 , which is a two-site chain consisting of the impurity and the first conduction electron site, that can simply be diagonalized numerically. In the next step, one site is added and coupled to the previous iteration procedure, and then the matrix is diagonalized. This is related to take into account excitations on a reduced energy scale due to to the fact that the coupling falls off exponentially and is proportional to Λ−N/2 . For diagonalized Hamiltonian HN the basis for HN +1 are expressed as |r; siN +1 = |riN ⊗ |s (N + 1)i ,

(3.13)

where |riN is the eigenstates of HN , and |s (N + 1)i = {|0i , |↑i , |↓i , |↑↓i} is a basis associated to the added site. Using the above basis (3.13), the matrix elements of the Hamiltonian of the Wilson chain with N + 1 sites are given by HN +1 (rs, r´s´) =N +1 hr, s| HN +1 |´ r, síN +1 .

(3.14)


25

Now one repeats the diagonalization procedures for HN +1 . Since the number of many-body states in HN grows by a factor of 4N , the resulting matrix becomes very big after a few iterations. For this reason, one needs to truncate the number of states in each iteration keeping states with the lowest energies. Taking more states one gets better accuracy but it needs more computational time.

3.2

Calculation of physical properties The main applications and goals of NRG for a quantum impurity problem is

to calculate thermodynamics, statics and dynamics properties.

3.2.1

Thermodynamic and static properties

In numerical calculations, one can calculate the partition functions ZN for the sequence of truncated Hamiltonian HN as ZN (T ) =

X

e−HN /kB T =

X

N /k

e−Er

BT

,

(3.15)

r

with ZN (T ) ≈ Z (T ) when N goes to infinity. The entropy S and specific heat C can be obtained from the free energy F = −kB T ln Z and the internal energy U = hHi, ∂F , ∂T ∂U C= . ∂T

S=−

(3.16) (3.17)

The static susceptibility of the system is χ (T ) =

Z

kB T 0

hSz (τ )Sz (0)i dτ − kB T hSz i2 ,

(3.18)


26

where τ is an imaginary time (0 < τ < kB T ), Sz the z component of the impurity spin operator, and hSz (τ )Sz (0)i = T r eH/kB T eτ H Sz e−τ H Sz /Z. All of the above

static properties are for the total system which consists both the magnetic impurity

and conduction electrons. In order to obtain only the static properties of the magnetic impurity, one needs to calculate the static properties of the system with and without magnetic impurity and then subtract them from each others.

3.2.2

Dynamic properties

Another NRG application is to calculate dynamic properties of a quantum impurity problem. If all the eigenstates |ri and eigenvalues Er of the Anderson Hamiltonian are known, the density matrix ρ (T ) of the system at temperature T = 1/(kB β) can be expressed as ρ (T ) =

1 X −βEr |ri hr| , e Z (T )

(3.19)

and the spectral density of magnetic impurity can be given in the Lehmann representation as [28, 65] Aσ (ω, T ) =

1 X |Mr,´r |2 e−Er /kB T + e−Er´/kB T δ (ω − (Er´ − Er )) , Z (T ) r,´r

(3.20)

where Mr,´r = hr|fσ |´ ri is the relevant matrix elements. Here, r and r´ index the many-particle levels |ri and |´ ri with the energies Er and Er´ , respectively. One now can calculate transport properties from the spectral density. For instance, the conductance of a quantum dot is Z ∂f 2e2 X Aσ (ω, T ) . dω − G (T ) = h σ ∂ω with Fermi distribution function f .

(3.21)


3.3

27

NRG Ljubljana code All of the NRG codes use the same steps and strategy to solve quantum

impurity problems. In my thesis, I use NRG Ljubljana which is an open source numerical renormalization group code [91]. ”NRG Ljubljana” is a set of interconnected computer codes to perform NRG calculations for quantum impurity problems with the possibility to generalize it for multi-impurity and multi-channel problems as well as different kinds of quantum impurity problems. In general, adapting the NRG package for different problems to compute physical quantities is possible. It contains a lot of useful tools for analyzing the numerical results such as thermodynamic properties, magnetic and charge susceptibility, entropy and heat capacity, spectral functions and etc. I use free softwares such as Gnuplot and Xmgrace for easy interpretation and plotting the numerical output. I also use Perl and Python programming to write a script for manipulating the numerical results. ”NRG Ljubljana” is constructed to perform NRG calculations and reach an extraordinary degree of flexibility. For this reason, low level numerically concentrated parts of the code has been written in the C++ language, whereas the high level numerical code is programmed in a combination of functional and technical Mathematica code. The basis of the code is a SNEG Mathematica package [92] to perform numerical calculations for non-commutative second quantization operators. SNEG offers the natural notation for its output, and syntactically different but semantically the same techniques to enter the input expressions. In addition to simplifying the input for numerical codes, the SNEG package is also useful for executing some calculations and simplifications directly such as second-quantization operator commutators. SNEG also contains a lot of transformation rules that describe not only the algebra of operators but also a complete library of utility functions. SNEG library is the establishment of NRG Ljubljana code and can be used as a package for the second-quantization operators. The most important application of SNEG


28

library in the NRG package is to make a connection between the user and the numerical codes. This allows a strong separation among the coded in C++ and the coded in Mathematica and SNEG, not only for performance and efficiency but also particularly for maintainability of the NRG code. A next layer of the NRG code is a Mathematica package which describes a Hamiltonian, a basis of states, and physical projectors: with the assistance of SNEG package. The Hamiltonian and the projectors can be defined appropriately by means of well-known secondquantization expressions. This program sets up the Hamiltonian and makes ready the input for the NRG iteration calculations. For a better efficiency, the NRG iteration executes a special C++ program. Moreover, most of the time (nearly 90 percent) is spent in the LAPACK dsyev and dsyevr libraries to calculate eigenvalues and eignevectors of the Hamiltonian. The NRG code is appropriate for accomplishment of large scale NRG calculations on a computer cluster. I have implemented and installed NRG Ljubljana code on two different computer clusters in the Institute of Molecular Physics.

Chapter 4 Entanglement, the Kondo effect, and quantum phase transition in a triple quantum dot This chapter presents studies of competitions between the Kondo entanglement and the entanglements of three electrons confined in a triple quantum dot with a triangular geometry, in which one of the dots is attached to two non-interacting metallic electrodes. I also study the relation between quantum entanglement and charge transport as well as thermodynamic properties. At a temperature T lower than the Kondo temperature TK , different kind of Kondo effects between the dots and their environment take place. The possibility of quantum phase transitions between different Kondo ground state in the system is studied as well. In this situation, there is a quantum phase transition between the fully screened Kondo effect and underscreened Kondo effect with changing either the inter-dot coupling between side-coupled dots or the coupling between dots with the electrodes. The Chapter is organized as follows. Sec. 4.1 presents the main purpose of the chapter, which is to study the triple quantum dot with three electrons in the presence of a metallic environment. The system is modeled in Sec. 4.2. 29

Chapter 4. Entanglement and Kondo effect

30

The relation between the concurrence and spin-spin correlation functions in the thermal equilibrium and the absence of magnetic field is discussed in sections 4.3. In Sec. 4.4, the ground states of three spins and their corresponding concurrences are presented and discussed in details. The TQD system is also modeled with an effective four spin system in Sec. 4.5. The numerical results of the chapter are investigated and discussed in Sec. 4.6. It is shown how the the entanglement is related to thermodynamic and transport properties of the system. A special emphasis is put on entanglement between spins in the dots. It should be noted that this chapter presents the argumentation published in The European Physical Journal B 87, 145 (2014) [35], and some part of The European Physical Journal B 86, 1 (2013) [14].

4.1

Spins in triple quantum dots I here give some arguments on entanglement of electron spins in quantum

dots. In general, one of the main problems of quantum information in nanostructures is to generate and manipulate entangled states. To induce entanglement between electron spins in two isolated qubits, there are many possible ways. In experiments, however, the main problem is to keep the states entangled even in the presence of their environment, specially due to the charge fluctuations. For two electrons confined in a double quantum dot (DQD), it was found that the entanglement don’t change even in the presence of charge fluctuations when the temperature is sufficiently lower than the effective superexchange interaction between the dots [17, 34]. However, in the presence of metallic electrodes the formation of a Kondo singlet state is responsible for the change of entanglement between electron spins in the dots. More recently, it was found that in an isolated TQD with three localized electrons entanglement can be created from the doublet ground states [14]. The mixing between the states with an opposite spin direction always change the entanglement. Furthermore, broken the symmetry of TQD with an external electric


31

field can generate fully entangled states which can survive in a relatively wide temperature range. This can lead to create a robust pairwise entanglement generation at elevated temperatures. The main purpose here is to study the entanglement of such a TQD system coupled to two metallic electrodes, see Fig. 4.1. Whereas a similar investigation for DQD has found that at low temperatures and low magnetic field the Kondo effect suppresses the spin entanglement [17, 34], however here the Kondo effect can generate maximally entangled states. A generalize form of the concurrence can be obtained by the reduced density matrix using the well-known Wootters formula [93]. Since, there are significant charge fluctuations in the system, therefore the state may not be presented by the simple spin degrees of freedom. In practice, such charge fluctuations and the coupling to the environment are unavoidable. It is also intended to show how breaking the triangular symmetry of the system can change not only super-exchange interactions but also entanglement of electron spins. On the other hand, the coupling to the metallic electrodes can induce the Kondo effect which can change the inter-dot spin-spin correlations. I also want to show how the Kondo effect can destroy or induce entanglement between electron spins in the dots. The results are different from the DQD configurations in which the Kondo effect reduces the entanglement between the electron spins in the dots [17]. I calculate the concurrence to quantify quantum entanglement. The concurrence is expressed in terms of the spin-spin correlation functions.

4.2

Modeling of triple quantum dot system A triple quantum dot qubit system is considered in which one of the dot is at-

tached to two electrodes with non-interacting electrons, as is presented in Fig. 4.1. The system is modeled and studied by a three-impurity Anderson Hamiltonian:


t2

A

32

B t3

t1 C

L ΓL

R ΓR

Figure 4.1 Triple quantum dot system attached to the electrodes.

H = Hd + Hlead + Ht . The Hamiltonian for isolated three-coupled dots is given by Hd =

X i

ǫi d†i di + U

X i

† † † ni↑ ni↓ + t1 fAσ fCσ + t2 fAσ fBσ + t3 fBσ fCσ + h.c. , (4.1)

† where niσ = fiσ fiσ is the number operator for an electron with the spin σ at the

dot i ∈ {A, B, C}. An energy level of each dot is ǫi = ǫ; U shows the Coulomb interaction between two electrons in the same dot; and t1 , t2 and t3 are the interdot hopping. The dot C is coupled to the electrodes which is described by Ht =

X

(Vα c†αkσ fCσ + h.c.),

(4.2)

α,k,σ

with the tunnel matrix element Vα between the dot C and the electrodes, which is assumed to be momentum independent and real. This is related to considering that the conduction band states are in spherical harmonics around the impurity and the s-wave states play a main role in the scattering process of an electron from the impurity. Non-interacting electrons in the electrodes are described by the following Hamiltonian Hlead =

X

α,k,σ

(ǫk c†αkσ cαkσ + h.c.),

(4.3)


33

here ǫk is an electron spectrum for electrons with the wave-vector k. Additionally it is assumed that the electrodes have a constant flat density of states ρ = 1/(2D), with the half-bandwidth D = 1. The hybridization strength can be expressed as Γ = πρ (|VL |2 + |VR |2 ). The chapter is devoted to symmetric couplings such that VL = VR = V .

4.3

States for three spins The Kondo effect occurs when there is a spin with twofold degenerate ground

state which is coupled to a bath of electrons. Therefore, before studying the Kondo effect of the TQD system, it is worth to describe the ground state of the isolated TQD. To do this one needs to transform the Hubbard Hamiltonian Hd , Eq. (5.1), to an effective Heisenberg Hamiltonian. Such a transformation is justified because the on-site Coulomb interaction U is bigger than the inter-dot couplings and the probability of double occupancy in a dot is very small. The effective Hamiltonian reads [13, 14, 94] Hd,ef f = J1 SC · SA + J2 SA · SB + J3 SB · SC −

1 (J1 + J2 + J3 ) , 4

(4.4)

with the super-exchange coupling J1,2,3 = 4t21,2,3 /U . It is also worth to notice that throughout this chapter the last term in the above Hamiltonian is ignored. This term only shifts all the energy levels and does not change the physical properties of the system. The energy level spectrum of the TQD consists of quadruplet spin states, 3/2 1/2

QSz , with total spin S = 3/2, Sz = {±3/2, ±1/2}, and doublet spin states, DSz , with total spin S = 1/2, Sz = ±1/2 [13, 14, 94]. In fact, the TQD

spin states can be constructed by adding another electron to the singlet or triplet state of two electrons. The quadruplet spin states with Sz = {+3/2, +1/2} are


34

expressed as |Q1/2 i =

|↓A ↑B ↑C i + |↑A ↓B ↑C i + |↑A ↑B ↓C i √ , 3 |Q3/2 i = |↑A ↑B ↑C i ,

(4.5) (4.6)

and similarly the states |Q−1/2 i and |Q−3/2 i with the opposite spin direction Sz = {−3/2, −1/2}. Projecting into the singly-occupied, the doublet spin states with Sz = +1/2 can be written as (| ↑A ↓B i − | ↓A ↑B i) ⊗ | ↑C i √ , 2 2| ↑A ↑B i ⊗ | ↓C i − (| ↑A ↓B i + | ↓A ↑B i) ⊗ | ↑C i 2 √ |D1/2 i = , 6

1 |D1/2 i =

(4.7) (4.8)

1 2 and similarly the states |D−1/2 i and |D−1/2 i with the opposite spin direction Sz =

1 −1/2. The first doublet spin state |D1/2 i is constructed from the singlet state

between electrons in the dots A and B

3 1 1 D1/2 |SA · SB |D1/2 =− , 4

(4.9)

and a decoupled spin in the dot C such that

1 1 1 1 D1/2 |SC · SA |D1/2 = D1/2 |SC · SB |D1/2 = 0.

(4.10)

2 The second doublet spin state |D1/2 i is formed from the triplet states between the

dots A and B with the total spin Sz = 1 and 0 by adding an electron to the dot C. The spin-spin correlation functions between electrons in the side-coupled dots A and B for the second doublet spin state is

1 2 2 D1/2 |SA · SB |D1/2 = , 4

(4.11)

and between spin the dot C and other spins are

2 1 2 2 2 D1/2 |SC · SA |D1/2 = D1/2 |SC · SB |D1/2 =− . 2

(4.12)


35

The spin-spin correlations between the dots A and B for the doublet spin 1 2 states |D1/2 i and |D1/2 i are anti- and ferro-magnetic, respectively. For t1 = t3 , the

1 2 eigenvalues for |D1/2 i and |D1/2 i are

ED1 = −12t21 /U,

(4.13)

ED2 = −4(4t21 − t22 )/U.

(4.14)

One can see from the energy separation E∆ = ED1 − ED2 = J1 − J2 that the levels

1 cross at t1 = t2 for large U/Γ, with |D1/2 i as the ground state of the isolated TQD

2 for t2 > t1 and |D1/2 i for t2 < t1 . The states are degenerate at t2 = t1 which reflects

the magnetic frustration at that point, where hSA · SB i = hSB · SC i = hSA · SC i. The level crossing and its corresponding phase transition will be studied later. From spin-spin correlation functions, Eqs. (4.9) and (4.11), and the concurrence CAB = max {0, −2hSA · SB i − 1/2}, one can easily see that the concurrence

1 2 calculated for the states |D1/2 i and |D1/2 i is

1 CAB D1/2 = 1,

1 1 CAC D1/2 = CBC D1/2 = 0.

(4.15) (4.16)

This is in agree with entanglement monogamy [95, 96] which is a pure quantum mechanical phenomenon. According to the monogamy concept if two quantum objects, i.e. the spins A and B here, are maximally entangled they cannot be entangled at all with a third quantum object. In other words, when the quantum correlations of spins A and B are very strong one can infer that the spins A and 2 i B are not correlated any more with the rest of system. The concurrence for |D1/2

also reads 2 CAB D1/2 = 0,

2 2 1 CAC D1/2 = CBC D1/2 = . 2

(4.17) (4.18)

This satisfies the squared version of entanglement monogamy for three spins in 2 2 which CAC + CBC ≤ 1 [96, 97]. In fact, there is a transition between entangled


36

and unentangled ground states depending on whether t2 > t1 or t2 < t1 .

4.4

Effective four-spin model In this section I go on to show that the TQD system can be described by a

four-spin Heisenberg model. In fact, entanglement of the system can be understood from an effective four-spin model, in which the conduction band is represented using a single spin SD . To be precise, one can transform the Anderson Hamiltonian to an effective four-spin Heisenberg Hamiltonian. The effective four-spin Hamiltonian can be written as Hef f = J1 SA · SC + J2 SA · SB + J3 SB · SC + JD SC · SD ,

(4.19)

with the corresponding exchange couplings J1,2,3 = 4t21,2,3 /U . JD ≈ 8Γ/ (πU ) [98] mimics the antiferromagnetic Kondo exchange coupling. Using the effective Hamiltonian (4.19) one can derive many physical quantities analytically. One can solve the effective Hamiltonian (4.19) and obtain the ground states of the four spins. Lets me consider the symmetric case J1 = J3 , where CAC = CBC . In general, there are two distinct ground states depending on whether the exchange interaction JD is larger or smaller than a critical value JDc . The critical value JDc can be obtained analytically as JDc =

(J1 − J2 )(J1 + 2J2 ) . 2J2

(4.20)

For four spins one can calculate the spin correlation functions and the concurrence. There are two regimes of interest, depending on the ratio of the exchange coupling JD /JDc . On the one hand, for JD > JDc , two singlets forms at the CD bond and at the AB bond. In fact, the formation of the singlet at the CD bond mimics the Kondo singlet between the dot C and the electrodes. In this situation the spins at the bonds AB and CD are fully entangled with the concurrence CAB = CCD = 1. In turn, the spins at the bonds AC and BC are unentangled with


37

the concurrence CAC = CBC = 0 which means monogamy. On the other hand, for JD < JDc , the triplet forms at the AB bond. In this situation the concurrence CAB = 0, which means that the spins at the AB bond are unentangled. However, the exact values of CAC = CBC depend on the spin correlations between the spins at the CD band. The physical picture is therefore obvious, and indicates a transition between two distinct ground states which depends on whether the spins at the AB bond are in triplet or singlet configurations.

4.5

Numerical results I now present the NRG results obtained by means the Ljubljana code for the

TQD Anderson model in the strongly correlated regime with U = 10Γ, well within the Kondo regime U/ (πΓ) >> 1 [70], at temperature T = 10−6 D sufficiently lower than the Kondo temperature TK estimated as 2 × 10−4 D. The NRG results should describe the system for T 1 (such that the results to be essentially independent of D). The NRG calculations in this chapter are considered with a discretization parameter Λ = 2. “Twist parameter” is also used to improve the accuracy of the numerical calculations. In this technique, one performs calculations, i.e. a spectral function, for several different values of discretizations and then averages all of the results. In this technique, one introduces a continuous parameter z ∈ [0 : 1] which characterizes different discretizations. I here select four different values of the twist


0.2 0 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 1 0.8

=

(c)

0.6 0.4 0.2

(d)

0.4

0 0.5

δnA2 = δnB2 2 δnC

0.3 0.2

0.1

0.1

Susceptibility

δnA2 = δnB2 2 δnC

(l)

ln 2 Entropy

0.4

0.4 Simp

(f)

Simp

0.2 0 0.3

1/4

(m)

1/4

0.2

0.2 kBT χimp/(gμB)

0.1 0 1 0.8

(k)

0.6

0.2

Conductance

0 0.8

ln 2

0.6

0 0.3

Pparallel Pantiparallel

0.4

0.3

(e)

(j)

0.2

0.2 0 0.8

Susceptibility

0 0.5

(i)

0.6 0.4

Pparallel P antiparallel

Concurrence

0.4

0.2

(b)

CA B CA C=CB C

0.6

0.4 0 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 1 0.8

(h)

Charge fluctuations

0.6

0.8

Spin-spin correlation

CA B CA C=CB C

Probabilities

1

(a)

2

0 1 0.8

(g)

0.6 0.4

G/G0

Γc

0.2

0.2 0 -3 10

2

(n)

0.6 0.4

G/G0

t2c

kBT χimp/(gμB)

0.1

Conductance

Entropy

Charge fluctuations

Probabilities


Concurrence

1 0.8

38

10

-2

10 t2 /D

-1

10

0

0 -3 10

10

-2

10

-1

10

0

Γ/D

Figure 4.2 Left panel: concurrence, (a), spin-spin correlations, (b), probabilities for parallel and antiparallel spin configuration, (c), charge fluctuations, (d), entropy Simp /kB , (e), susceptibility kB T χimp (T )/(gµB )2 , (f), and conductance at temperature T = 10−6 D, (g), as a function of t2 /D for U/D = 0.1, Γ/D = 0.01, ǫ = −U/2, t1,3 /D = 0.01. Right panel (h)-(n): results as in the left panel, but as a function of Γ/D for t2 /t1 = 0.5.

parameter z to eliminate the oscillations in the spectral function. The truncation energy cutoff is Ecutoff = 10ωN , with the characteristic energy scale ωN at the N -th NRG iteration.


4.5.1

39

Concurrence

The NRG results for the concurrence of an electron pair in TQD are presented in Fig. 4.2(a) and (h) (the left and right top panel) in terms of an inter-dot coupling t2 /D and a lead-dot hybridization Γ/D. There are two different regimes: on one hand, the concurrence is minimal, CAB = 0 for t2 < t2c because the ground state is 2 |D1/2 i, Eq. (4.8), which corresponds to the spin triplet between the dots A and B.

On the other hand, for t2 > t2c the concurrence reaches its maximal value CAB = 1 1 which corresponds to the ground state |D1/2 i, Eq. (4.7). This describes the spin

singlet between the dots A and B and a separated spin at the dot C. This means entanglement monogamy, in which the concurrence between the other spins are zero CAC = CBC = 0, whereas the spins in the dots A and B are fully entangled. Note that entanglement studied here is approximately identical to entanglement of the ground state of the isolated TQD considered in the previous section. However, the level crossing point in this case mainly depends on the hybridization Γ/D, Eq. (4.20), and always t2c < t1 . Two distinct phases are separated by a first order quantum phase transition [98–100]. Therefore the changes in the inter-dot coupling t2 /D or the lead-dot hybridization Γ/D can switch on/off entanglement, as was already predicted from the effective four-spin model, Eq. (4.20). In the following I show how concurrence properties are related to other physical quantities and the Kondo effect.

4.5.2


The spin-spin correlation functions hSi · Sj i are expressed in terms of the inter-dot coupling t2 /D and hybridization Γ/D, Fig. 4.2(b) and (i). The spin-spin correlation hSA · SB i (hSA · SC i = hSB · SC i) changes sharply from 1/4 to −3/4

1 2 (1/2 to 0) due to two distinct ground states |D1/2 i and |D1/2 i, as discussed in

Eqs. (4.9) and (4.11). This sharp transition in the spin-spin correlation functions is the signature of a first-order quantum phase transition, as expected from two distinct ground states. The formation of the spin singlet and triplet is evident, but due to the coupling to the electrodes the spin-spin correlation functions


40

are not saturated to their maximal or minimal values. The study of the spinspin correlation functions illustrates a robust singlet-triplet transition between the spins A and B. For t2 > t2c , the spin-spin correlation hSA · SB i decreases with increasing t2 /D due to the charge fluctuations. Similarly, for Γ < Γc , the spin-spin correlation hSA · SC i = hSB · SC i decreases with increasing Γ/D. In fact, all of the concurrences mentioned above can be obtained directly from the spin spin correlation functions via CAB = max {0, −2hSA · SB i − 1/2} / P↑↓ + P|| . D E D E ↑ ↑ ↓ ↓ ↓ ↑ ↑ ↓ P|| = PA PB + PA PB and P↑↓ = PA PB + PA PB are the probabilities that two

electrons in side-coupled dots form singlet and triplet, respectively, see Figs. 4.2(c) and (j).

4.5.3

Charge fluctuation

There are notable differences in behaviors of charge fluctuations δn2i = hn2i i − hni i2 plotted as a function of t2 /D and Γ/D, Figs. 4.2(d) and (k). On

one hand, the charge fluctuation δn2A = δn2B and increases with increasing t2 /D, whereas the charge fluctuation δn2C is approximately independent from t2 /D,

Fig. 4.2(d). On the other hand, the charge fluctuation δn2C increases with increasing hybridization Γ/D, while the charge fluctuations δn2A = δn2B decreases Fig. 4.2(k). The large change in the charge fluctuation signals the transition from a pure spin consideration. However, for the system considered here, the concurrence is independent from charge fluctuation due to renormalization factor P↑↓ +P|| (the probability for singly occupied spin states) in the concurrence formula. The single occupancy probability of the dots decreases and the double occupancy probability increases with increasing t2 /D, see Fig. 4.2(d). Surprisingly, large charge fluctuations do not affect the concurrence as well as its sharp quantum phase transition, even for very large t2 /D and Γ/D.


4.5.4

41

Entropy and susceptibility

Now I would like to present the impurity contribution to the total electronic entropy and susceptibility, Figs. 4.2(e), (f), (l) and (m). The entropy and susceptibility for an unscreened impurity with total spin S are Simp /kB = ln(2S + 1) and kB T χimp (T )/(gµB )2 = S (S + 1) /3, respectively [4, 18, 58]. For a free unscreened impurity with S = 1/2 in the local moment regime, one therefore expects Simp /kB = ln(2) and the Curie law with kB T χimp (T )/(gµB )2 = 1/4. The values of the entropy and susceptibility are smaller than the expected values due to the coupling to the electrodes, reflecting the effect of the charge fluctuation δn2C . The entropy and susceptibility however are zero for the screened impurity in the strong coupling regime, see Fig. 4.2(e), (f), (l) and (m). At low temperature, the contributions of all excited states on the entropy and susceptibility are projected out. Thus Simp /kB and kB T χimp (T )/(gµB )2 approach to their values in 1 the TQD doublet ground state. Therefore, the spin in the ground state |D1/2 i is

2 screened completely, whereas the spin in the ground state |D1/2 i is a free S = 1/2

disconnected from the electrodes. The transition from the full screened to the underscreened Kondo effect is a first-order quantum phase transition. In fact, in the case of t2 > t2c , the electron in the dot C forms the fully entangled Kondo state with electrons in the electrodes. For this reason the dot C is effectively decoupled from the electrons in the side-coupled dots A and B. In the case of t2 < t2c , the ferromagnetic correlations between the side-coupled dots A and B create an effective S = 1 local moment at low temperatures which can form a partial Kondo screening [100] with a residual uncompensated S = 1/2 local moment, the wellknown underscreened S = 1 Kondo efffect. The observation of different types of Kondo effects is one of the main results in this chapter. In general, the inter-dot coupling t2 (or hybridization Γ/D) clearly drives a quantum phase transition: For large t2 , both spins A and B form a singlet with zero total spin S = 0, while small t2 locks the spins A and B into an S = 1 local moment, which is underscreened by conducting electrons, leading to the doublet ground state with residual S = 1/2.


4.5.5

42

Conductance

One of the direct measurable physical properties of the TQD system is the conductance, and all the physics discussed above are clearly manifested in the transport properties. The conductance as a function of t2 /D and Γ/D is presented in Figs. 4.2(g) and (n). The zero-bias conductance G through the dots changes discontinuously at the transition point t2c . The conductance drops abruptly from the unitary limit G = G0 to G = 0 in the local moment case when the inter-dot coupling t2 (or hybridization Γ/D) is decreased. This is a first order quantum phase transition. The system in the strong coupling phase behaves as the singleimpurity Anderson model with G = G0 . The conductance behaviour is the same as parallel double quantum dots, where there is a sharp transition from the singlet to the triplet states [15, 101]. The main difference is that in the parallel double quantum dot configuration transition is from the non-Kondo (free-moment) to underscreened S = 1 Kondo effect, whereas in TQD considered here the transition is from the fully screened to underscreened S = 1 Kondo effect. One can switch on/off the transport with changing the inter-dot coupling t2 /D and the hybridization Γ/D. Comparing the conductance and concurrence, one can easily see that the spins A and B are entangled with the unitary conductance G = G0 and otherwise are unentangled with zero conductance, which can be used as an entanglement probe. All of the above results, the ground state properties of the TQD system, are summarized in the table below:

Chapter 4. Entanglement and Kondo effect Topology Type of Kondo effect Ground state Concurrence CAB

43

t2 < t2c (Γ < Γc ) Underscreened E 2 D1/2 0

t2 > t2c (Γ > Γc ) Fully screened E 1 D1/2 1

Concurrence CAC = CBC

1/2

0

Spin-spin correlation hSA · SB i

1/4

-3/4

hSA · SC i = hSB · SC i

-1/2

0

Entropy Simp

ln(2)

0

Susceptibility T χimp (T )

1/4

0

0

1

Conductance G/G0

4.5.6

Phase diagram

The main result of this chapter is presented in Fig. 4.3(a): the phase diagram of the fully screened Kondo effect and the underscreened S = 1 Kondo effect as a function of the parameters U/Γ and t2 /t1 . The region bellow the red line corresponds to the zero concurrence CAB = 0, and the region above to CAB = 1. The red line presents the quantum phase transition between the two phases, where t2 = t2c . Transitions between two regions are rather sharp even for large charge fluctuations, with Γ > U . The diagram therefore shows that the quantum phase transition persists even for weak electron correlations. The strength of an effective exchange coupling J2,ef f between the spins A and B governs the final ground state of the system and a level-crossing quantum phase transition with a change of entanglement between CAB = 0 and CAB = 1. In general the effective exchange coupling J2,ef f is a sum of two different exchange couplings: the super-exchange coupling J2 ≈ 4t22 /U , such that the local S = 0 singlet state is lower in energy than the local S = 1 triplet state, and the Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction JRKKY , which are mediated by the Kondo singlet formed between the dot C and the electrodes [15, 58, 102]. In fact, there is a competition between the effective exchange coupling J2,ef f and the Kondo exchange coupling JK , which for the single impurity Anderson model (2.18) can be written as JK = 2Γ/π[1/(ǫ +


44

U ) − 1/ǫ] [103]. There are therefore two different regions in the phase diagram: for 1.0

1.2

CAB =1 J2/J1

0.5

0.6

C

0.55

CAB =0

(a)

0.4

0.4 0.2

(b)

CAB =0

0.6

A

J2

B J3

J1

0.4

0.45

t2/t1

0.8

CAB =1

0.8

< S 2 >=0.65

0.6

1.0

C

0.2

D

JD

0.0

0.0 0

4

8

12

16

20

U/Γ

24

28

32

0

1

2

3

4

5

6

J1/JD

Figure 4.3 (a) Phase diagram obtained from the NRG calculations in the (U/Γ, t2 /t1 ) plane for t1,3 /D = 0.01 and fixed U/D = 0.1. Full red line (connecting calculated t2c /t1 , bullets) separates CAB = 1 (the Kondo phase) and CAB = 0 regions between two ground state configurations. Dashed lines represent constant values of the local moment hS2C i. (b) Phase diagram obtained p from the 4-spin model in the (J p1 /JD , J2 /J1 ) plane, as an analogue of (a) with the full red line representing J2c /J1 .

J2,ef f > JK , the region above the red line, where the antiferromagnetic correlation is dominated and the electrons form the entangled singlet state; and for J2,ef f < JK , the region below the red line, where the ferromagnetic correlation wins and the electrons are in the unentangled triplet state. A more subtle behavior arises in the strongly correlated regime U >> Γ, where the charge fluctuations of the system are negligible. In this regime, J2c approaches J1 which is close to the level-crossing point for the isolated TQD. To gain further physical insight, the local moment hS2C i of the dot C is shown

with dashed lines, Fig. 4.3(a) . hS2C i reaches the value 3/4 in the local moment regime for a sufficiently large U/Γ. It is obvious that for a large Γ (i.e. a smaller U/Γ) the local moment hS2C i decreases due to large charge fluctuations. The phase diagram obtained from the NRG calculations can be understood p from the four-spin model. The phase diagram (J1 /JD , J2 /J1 ) associated with the four-spin model is presented in Fig. 4.3(b). The full red line shows the transition

between entangled and unentangled states. Here the parameter JD mimics an effective Kondo coupling JK of TQD to the electrodes which is proportional to


45

Γ/U . For small J2 /J1 but large JD /J1 the spins in the dots A and B are fully entangled when dots C and D are in the singlet state, which can be an example of the monogamy properties of entanglement [95]. In fact, the critical points p p calculated for both models t2c /t1 ∝ U/Γ and J2c /J1 ∝ J1 /JD behave the

same. Therefore one can estimate J2c ∝ J12 /JK .

4.5.7

Molecular structure and the Kondo effect

The t2 < t2c case corresponds to the ferromagnetic correlations between spins in the side coupled dots A and B. In this situation, the molecular structure of the three dots and the Kondo effect of the central dot C compete with each other. 1 The ground-state of TQD is |D1/2 i. A flip of the spin at the central dot C makes a

new state that contains a finite projection on excited high-energy states of TQD. This increases the competition between the Kondo regime of the central dot C and the ferromagnetic correlation of spins in the side coupled dots A and B [104]. In the t2 > t2c case, the correlation between spins in side-coupled dots A and B is anti-ferromagnetic, and the system is in the usual S = 1/2 single impurity Kondo effect because the spin at the central dot C is decoupled and the ground 2 state is |D1/2 i. In this situation the spin configuration does not compete with the

Kondo regime. The flip of the central dot spin creates a state that has the equal energy as the original one, which strengthens the Kondo effect.

4.5.8

Kosterlitz-Thouless quantum phase transition

The quantum phase transition considered so far has been the first order in the case of the symmetric coupling with t1 = t3 , where there is no any mixing between singlet and triplet spin states. The model has contained a simple level crossing at the critical point t2c which marked the transition between the screened and the underscreened doublet ground states. It has been shown that entanglement and phase transition were robust even for large charge fluctuations. It is interesting to consider a broken symmetry configuration with t1 6= t3 , where one expects the


46

Kosterlitz-Thouless quantum phase transition due to exponential variation of the spin-spin correlation [87, 102]. Let me now prove how the Kosterlitz-Thouless quantum phase transition occurs for such a system. The effective Kondo Hamiltonian analogous to Eq. (4.19) can similarly be expressed as [102] Hef f = J1 SC · (SA + SB ) + J2 SA · SB + JK SC · s(0) + δSA · SB ,

(4.21)

where the last term corresponds to the asymmetry with δ = 4 (2tx + x2 ) /U and x = t1 − t3 . The ground state of the isolated TQD comprises a pair of doublet states, denoted as [14, 102] |DS±z i = ±α|DS1 z i + β|DS2 z i, where α2 = 1 − β 2 =

(4.22)





J1 − J2 + δ/2 1 . 1+ q 2 2 2 (J1 − J2 + δ/2) + 3δ /4

(4.23)

The effective Kondo coupling JK vanishes continuously as [102] JK,ef f

4 2 = JK 1 − α . 3

(4.24)

In this case, a Kosterlitz–Thouless type transition occurs, related to the exponentially decay of the Kondo temperature (the Kondo scale) TK ≈ exp (−1/JK,ef f ) with increasing J2 [102, 105, 106]. This means that the transition from the disordered phase at high temperature in the local moment regime to the ordered phase at low temperature in the Kondo regime is exponential, as expected from the standard Kosterlitz–Thouless transition [85]. Notice that the square root under the exponential is missing, as compared to the standard Kosterlitz–Thouless transition. In the Kondo model this is due to the special bare values of the coupling constants corresponding to the isotropic coupling. In a case of an anisotropic coupling the initial condition is violated and the standard Kosterlitz–Thouless is obtained [107]. The transition is thermally induced and therefore is expected to


47

depend on the Boltzmann factor exp (−1/T ). Moreover, to calculate the correlators, one needs to take an average by integrating over the Gaussian distribution of each correlator given by the Boltzmann factor. This argument can indicate the cause of the exponential decay in the transition. The Kosterlitz–Thouless transition in the Kondo model corresponds to a situation when JK,ef f changes the sign. For JK,ef f > 0 the system is in the ferromagnetic Kondo model, where the impurity and the conduction electrons are coupled ferromagnetically, whereas for JK,ef f < 0 the system is in the antiferromagnetic Kondo model. The Koster√ litz–Thouless transition point occurs when JK,ef f = 0 (α = 3/2), separating the ferromagnetic and antiferromagnetic Kondo effects. In general, the continuous transitions in quantum impurity problems can be categorized into transitions with power-law behavior (typically second-order) and those with exponential behavior (Kosterlitz-Thouless) [4]. The effect of inter-dot symmetry breaking is shown in Fig. 4.4, where t1,3 = t0 ± δt. The parameters are the same as in Fig. 4.2, with t0 /D = 0.01, U/D = 0.1, and Γ/D = 0.01. The the main effect of finite δt with increasing t2 /t0 is a smooth transition in the concurrence CAB from 0 to 1, Fig. 4.4(a), and the spin-spin correlation hSA · SB i from −3/4 to 1/4, Fig. 4.4(b). The schematic phase diagram, Fig. 4.3, still holds for the asymmetry case, but with the phase boundary now to be a smooth transition. In the case of broken symmetry with J1,3 = J0 ± δJ, the effective four-spin model can adequately reproduce the spin-spin correlations when J2 goes to 0 see the inset in Fig. 4.4(b). Note, the dependence of the spin-spin correlations on J2 calculated in the effective four-spin model does not change smoothly, it shows an abrupt transition with the gap closing even for large values of δJ, Fig. 4.4(c). The difference between the effective model and the NRG results can be due to the fact that in the effective four-spin model the change of the spin-spin correlations is due to the change of the ground state, whereas the NRG calculations of the correlations show the exact results for the system which is a quantum many-body phenomena.


1.0

CAB

1

(a)

0.8

48

2 n=5

t1,3=t0 ± δt

0.6

δt

δt=0 δt=n 0.05t0

0.4 0.2

0.2

J1,3 =J0 ± δJ

0

0 0.2

−0.2

-0.4

0

(b)

−0.2

δJ=0 δJ=n 0.1J0 n=5

(c)

-0.6

1 2

-0.8 0

1

2

J2/J0

-0.4

δt

-0.6 -0.8 0.1

1

10

t2 /t0 Figure 4.4 (a) Concurrence obtained from the NRG calculations CAB for the symmetry breaking case with t0 /D = 0.01 for several values of δt, in increments 0.05t0 (full black lines), compared to δt = 0 (full red line) and with other parameters as in Fig. 4.2. (b) Spin-spin correlations hSA · SB i corresponding to CAB from Fig. 4.3(a). (c) Results for hSA · SB i as obtained for the simplified 4spin model for J0 /JD = 10. Full black lines represent results for δJ in increments 0.1J0 .

4.6

Summary The pairwise entanglement between two electrons in a triangular TQD with

three electrons, in which one of them is attached to two metallic electrodes, is quantitatively studied. It is shown that two many-body phenomena compete with each others, the Kondo effect and the inter-dot exchange interactions. In contrast to DQD configurations where the Kondo effect destroy entanglement [17], in the TQD configuration, the Kondo effect can generate entanglement between the


49

electrons in the dots A and B which can be identified in charge transport measurements. Two different cases arises depending on the relation between inter-dot exchange couplings J2 and J1 . In one case, for J2 > J2c , the central dot C and the electrodes are fully entangled due to the screened S = 1/2 Kondo effect. In the other case, for J2 < J2c , electrons in the dots A and B are unentangled due to underscreened S = 1 Kondo effect. Consequently, one can abruptly switch entanglement between fully screened S = 1/2 and underscreened S = 1 Kondo effects, which is a first-order quantum phase transition. In the case of broken inter-dot symmetry between the side-coupled dots and the central dot C, instead of the sharp transition in the symmetry case, there is a smooth transition between entangled and unentangled states. The results show that the phase transition in this case is of Kosterlitz-Thouless type. At the end, I discuss the possibility of future research strengths. The effect of a static magnetic field on the system considered here is an open question. A possible future research direction could therefore arise whether entanglement and transport in the triple quantum dot system can be manipulated with the magnetic field. It has been shown that magnetic field can generate and manipulate quantum entanglement for an isolated triple quantum dots [14] due to Zeeman splitting. The Kondo effect can be expected by tuning to a quadruple-doublet degeneracy point by applying a magnetic field [14]. This is similar to two-level system where a Kondo effect observed in singlet-triplet degeneracy point by applying a magnetic field [108, 109]. Motivated partly with this, one can expect that magnetic field do not only destroy the Kondo effect as it is well-known but also can generate different kinds of the Kondo effect because of changing the ground state of the system with the magnetic field.

Chapter 5 Friedel-Luttinger sum rule and Kondo effect in a triple quantum dot In the previous chapter I have studied the physical properties of a system of three tunnel-coupled quantum dots (TQD) in a triangular geometry such that one of the dots is coupled to two metallic leads, see Fig. 5.1. The main interest has been the behavior at half-filling such that each dot was singly occupied. Three electron states of the isolated TQD comprised two lowest doublets with the total spin S = 1/2 and quadruple states with the total spin S = 3/2. The study has been devoted to interplay between the Kondo effect and the inter-dot magnetic correlations when the ground states are always one of the doublet states. An interesting issue in the previous chapter has been the zero-bias conductance G through the dots which has changed abruptly from the unitary limit of G/G0 = 1 in the screened Kondo phase to G/G0 = 0 in the underscreened Kondo phase. An obvious question arises: is there a general formula to describe the zero-bias conductance for both screened and underscreened Kondo phases? The Friedel sum rule relates an electron occupancy of the impurity to the phase shift of scattered electrons [18]. This is one of the few exact results in solid

51

Chapter 5. Sum rule and Kondo effect

52

state physics, with a vast range of applications in the fields of scattering theory, magnetic and nonmagnetic impurities in metals [110], the Kondo effect [18], and coherent transport through quantum dots [111, 112] and molecules [110]. The question is whether the Friedel sum rule fulfills for the underscreened Kondo effect. For a two-level system, it was already shown that the zero-bias conductance can be expressed only in terms of the dot occupancy according to a Friedel-Luttinger sum rule, which is applicable to both screened and underscreened Kondo effects [20, 21]. The Friedel-Luttinger sum rule was derived as a consequence of a generalization of Luttinger’s theorem to the underscreened Kondo phase. In fact, Luttinger’s theorem demonstrates that number of states enclosed by the Fermi surface is equal to the number of particles [113]. In turn this chapter demonstrates how in the TQD system the essential physics of the screened and underscreened Kondo phases can be captured by the Friedel-Luttinger sum rule. The main purpose of this chapter is to consider a problem of electronic correlations and a role of many-particle states in coherent transport through the TQD system in all range of electron fillings. To this end, one first needs to study electronic structures for the isolated TQD which determine the spin of the ground state for any number of electrons, i.e. with S = 1/2 and 1. Then it is possible to see the condition for local moment formation as a prior presumption for the Kondo screening. In fact, various types of the Kondo effects clearly are expected to take place with the screening of the local moment from spin S to spin S´ = S − 1/2.

Depending on whether S´ is greater than or equal to zero, the spin S is said to be screened when S´ = 0 or underscreened when S´ ≥ 1/2. The residual moment S´ and the surrounding Fermi sea forms a singular Fermi liquid [114]. A fundamental characteristic of a singular Fermi liquid is that the low-energy properties are dominated by singularities as a function of energy and temperature, resulting in a breakdown of Fermi-liquid picture [115]. The underscreened Kondo problem is one of the simplest example of the singular Fermi liquid behavior [114]. Additionally, the regular Fermi liquid theory (also called the Landau–Fermi liquid theory) shows that the interacting Fermi liquid can be constructed from some effective particles (quasi-particles) that behave as non-interacting fermions although their masses


53

and other properties are different from the non-interacting fermion systems. One can expect a quantum phase transition between regular- and singular-Fermi liquid ground states. The question is whether the quantum phase transition and different Fermi liquid ground states can be understood in terms of the Friedel-Luttinger sum rule. In fact, the method, model, and Hamiltonian are the same ash the previous chapter. The calculations are performed with the NRG Ljubljana code [116]. The chapter is organized as follows. In Sec.5.1, electronic structures of the isolated TQD for all electron fillings along with their corresponding correlators are presented. In Sec.5.2, the numerical results of the correlators for the TQD system are shown. In Sec.5.3, the Friedel-Luttinger sum rule is derived. In Sec.5.4, the numerical results of conductance are discussed. Finally, in Sec.5.5, the conclusion is presented.

A

t2

B t3

t1 C

L ΓL

R ΓR

Figure 5.1 Triangular triple quantum dots attached to the leads. In this chapter I consider the system in the inter-dot symmetric case with t1 = t3 .


5.1

54

Isolated triple dot The isolated triple quantum dots (TQD) can be modeled by an extended

Hubbard model H=

X i

ǫi fi† fi + U

X

niσ ni´σ + U1

i,σ6=σ ´

X

niσ nj σ´

i6=j,σ´ σ

† † † + t1 fAσ fCσ + t2 fAσ fBσ + t3 fBσ fCσ + h.c. ,

(5.1)

† where niσ = fiσ fiσ is the electron number operator with spin σ at the dot i ∈

{A, B, C}. An energy level of each dot is ǫi = ǫ; U and U1 are the intra- and the inter-dot Coulomb integrals; t1 , t2 and t3 are the inter-dot hoppings. Throughout this chapter, TQD is analyzed for the symmetric coupling with t1 = t3 , see Fig. 5.1. With one energy level on each dot, TQD can be filled up to six electrons. I here determine the spin states of TQD which is charged from one to six electrons to gain information about the ground states of TQD. The local magnetic moment of the ground state is the main priority for the Kondo screening. For this reason, the ordering of energy levels and spin states are analyzed in details. In fact, once the ground state properties of TQD are established, one can easily understand how the Kondo screening occurs. It should also be noted that the three electron states have discussed explicitly in the previous chapter.

5.1.1

One and five electrons

Due to the small size of the dots, their intrinsic level spacing is large enough that each dot is assumed to contain just a single energy level relevant to transport. Thus for one electron in TQD one can consider only three possible charge states (1, 0, 0), (0, 1, 0), (0, 0, 1), denoted by the kets |Ai, |Bi, |Ci respectively. With the single electron basis {|σA i , |σB i , |σC i}, the corresponding Hamiltonian Hσ for an


55

electron with spin σ in TQD is given by 



ǫ t2 t1     Hσ = t 2 ǫ t 1  .   t1 t1 ǫ

(5.2)

Without magnetic flux in TQD, all ti are real, and I take them all to be positive. The energy levels and states of the singly occupied TQD are expressed as 1 E1 = −t2 + ǫ, |ψ1 i = √ (− |σA i + |σB i) , 2

E2 =

E3 =

t2 − ∆ + 2ǫ (t2 − ∆) |σA i + (t2 − ∆) |σB i + 4t1 |σC i q , , |ψ2 i = 2 4t1 2 (t2 − ∆)2 + 16t21 t2 + ∆ + 2ǫ (t2 + ∆) |σA i + (t2 + ∆) |σB i + 4t1 |σC i q , , |ψ3 i = 2 2 2 4t1 2 (t2 + ∆) + 16t1

where ∆ =

(5.3)

(5.4)

(5.5)

p 8t21 + t22 . One can see that the order of the levels depends on the

sign of the inter-dot coupling t2 . There is a level-crossing (E1 = E2 ) between

two lowest states for t2 = t1 . Therefore, the ground state of the system is |ψ1 i or |ψ2 i depending on the inter-dot coupling values, i.e., for t1 /D = 0.01 and t2 /D = 0.005 is E1 = −t2 + ǫ, whereas for t1 /D = 0.005 and t2 /D = 0.01 is E2 = (t2 − ∆ + 2ǫ) /2. Lets now start to study five-electron states, when U1 = 0. Since the maximal number of electrons in TQD is six, one can easily figure out the five-electron configurations as those of a single hole in the six electron background. A hole is defined as the lack of an electron with respect to the completely filled TQD with six electrons. The single-hole Hamiltonian can be obtained directly from the single-electron Hamiltonian by changing the inter-dot hopping signs from ti to −ti , and changing ǫ to 5ǫ + 2U .


5.1.2

56

Two and four electrons

Let me now analyze the case with two electrons confined in TQD. Combination of two electrons can carry a total spin of S = 1 or S = 0, depending on whether they are in the triplet or the singlet configurations. The Hamiltonian HS for two electrons in the singlet (S = 0) with the basis |S1 i = |S2 i =

√1 2

(|↓A ↑C i + |↓C ↑A i), |S3 i =

√1 2

√1 2

(|↓A ↑B i + |↓B ↑A i),

(|↓B ↑C i + |↓C ↑B i), |S4 i = |↓A ↑A i, |S5 i =

|↓B ↑B i and |S6 i = |↓C ↑C i reads   √ √ 2ǫ + U1 t2 t1 2t1 2t1 0   √ √    t2 2t1 0 2t1  2ǫ + U1 t1   √ √    t1 t1 2ǫ + U1 0 2t2 2t2  . HS =  √   √  2t1 2t1 0 2ǫ + U 0 0    √ √    2t1 0 2t2 0 2ǫ + U 0    √ √ 0 2t1 2t2 0 0 2ǫ + U

(5.6)

For simplicity and to solve the Hamiltonian analytically, one can neglect the double occupancy and take into account the 3 × 3 upper left corner of HS which corresponds to the three singly occupied configurations. This is justified for a sufficiently large Coulomb repulsion U . It should be noted that the Coulomb repulsion U is responsible for super-exchange interaction J = 4t2 /U [6]. When U1 = 0 the energy levels and states are given by 1 ES,1 = −t2 + 2ǫ, |ψS,1 i = √ (− |S1 i + |S2 i) , 2

ES,2 =

ES,3 =

t2 − ∆ (t2 − ∆) |S1 i + (t2 − ∆) |S2 i + 4t1 |S3 i q , + 2ǫ, |ψS,2 i = 2 2 2 4t1 2 (t2 − ∆) + 16t1 t2 + ∆ (t2 + ∆) |S1 i + (t2 + ∆) |S2 i + 4t1 |S3 i q . + 2ǫ, |ψS,3 i = 2 2 2 4t1 2 (t2 + ∆) + 16t1

(5.7)

(5.8)

(5.9)

Two electrons in the triplet state with the total spin S = 1 can have three


57

z-component spin {Sz = 0, ±1}. Due to the Pauli exclusion principle, two electrons with the same spin direction cannot occupy the same dot. Therefore, there are three possible triplet configurations TijSz =1 = |↑i ↑j i, TijSz =−1 = |↓i ↓j i and S =0 T z = √12 (|↑i ↓j i − |↓i ↑j i), where the subscript i and j, i 6= j, denote the dots ij A, B and C. The corresponding triplet Hamiltonian is given by 

2ǫ + U1

  HT =  −t2  −t1

−t2

−t1

2ǫ + U1

−t1

−t1

2ǫ + U1



  . 

(5.10)

When U1 = 0 the energy levels and states of TQD with two electrons are expressed by 1 ET,1 = t2 + 2ǫ, |ψT,1 i = √ (|TAB i + |TCA i) , 2

ET,2 = −

ET,3 = −

(5.11)

t2 − ∆ (t2 − ∆) |TAB i − (t2 − ∆) |TBC i + 4t1 |TCA i q (5.12) , + 2ǫ, |ψT,2 i = 2 2 2 4t1 2 (−t2 + ∆) + 16t1 (t2 + ∆) |TAB i − (t2 + ∆) |TBC i + 4t1 |TCA i t2 + ∆ q + 2ǫ, |ψT,3 i = (5.13) . 2 4t1 2 (−t2 − ∆)2 + 16t21

Therefore, the energy levels written in this basis is identical to the single-electron energy levels, except that all inter-dot coupling acquire a negative sign. This holds when ǫ = 0 and U1 = 0. Lets now start to investigate four-electron states. As the maximal number of electrons in TQD is six, one can interpret the four-electron configurations as those of two holes. The two-hole singlet Hamiltonian is the same to that of the two-electron singlet. However, one needs to replace the energy of two-electron complexes with the energy of two-hole complexes, and change the sign of the tunneling connecting the singly occupied configurations. The sign of elements √ 2ti connecting the singly and doubly occupied configurations does not change,


58

which breaks the particle-hole symmetry. On the other hand, the two-hole triplet Hamiltonian is identical to the single-electron Hamiltonian, differing from it just in diagonal terms which change from ǫ to 4ǫ + U .

5.1.3

Correlators

I here present the expectation values of correlators for the ground states of isolated TQD. Since two cases t1 > t2 and t1 < t2 exhibit different ground states, I treat two cases separately. As shown in the previous chapter for a half-filled TQD, the t1 > t2 case corresponds to ferromagnetic correlations between electrons in the AB bond, whereas the t1 < t2 case to antiferromagnetic correlations. For this reason, two cases t1 > t2 and t1 < t2 are called ferromagnetic and antiferromagnetic cases, respectively. First the total electron charge hntot i and the local electron occupancy of each dot hni i (i = A, B, C) is analyzed to clarify precisely the charge distribution as a function of the gate-voltage ǫ + U/2. The charge distribution is expected to be inhomogeneous due to the asymmetry in the inter-dot electron tunnelings t1 6= t2 . The charge inhomogeneity affects significantly the magnetic properties. The square of the total spin hS2tot i = S(S + 1), where Stot =

P

Si represents

the total spin of TQD, and the local spins hS2i i = Si (Si +1) are shown to clarify the

formation of the local magnetic moment. The correlator hS2tot i = 2 is associated

to the total spin S = 1 where the underscreened Kondo effect is expected, and hS2tot i = 3/4 is associated to the spin S = 1/2 where the fully screened Kondo effect is expected. The correlator hSi · Sj i indicates an antiferromagnetic or a ferromagnetic coupling between the spins. The spin-spin correlator is negative for the antiferromagnetic coupling with a minimal value −3/4 for the singlet state, whereas it is positive for the ferromagnetic coupling with a maximum value 1/4 for the triplet state. The value of hSi · Sj i changes passing from the triplet to the singlet state.


59

6 5

(a) nA

ni

4 3

t1 t2

n tot = nB nC

2 1 0 2

(b)

2

2

SA

Si

2

1.5

S tot = S 2B 2

SC

1

0.5

Si . S j

0 0.2

(c)

0

-0.2 SA.

-0.4

SC

SA. S B = SB. SC

-0.6 -0.8 -0.2

-0.1

0

0.1

0.2

+ U/ 2 Figure triple dot: total hntot i and local hni i charges,

isolated

5.2 Correlators for total S2 tot and local S2 i spins and spin-spin correlations hSi · Sj i as a function of gate-voltage ǫ + U/2 for U/D = 0.2, t1 /D = 0.01 and t2 /D = 0.005.

The correlators are expected to show a series of jumps, which correspond to level crossing when an extra electron is introduced to TQD. Actually, the crossing point appears when two quantum states with consecutive fillings are energetically accessible in the same time. Therefore, one can find the positions of the crossings with the equation E(ntot ) = E(ntot ± 1) which means that a total energy with ntot electrons becomes equal with an energy for ntot ± 1 electrons. Figs. 5.2 and 5.3 show the expectation values of various operators for two different cases t1 > t2 and t1 < t2 . It is assumed that the Fermi energy ǫF = 0 and the levels ǫ in all dots are simultaneously shifted by the gate voltage. Notice that


60

6 5

(a) nA

ni

4 3

t

n tot = nB nC

1

t

2

2 1 0 2

(b)

2

2

SA

Si

2

1.5

S tot = S 2B 2

SC

1

0.5

Si . S j

0 0.2

(c)

0

-0.2 SA.

-0.4

SC

SA. S B = SB. SC

-0.6 -0.8 -0.2

-0.1

0

0.1

0.2

+ U/ 2 Figure isolated triple dot: total hntot i and local hni i charges, Correlators

for

5.3 total S2tot and local S2i spins and spin-spin correlations hSi ·Sj i as a function of gate-voltage ǫ + U/2 for U/D = 0.2, t1 /D = 0.005 and t2 /D = 0.01.

the plots have asymmetric shapes with respect to the middle of the Hubbard gap at ǫ + U/2 = 0 due to the particle-hole asymmetry. In the following, I discuss the correlator dependencies for all electron fillings. I first describe the expectation values of the correlators in the case of one electron in TQD. Comparing the ground state energies of one electron with no electron in TQD, one finds the position of gate-voltage where the first electron enters TQD. For the t1 > t2 case the crossing point is at ǫ + U/2 = |t2 | + U/2, whereas for the t1 < t2 case the crossing point is at ǫ + U/2 = (U − |t2 | + ∆) /2.

For both cases, the length of the total spin hS2tot i is 3/4 due to one free spin in


61

TQD. It should be noted that for the case t1 < t2 one electron is localized in dots A and B while the dot C is empty, and one expects to have a so-called “dark-state” that completely blocks the electron transport through the TQD system. Next I present the correlators for two electrons in TQD. Comparing the ground state energies of both one and two electrons in TQD, one obtains the position of the gate-voltage in which a second electron enters TQD. For the t1 > t2 case the crossing point is at ǫ + U/2 = (U − |t2 | + ∆) /2, whereas for the t1 < t2 case is at ǫ + U/2 = |t2 | + U/2. For both the t1 > t2 and t1 < t2 cases, the square of

the total spin correlator hS2 i is 2 and the spin-spin correlators hSi · Sj i are positive due to the triplet ground state. In this situation, by coupling TQD to the leads one can expect underscreened S = 1 Kondo effect. This is discussed explicitly in the following sections.

The case with three electrons has been discussed in the previous chapter. Lets me remind that in this situation the ground states is doublet with total spin S = 1/2. Depending on the super-exchange coupling, the ground state can be 1 2 either the doublet |D1/2 i or the doublet |D1/2 i. The coupling between the spins in

the dot A and B can be antifferomagnetic with hSA · SB i = −3/4 or ferromagnetic with hSA ·SB i = 1/4, respectively. Comparing their ground state energies for three

and two electrons one gets the crossing point at ǫ + U/2 = (16|t1 |2 − 4|t2 |2 ) /U +

(U − |t2 | − ∆) /2 for the t1 > t2 case and ǫ + U/2 = 12|t1 |2 /U + (U − |t2 | − ∆) /2 for the t1 < t2 case. After that the situation for four electrons is similar to two-electron case, however now the ground state is formed as singlet. In both the t1 > t2 and t1 < t2

cases, the square of the total spin correlator hS2 i is zero and also the spin-spin correlators are negative due to the singlet ground state. In this situation one can easily expect that in the TQD system there is no Kondo effect because there is no any local magnetic moment for screening. Finally the five-electron problem is considered which is the similar as the one-electron problem, but now the energy spectrum is reversed. For both t1 > t2 and t1 < t2 cases, the total spin correlator hS2tot i is 3/4 which indicates one free


62

spin. In this situation, by coupling TQD to the leads one therefore can expect S = 1/2 Kondo effect.

5.2

NRG results of correlators In the following I present and discuss numerical results obtained by means

of the NRG code for correlation functions of the TQD system as a function of the gate-voltage which shifts the position of ǫ + U/2. All the results are shown in the strong coulomb repulsion regime U/Γ = 20 at low temperature T /D = 10−13 , which ensures that the system is in the Kondo regime. The signatures of the Kondo effects are not only the transport properties but also various correlators. I therefore study thermodynamic values of various operators. Figs. 5.4 and 5.5 present the correlators plotted versus gate-voltage for two different cases t1 > t2 and t1 < t2 . The correlators show a series of jumps, which correspond to level crossing between two different electron ground states. It is seen that the electron concentration hni i increases with decreasing ǫ + U/2. The electron concentration of dots A and B show some sharp jumps. Meanwhile, the electron concentration of the dot C remains a smooth function of ǫ + U/2. The plots of correlators versus the gate-voltage have also asymmetric shapes due to the particle-hole asymmetry.

5.2.1

Ferromagnetic case

In Fig. 5.4, the correlators of the t1 > t2 case are presented. From the right-hand side the first jumps in the correlators correspond to the level-crossing through the one-electron ground state with charge fluctuations between the states with zero and one electron. In this situation, the total electron correlator hntot i shows a sudden jump at ǫ + U/2 ≈ 0.11 = (U − |t2 | + ∆) /2, where an electron enters suddenly in the dots A and B. The sharp jump in the correlator hni i finds the sharp transition in their counterparts, namely in the correlators


63

6 n tot

5

nA = n B nC

4

ni

(a) t

1

t

2

3 2 1 0 2 2

Si

SA

= S 2B 2

SC

1

(SA + S B

(

2

1.5

(b)

2

S tot

2

0.5

Si . S j

0 0.2

(c)

0

-0.2 -0.4 SA. SC

-0.6 -0.8 -0.2

-0.1

0

SA. S B = SB. SC

0.1

0.2

+ U/ 2

2 Figure 5.4 Correlators: total hn i and local hn i charges, total Stot and tot i

local S2i spins and inter-dot spin-spin correlations hSi · Sj i as a function of gate-voltage ǫ + U/2 derived by NRG method for U/D = 0.2, Γ/D = 0.012, t1 /D = 0.01 and t2 /D = 0.005.

hS2tot i, (SA + SB )2 = SAB (SAB + 1) and hSi · Sj i. One can also see that hS2i i

saturates in the middle of the plot at half-filled TQD and it is closed to 3/4, in-

dicating one free spin in each dot. The length of the total spin hS2tot i achieves its maximal value about 2 when there are two electrons in TQD. In this situation underscreened S = 1 Kondo effect is expected. However, the length of the total

spin (SA + SB )2 for the AB bond reaches its maximal value for nearly half-filled TQD, the center of the plot. The second sharp transitions in correlators are at

ǫ + U/2 = − (16|t1 |2 − 4|t2 |2 ) /U − (U − |t2 | − ∆) /2 ≈ −0.07, which corresponds to the charge fluctuations between the three-electron state and the four-electron


64

ground state. In this situation, both of the spin squares hS2tot i and (SA + SB )2

change sharply from their maximal to minimal values, while the spin-spin correlators hSA · SB i change abruptly from ferromagnetic to antiferromagnetic. In this situation, one can anticipate a quantum phase transition between fully screened and underscreened Kondo effect. Lowering the gate-voltage further, the length of the total spin is strongly reduced and achieves its minimal value hS2tot i = 0.02 where there are four electrons in TQD. This is a bit above the value 0 for square of the total spin of four electrons with singlet ground state, which can expect from strong charge fluctuations. Lowering the gate-voltage more, hS2tot i reaches 3/4 for five electrons in TQD. This indicates the local moment formation for five electrons in TQD, where one can expect fully screened S = 1/2 Kondo effect. Moreover, the correlator hSA · SB i in the singlet (triplet) state is a bit below the value −3/4 (1/2) for two localized electrons in the singlet ground state due to strong charge fluctuations. Because of these charge fluctuations, the other correlators, i.e. hS2i i, are also smaller than their expected values for the ground state of the isolated TQD.

5.2.2

Antiferromagnetic case

Let me turn to the case t1 < t2 presented in Fig. 5.5. The behavior of correlator hni i in this case is similar to the previous case. One can also see that the correlator hS2tot i reaches its maximal value about 2 for two electrons at TQD.

Therefore in two-electron state one can expect underscreened S = 1 Kondo effect for both the ferromagnetic and antiferromagnetic cases. There are two sharp transitions in correlators on the right-hand side of the figure which correspond to the fluctuations between the one-electron and the two-electron states at ǫ + U/2 = |t2 | + U/2 ≈ 0.11, as well as between the two-electron and the three-electron states at ǫ + U/2 = 12|t1 |2 /U + (U − |t2 | − ∆) /2 ≈ 0.09. In fact, there are sharp transitions in hS2tot i at ǫ + U/2 ≈ 0.09 and ǫ + U/2 ≈ |t2 | + U/2, where a

quantum phase transition between fully screened and underscreened Kondo effect are expected. The underscreened Kondo effect corresponds to two-electron ground


65

6 n tot

5

nA = n B nC

4 ni

(a) t1 t2

3 2 1 0 2 2

2

SC

1

(SA + S B

(

2

2

S A = SB

1.5 Si

(b)

2

S tot

2

0.5

Si . S j

0 0.2

(c)

0

-0.2 -0.4

SA. SC

SA S. B = SB. SC

-0.6 -0.8 -0.2

-0.1

0

0.1

0.2

+ U/ 2

2 Figure 5.5 Correlators: total hn i and local hn i charges, total Stot and tot i

local S2i spins and inter-dot spin-spin correlations hSi · Sj i as a function of gate-voltage ǫ + U/2 derived by NRG method for U/D = 0.2, Γ/D = 0.012, t1 /D = 0.005 and t2 /D = 0.01.

state with the total spin S > 1/2 in the range of 12|t1 |2 /U + (U − |t2 | − ∆) /2