Numerical Renormalization Group studies of Quantum Impurity Models

Numerical Renormalization Groupstudies of

Quantum Impurity Modelsin the Strong Coupling Limit

Michael Sindel

Munchen 2004

Numerical Renormalization Groupstudies of

Quantum Impurity Modelsin the Strong Coupling Limit

Michael Sindel

Dissertation

an der Fakultat fur Physik

der Ludwig–Maximilians–Universitat

Munchen

vorgelegt von

Michael Sindel

aus Rothenburg ob der Tauber

Munchen, den 22. Oktober 2004

Erstgutachter: Prof. Dr. Jan von Delft

Zweitgutachter: Prof. Dr. Walter Hofstetter

Tag der mundlichen Prufung: 27. Januar 2005

Contents

Abstract xii

I General Introduction 1

1 Introductory remarks 3

2 Quantum dot (QD) basics 7

2.1 Experimental realization of QDs . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Level quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Charging effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Tunability of QDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Experimental consequences on transport . . . . . . . . . . . . . . . . . . . 10

2.6 Kondo effect in QDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Introduction to the Kondo effect 15

3.1 The Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Unitary transformation of the AM . . . . . . . . . . . . . . . . . . . 17

3.1.2 Mapping onto an effective model: Schrieffer-Wolff transformation . 18

3.2 The Kondo effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 The historical origin - the resistivity minimum in bulk . . . . . . . 21

3.2.2 The Kondo model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.3 Kondo’s explanation . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.4 General properties of the Kondo effect . . . . . . . . . . . . . . . . 29

3.2.5 Poor man’s scaling (PMS) . . . . . . . . . . . . . . . . . . . . . . . 30

4 Wilson’s Numerical Renormalization Group (NRG) 33

4.1 Physical quantities computed with NRG . . . . . . . . . . . . . . . . . . . 38

4.1.1 Thermodynamic quantities . . . . . . . . . . . . . . . . . . . . . . . 38

4.1.2 Dynamic quantities: A(ω, T ) and χ′′(ω, T ) . . . . . . . . . . . . . . 39

II NRG-studies of Quantum Impurity Models 44

vi CONTENTS

5 QDs coupled to ferromagnetic leads 475.1 Effect of a finite spin polarization in the leads on the Kondo resonance . . 48

5.1.1 Quantitative determination of the compensation field Bcomp . . . . 545.1.2 Experimental relevance . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2 Gate controlled spin splitting in the Kondo regime . . . . . . . . . . . . . . 575.2.1 Comprehensive study dealing with QDs coupled to ferromagnetic leads 64

6 Non-monotonic occupation in two-level QDs 65

7 Frequency-dependent transport in the Kondo regime 71

8 Kondo correlations in optical experiments 77

9 Summary and outlook 91

III Appendix 93

A Derivation of the NRG-equations 95A.1 Manipulation of the CB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

A.1.1 Transformation of H`d . . . . . . . . . . . . . . . . . . . . . . . . . 98A.1.2 Transformation of HCB . . . . . . . . . . . . . . . . . . . . . . . . . 99

A.2 Mapping of the CB onto a semi-infinite chain . . . . . . . . . . . . . . . . 101A.3 Iterative numerical diagonalization . . . . . . . . . . . . . . . . . . . . . . 103A.4 Results of the iterative diagonalization . . . . . . . . . . . . . . . . . . . . 105

B How to obtain continuous dynamic functions 107B.1 The broadening of the δ-functions . . . . . . . . . . . . . . . . . . . . . . . 107B.2 How to combine information from different iterations . . . . . . . . . . . . 109

C Symmetries in the NRG-scheme 113

D The density matrix Numerical Renormalization Group (DM-NRG) 115

E Kramers-Kronig (KK) relation 119

F Kubo formalism 123F.1 Derivation of the Kubo formula . . . . . . . . . . . . . . . . . . . . . . . . 124

IV Miscellaneous 129

Bibliography 131

List of Publications 137

Acknowledgements 138

Table of Contents vii

Deutsche Zusammenfassung 140

Curriculum vitae 143

viii Table of Contents

List of Figures

2.1 Lateral and vertical QDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Tunability of QDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Coulomb oscillations and the Coulomb staircase . . . . . . . . . . . . . . . 112.4 Differential conductance of a Kondo QD . . . . . . . . . . . . . . . . . . . 122.5 Kondo effect in QDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Relevant parameters of a QD . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Unitary transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Virtual excitations in a singly occupied QD . . . . . . . . . . . . . . . . . . 203.4 Anomaly of the electrical resistivity . . . . . . . . . . . . . . . . . . . . . . 223.5 First order processes in a Kondo QD . . . . . . . . . . . . . . . . . . . . . 253.6 Second order processes in a Kondo QDs . . . . . . . . . . . . . . . . . . . . 263.7 Poor man’s scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1 Flow of couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Flow diagram of eigenenergies . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 QD occupation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.4 Dynamic quantities calculated with NRG . . . . . . . . . . . . . . . . . . . 42

5.1 Contact between QD and arbitrary material . . . . . . . . . . . . . . . . . 485.2 DoS of a polarized lead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Split Kondo resonance of a QD contacted to ferromagnetic leads . . . . . . 555.4 Zero-field splitting of Kondo resonance in a carbon nanotube quantum dot 565.5 Gate voltage dependence of the splitting of the Kondo resonance . . . . . . 575.6 Spectral function of a Kondo QD in presence of ferromagnetic leads . . . . 635.7 Sketch of different DoS types . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8.1 PL-experiment with self assembled QDs . . . . . . . . . . . . . . . . . . . 798.2 Optical transition between a ’Kondo’ and a ’non-Kondo’ state . . . . . . . 80

A.1 Depiction of arbitrary leads DoS . . . . . . . . . . . . . . . . . . . . . . . . 97A.2 NRG-iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

B.1 Possible transitions for T = 0 and T 6= 0 . . . . . . . . . . . . . . . . . . . 108

x List of Figures

B.2 Logarithmic gaussian vs. ’usual’ gaussian . . . . . . . . . . . . . . . . . . . 109B.3 Combining different clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 111B.4 Matrix elements of neighboring clusters . . . . . . . . . . . . . . . . . . . . 112

D.1 DM-NRG vs. NRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

E.1 KK-transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

F.1 A biased QD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

List of Tables

9.1 Implemented models in the NRG-code . . . . . . . . . . . . . . . . . . . . 92

xii Abstract

Abstract

In this thesis we summarize a number of theoretical studies dealing with various propertiesof quantum dots. Small quantum dots with a large level spacing are very well describedby the Anderson impurity model. In modern quantum dot experiments all parameters ofthis model can be tuned via external gate voltages. Thus, it should be possible to checkour theoretical findings experimentally. We are particularly interested in temperaturesT smaller than the so-called Kondo temperature TK , T < TK . The Kondo temperatureTK is an energy scale below which a local spin inside the dot interacts strongly with theconduction electrons in its neighboring leads. Consequently we employ Wilson’s numericalrenormalization group method [1], a numerical technique which allows for an accuratecalculation of properties of quantum dots in the Kondo regime.

We start with a general introduction to the physics of quantum dots, an introductionto the Kondo effect and the numerical renormalization group method (Part I).

The main part of this thesis, Part II, is divided into several studies:(i) We analyze the properties of the Kondo resonance of a quantum dot that is coupled toleads with a finite spin polarization. We find that this polarization suppresses and splitsthe Kondo resonance. We extend our study to a dot that is coupled to a lead with anarbitrary density of states and study the gate-voltage dependence of the Kondo resonance.(ii) We investigate the filling scheme of a spinless two-level Anderson model as a functionof gate voltage. We identify parameters where the two levels do not fill monotonicallywhen being lowered relative to the Fermi energy of the leads. For asymmetrically coupledlevels, we even find an occupation inversion of the two levels (i.e. the upper level has abigger occupation than the lower level) for a specific gate voltage region. We explain thisbehavior by means of the self consistent Hartree approach.(iii) We calculate the finite frequency conductance of a Kondo quantum dot within theKubo formalism. An analytical formula, which establishes a relation between the frequencydependent conductance and the (local) equilibrium spectral function, is derived. By meansof the fluctuation dissipation theorem we establish a relation between current noise andthe equilibrium spectral function.(iv) Motivated by emission experiments in self-assembled semiconductor quantum dots [2]we study optical transitions between ’Kondo’ and ’non-Kondo’ states. For this sake the’standard’ Anderson model is extended. We find that the emission and absorption spectrumcan be nicely understood by analogy to the X-ray edge absorption problem.

In Part III we summarize technical details that have been of relevance in this thesis.

xiv Abstract

Part I

General Introduction

2

Chapter 1

Introductory remarks

Many-body phenomena [3] are of central interest in many fields of modern condensedmatter physics. Those phenomena are challenging both from an experimental and from atheoretical point of view.

In typical bulk materials, it can often be difficult to distinguish whether a measuredeffect stems from single particle physics or is really due to correlation effects. Quantumdots (QDs) (small electron droplets which are confined in all spatial dimensions) on theother hand, are ideally suited to investigate the interplay of single-particle and many-bodyphysics in a controlled fashion. In typical QDs many-body effects become observable attemperatures T below ∼ 1K. Consequently, a lot of effort was invested to reach theselimits experimentally. In QDs it is possible to study correlation effects systematically. Inparticular, it is possible to tune QDs such that one of the simplest strongly correlatedmodels, the Kondo model (KM), can be realized experimentally.

New theoretical methods need to be used to investigate regimes where many-body ef-fects are important. It turns out that mean-field theories are not capable of describingall effects that arise from many-body correlations properly. The quest for new theoreticalmethods, such as the numerical renormalization group method, lead to renormalizationtechniques which allow for a quantitative description of effects where correlations are im-portant. The fruitful interplay between experimental and theoretical developments allowedfor a considerable progress in Kondo physics, the physics related with the Kondo model,within the last years.

This thesis deals with some low-temperature properties of different QD-systems. We areparticularly interested in zero-temperature properties of those systems. Since correlationeffects are crucial in this regime we employ Wilson’s numerical renormalization group(NRG) method to tackle this problem.

The thesis is divided into four parts:

A pedagogical introduction to the field is given in Part I. After we provide basic knowl-edge about QDs in Chapter 2, we give a detailed introduction to the Kondo problem inChapter 3. In Chapter 4 we discuss the method that is heavily used in this thesis, theNRG-method.

4 1. Introductory remarks

The second part of this thesis, Part II, contains several studies that were carried outand published during my PhD-studies.In Chapter 5 we consider a QD coupled to leads with a finite spin polarization and ana-lyze the delicate effect of a spin polarization in the leads on the Kondo resonance. In thefirst part of this Chapter we are interested in the consequences of this polarization on theKondo resonance while keeping the gate voltage fixed (Section 5.1). In the second part(Section 5.2), on the other hand, we focus on the gate voltage dependence of the Kondoresonance for a QD coupled to leads with a particular density of states (DoS). We find thatthe local spin splitting of a QD coupled to leads of that type can be controlled by meansof an external gate voltage.Chapter 6 deals with the filling of a spinless two-level QD. We observe a non-monotonicfilling scheme for a particular region in parameter space and explain this behavior via asimple self-consistent Hartree approach. We identify gate voltage regions where the QDoccupation is even inverted, i.e. the occupation of the energetically higher lying level isbigger than the occupation of the energetically lower lying level, given the two levels arecoupled with a different strength. The generalization of this study to the spinfull case iscurrently in preparation.The finite frequency conductance of a Kondo QD is investigated in Chapter 7. We use theKubo formalism to compute the frequency-dependent conductance of a QD in the Kondoregime. We identify two possibilities to measure the equilibrium spectral function of ageneric QD, namely via (i) a finite frequency conductance measurement or via (ii) a cur-rent noise measurement.We study optical transitions between a ’Kondo’ and a ’non-Kondo’ state in Chapter 8.Transitions of this type have recently come into experimental reach in self assembledQDs [2]. In contrast to the previous Chapters, where various transport properties of QDswere considered, we focus on optical properties in Chapter 8. We use Fermi’s Golden Ruleto calculate the line shape related to the transitions mentioned above. The findings canbe nicely explained by an analogy to the X-ray edge problem.

Part III, the Appendix, contains technical details relevant for the studies carried outin Part II. In Appendix A, the general NRG-mapping of a single-level QD coupled to alead with a spin- and energy-dependent DoS is performed.Dynamic quantities, such as the widely used spectral function, are rather difficult to cal-culate for the following two reasons: firstly, to realize a continuous function, δ-functionsneed to be broadened properly. Secondly, the different energy scales of the NRG-iterationhave to be combined properly to obtain a function valid on all energy scales. Both issuesare addressed in Appendix B.To keep the numerical effort of the NRG-iteration tolerable and to get rid of artificialperturbations, we introduce some relevant and useful symmetries of the NRG-procedurein Appendix C. Their use is crucial as we are usually dealing with a Hilbert space ofconsiderable dimension (its dimension is typically ∼ 104).Problems that inherit more than a single energy scale (e.g. systems in presence of a magnetic

5

field B) have to be treated with a generalized NRG-procedure, known as the ’DM-NRG’-method. This method, invented by Hofstetter [4], generalizes the ’traditional’ NRG-methodwhich incorporates only a single energy scale. We introduce the ’DM-NRG’-method in Ap-pendix D.The Kramers-Kronig relations are well established relations for causal functions. Since weare mostly interested in functions that have a sharp feature around the Fermi energy (e.g.the Kondo resonance of width ∼ TK), it turns out that their numerical implementation israther tricky. Following Bulla [5] we introduce an accurate method to perform this trans-formation in Appendix E.In Appendix F we present a general way for the calculation of the conductance throughan interacting region. We derive equations for the conductance based on the Kubo formal-ism [6]. In particular, these relations allow for the accurate determination of the frequency-dependent conductance of a ’Kondo-QD’ as used in [7] (see also Chapter 7).

The last part, Part IV, contains miscellaneous. It contains the bibliography, a listof publications, the acknowledgements, “Deutsche Zusammenfassung” and finally the cur-riculum vitae.

6 1. Introductory remarks

Chapter 2

Quantum dot (QD) basics

The tunability of QDs makes them ideal devices to study quantum impurity problemsexperimentally. Due to the small spatial dimensions of QDs the energy levels inside theQDs are quantized. QDs with large charging energies (up to 1.5meV ' 15K; see e.g. [8])can be routinely manufactured, thus charging effects are important below temperaturesT ∼ 15K. Therefore: QDs reveal both charge and energy quantization.

2.1 Experimental realization of QDs

Quantum mechanical effects occur when the system size L is of the order of the de Broglie

wavelength1 λF = hm∗vF

and the thermal wavelength λT =√

h2

2m∗kBTof the electrons,

i.e. L ∼ λF , λT . Here m∗ denotes the effective electron mass (at the Fermi energy) andvF the Fermi velocity. Due to a much smaller Fermi velocity semiconducting materialshave a significantly larger de Broglie wavelength (λF ∼ 100nm) than metallic materials(λF ∼ 0.1nm). Accordingly, quantum mechanical effects are observable in semiconductorQDs with a diameter L ∼ 100nm, a length scale that can be routinely realized with today’sfabrication techniques.

The most frequently used tunable QDs - a property that self-assembled QDs do notshare - are lateral and vertical QDs, see Fig. 2.1.A lateral QD, see Fig. 2.1(left panel), is defined by metallic gates on top of a semiconductorheterostructure (typically GaAs/AlGaAs). Near the interface of these two semiconductingmaterials a two-dimensional electron gas (2DEG) forms. When a negative voltage is appliedat the top gates a small electron droplet, a QD, is formed in the 2DEG which itself is belowthe surface of the heterostructure. In addition, the gates allow for an accurate control ofthe tunnel barriers between the QD and the neighboring reservoirs (i.e. the source anddrain region), the shape of the QD and the electro-chemical potential inside the dot.A vertical QD, on the other hand, is etched out of a double-barrier heterostructure [9]and finally coated by metal gates, see Fig. 2.1(right panel). The number of electrons in a

1Which is identical to the Fermi wavelength here.

8 2. Quantum dot (QD) basics

vertical dot can be controlled very accurately by side gates without changing the tunnelbarriers. Lateral QDs do not share this property: a variation of the electron number inlateral QDs results in a variation of the tunnel barriers as well.Transport measurements through a QD can be performed by coupling it (via adjustable

Figure 2.1: Left: Atomic force microscopy picture of two coupled lateral QDs (brightcentral circles), serving as an ’artificial’ molecule [10]. A negative voltage in the gates A(source), B (drain), 1 and 2 leads to a partial depletion of the 2DEG (which is below thesurface of the heterostructure). Right: Vertical QDs are realized by coating a pillar withmetallic gates [11]. The side gates allow for an accurate control of the number of electronsin the island without changing the tunnel barriers. Transport measurements in QDs areperformed by applying a finite bias voltage VSD (a source-drain voltage) across the dot.

tunnel barriers) to electron reservoirs whose role it is to feed the QD with electrons. Whena finite bias voltage VSD is applied across the QD a current (which depends sensitively onthe parameters of the QD) might be driven through the QD.

2.2 Level quantization

The energy difference between neighboring eigenenergies of a system, the level spacing δE,is set by the system size L. Qualitatively, a decrease in system size results in an increasinglevel spacing [12].A lateral QD traps electrons in a disk of diameter L. A crude approximation of thisgeometry is a square of side length L, thus the level spacing is δE ∝ 1/L2. More preciselythe level spacing of such a QD [13] is given by

δE ' 1/ρL2, (2.1)

2.3 Charging effects 9

where ρ = kF

~vFis the (material dependent) DoS (at the Fermi energy) of the 2DEG. Note

that metallic systems have a much bigger DoS as compared to semiconducting materi-als. Thus, for fixed L, the level spacing of metallic QDs is much smaller than that ofsemiconducting QDs.

2.3 Charging effects

Due to the spatial confinement of the electrons inside the QD, Coulomb repulsion betweenall electrons inside the QD is an important energy scale. The energy penalty that has tobe paid when an additional electron enters the QD (initially occupied with N electrons),N → N +1, is called the charging energy EC of the QD, EC = e2/2C (C denotes the totalcapacitance of the QD). A good estimate of the total capacitance of a disk of diameter Lis C ∼ ε0L [13], with the dielectric constant ε0, thus

EC ' e2

2ε0L. (2.2)

Since the level spacing δE has a different L dependence as the charging energy EC , seeEqs. (2.1) and (2.2), one can (in principle) tune the ratio of these two energy scales exper-imentally by choosing L appropriately,

EC

δE' πe2

ε0~vF

(L

λF

). (2.3)

In typical QDs, the diameter L is larger than λF (L > λF ), thus charging energy EC is thedominant energy scale (EC > δE).2 Semiconductor QDs with a diameter L ∼ 100nm havetypically the following parameters: EC ≈ 1.5meV (as mentioned before), δE ≈ 0.1meV.At sufficiently low-temperatures charging effects result in Coulomb blockade behavior, i.e.the QD is charged one by one when its local electro-chemical potential is lowered relativeto the Fermi energy of the reservoirs.

One remark should be made here: the interaction between electrons inside a QD is notsolely due to charging effects. The first correction to EC is due to exchange interaction ES,an energy scale that arises from spin-spin interaction. Similar to Hund’s rule in atomicphysics a QD can lower its energy by maximizing its total spin.

2.4 Tunability of QDs

The success of QDs is associated with their tunability. As mentioned in Section 2.1,different gate voltages allow for a precise control of a variety of relevant parameters of theQD, such as the coupling strength between the QD and its surrounding reservoirs.

Here, we comment on gates that (i) allow for a rigid shift of the discrete levels insidethe QD and (ii) enable one to perform transport measurements (as a function of the bias

2The prefactor in Eq. (2.3), πe2

ε0~vF, is of order one in typical materials.


voltage VSD) between the source and drain region. A schematic illustration is given inFig. 2.2, where the left (right) reservoir plays the role of the source (drain) contact.A back gate (tunable via the gate voltage VG) influences the electro-chemical potentialinside the QD. It can thus be used to realize task (i). Task (ii) can be realized by directlyapplying a bias voltage between the left/right reservoir, see Fig. 2.1(b). We will henceforthuse the word lead synonymous for reservoir. The chemical potentials of the left/right lead,µL/µR, obey the relation µL = µR + eVSD.Since the electro-chemical potential in a QD (containing N electrons) µdot(N) can be tunedby means of VG, one can control the number of electrons inside the QD for temperaturesT < EC . For VSD = 0 (µ = µL = µR) and µdot(N + 1) > µ > µdot(N) the QD is filled withN electrons. The (N +1)-th electron can not enter the QD as long as µdot(N +1) > µ, i.e.transport is blocked [Coulomb blockade, Fig. 2.2(a)]. In typical semiconductor QDs withEC > δE one can measure the level-spacing δE of a QD by continuously increasing thesource-drain voltage VSD, while keeping VG fixed. Any time a new discrete level of the QDenters the transport window (of width eVSD) a peak in the differential conductance can beobserved, see Fig. 2.2(b).

��

��

��

��

��

��

��

��µdot(N)

µdot(N+1)

+δEEC

��

��

��

��

� � � � � � � � � � � �

��

��

��

(b)(a)

eVG

eVSD

L R L R

Figure 2.2: (a) The gate voltage VG allows for a rigid and linear shift of the electro-chemicalpotential of the n-th electron in the QD µdot(n), n ∈ N. For T < EC the QD is filled withN electrons, if the corresponding chemical potential µdot(N) lies below the lowest chemicalpotential of the leads µdot(N) < min{µL, µR} and µdot(N + 1) > min{µL, µR}, (b) Thechemical potential of the left (L) and right (R) lead µL/R can be changed relative to eachother by tuning the source drain voltage VSD. The sketch shows how the discrete energylevels in a QD can be probed by increasing the bias voltage VSD. The excited states (ofspacing δE) can be seen as peaks in differential conductance measurement.

2.5 Experimental consequences on transport

In this Section we show that both energy scales EC and δE can be extracted from dif-ferential conductance measurements. As discussed above, a differential conductance mea-

2.5 Experimental consequences on transport 11

surement can be used to probe the internal structure of the QD. A measurement of theCoulomb oscillations (dI/dVSD vs. VG, VSD → 0) allows for the estimation of EC , seeFig. 2.3(left panel). The level spacing δE, on the other hand, can be determined from ameasurement of dI/dVSD vs. VSD (with fixed VG), known as the Coulomb staircase.

Note that the peaks in the differential conductance measurements have a finite width(each local level is broadened) for the following two reasons: the local level (i) is coupled toreservoirs with strength Γ and (ii) it is thermally broadened ∼ kBTexp as the experiment iscarried out at finite temperature Texp. Obviously discrete peaks in differential conductancemeasurements are no longer observable once the total width of the level ∼ (Γ + kBT )exceeds the energy scale of interest (EC or δE), see Fig. 2.3(left panel). Thus, in weaklycoupled QDs (Γ ≈ 0.01meV) with level spacings δE ≈ 0.1meV (∼ 1K) and a chargingenergy EC ≈ 1.5meV, discrete peaks in the differential conductance are observable up totemperatures of several hundred mK. Note that an increase in the coupling Γ enhancesquantum fluctuations in the QD leading to a suppression of the discrete peaks in thedifferential conductance.The charging diagram, see Fig. 2.3(right panel), gives a compact specification of a QD. Toillustrate the differential conductance dI/dVSD as a function of both VG and VSD one usesa color scheme. From the position of the maxima of the differential conductance in thecharging diagram, one can extract the relevant parameters EC and δE.

Figure 2.3: Left: Conductance as a function of gate voltage VG in lateral QDs in the limitVSD → 0 [14]. The conductance peaks are separated by EC (EC À δE). As expected,the Coulomb blockade peaks get washed out when the temperature T is increased towardsEC . Right: Charging diagram of a QD [courtesy of A.K. Huttel (LMU)]. The differentialconductance is plotted as a function of both VG and VSD. For VSD = 0 the Coulomboscillations (with period ∼ EC) can be observed. Fixing VG and varying VSD, illustratedin Fig. 2.2(b), allows for the determination of δE, known as the Coulomb staircase.


2.6 Kondo effect in QDs

Under certain circumstances (as will be shown below) a many-body resonance, known asthe Kondo resonance, can develop in QDs. The following two requirements are necessaryfor the observation of this resonance: (i) the system is tuned to the local moment regime(LMR)3 and (ii) the experiment is carried out at a temperature smaller than the so-calledKondo temperature TK , which in turn depends exponentially on the parameters of the QD,cf. Eq. (3.29).

Based on theory, it was predicted in 1988 that the Kondo effect should be observablein QDs [16, 17]. However, it took roughly another ten years before its experimental ob-servation by Goldhaber-Gordon et al. [18]. In Fig. 2.4(a) the setup of this experiment isshown. The QD is coupled (with strength Γ) to two (left and right) leads with identicalchemical potential µL = µR = µ. In order to satisfy the condition Texp < TK , several gatescan be used to tune the QD parameters εd, Γ and U (this quantities that will be explainedin Chapter 3) such that this requirement is fulfilled. This tunability allowed for manycontrolled ’Kondo-experiments’ in the course of the last years, e.g. [19, 20]. Consequently,a better understanding of this non-trivial many-body effect was established within the lastyears.

Figure 2.4: (a) Image of the device used by Goldhaber-Gordon et al. [19] to measure theKondo effect in QDs. This experiment can be very well described by the Anderson model,a model that will be introduced in Section 3.1. (b) Differential conductance dI/dVSD vs.VSD of a QD for temperatures ranging from 15mK up to 900mK [20]. The arrow indicatesan increase in dI/dVSD at VSD ' 0 upon lowering the temperature T (a fingerprint of theKondo effect). Note that the differential conductance dI/dVSD vs. VSD clearly mimics theKondo resonance in the local DoS for sufficiently low-temperatures.

3The LMR is realized if the topmost nonempty level of the QD contains a single electron, acting like afree spin with n↑ = n↓ ∼ 1

2 [15].

2.6 Kondo effect in QDs 13

From a theoretical point of view the key quantity that governs this many-body res-onance is the local DoS. In particular, the sharp resonance in the local DoS pinned atthe Fermi energy of the leads, is responsible for an increase in the linear conductance (forT . TK) as found experimentally in Refs. [19, 20]. Fig. 2.4(b) shows the temperaturedependence of the differential conductance (which is essentially the local DoS) observed inan experiment carried out at TU Delft (Netherlands) [20].From the Friedel-sum rule [21], we know that the height of this resonance is ∝ 1

Γ. The

linear conductance G(T ) is consequently expected to saturate in the limit T → 0 to itstheoretical limit, known as the unitary limit (which is 2e2/h as observed in Ref. [20]; seelower inset in Fig. 2.5), resulting in a perfect transmission of an incoming electron.

The Kondo effect in QDs is usually identified by a logarithmic increase in the linearconductance G(T ) as a function of decreasing temperature (for temperatures ∼ TK), seelower inset in Fig. 2.5. As expected, for temperatures well below TK , a saturation ofG(T ) appears. One experiment showing this signature very nicely was carried out at TUDelft [20]. In this experiment the linear conductance through a QD contacted to leads(with a single channel) really reached the unitary limit of G = 2e2/h, see Fig. 2.5.

Figure 2.5: A measurement of the linear conductance G(T ) vs. gate voltage upon loweringthe temperature Texp (carried out by van der Wiel et al. [20]). For those gate voltages wherethe QD is in the LMR (see inset in the top left corner), a monotonic increase in G(T ) isobserved for a decrease in the temperature T (indicated by two ’up’ arrows). In both’Kondo-valleys’ the theoretical maximum of 2e2/h is almost reached. The logarithmicincrease in G(T ) upon lowering T in a ’Kondo-valley’ is shown in the lower inset. Theconductance in the valley between the two ’Kondo-valleys’, i.e. in a ’non-Kondo’ valley,decreases as T is lowered.


In contrast to the Kondo effect in bulk materials, where the resistivity increases belowTK , in QDs the conductivity increases for T < TK . Kondo correlations lead to an oppositebehavior in QDs as compared to bulk systems.The reason for this behavior is the following: in presence of magnetic impurities the scat-tering is strongly enhanced for temperatures T . TK . Since QDs are contacted to a left anda right lead an increasing scattering enhances the forward scattering drastically. Thus, thetransmission through the QD increases, its resistivity decreases. In bulk materials, on theother hand, a magnetic impurity is connected to various ’channels’. Consequently an in-creasing scattering does not result in a decreasing resistivity, as the electrons are scatteredin all possible directions.

To summarize the findings of this Chapter: the level quantization that appears inQDs constitutes an analogy between ’real’ atomic systems and QDs; consequently QDsare often referred to as ’artificial’ atomic systems. Since a detailed understanding of two-level systems is necessary for the realization of recently proposed quantum computingdevices [22], QD-physics recently gathered additional interest. In particular, intense studieson coupled QDs, ’artificial molecules’ [Fig. 2.1(a)], are carried out at the moment.The analogy between QDs and ’real’ atoms was strengthened when Kondo physics wasmeasured in QDs (1998) [18, 23, 24], roughly ten years after it was predicted theoretically[16, 17]. Kondo physics is of central interest throughout this thesis. Modern many-bodymethods are required to capture the physics of strongly correlated electrons in the Kondoregime properly.Before we focus on Kondo physics, however, we introduce a theoretical model, the Andersonimpurity model, that describes QDs extremely well. It is well-known that this model iscapable of Kondo physics. Thus, it allows for an accurate study of various interestingaspects of this many-body phenomenon.

Chapter 3

Introduction to the Kondo effect

This Chapter is divided into two parts: in the first part we introduce a theoretical modelthat describes the physics of QDs extremely well. The second part of this Chapter isdevoted to Kondo physics. Some aspects of the Kondo effect are addressed there.

3.1 The Anderson model

In this Section we introduce the Anderson model (AM), a model (suggested by P. W.Anderson in 1961 [25]) that describes the physics of single impurities hosted in a metal.This model had a revival when it was realized, roughly 15 years ago, that it is the effectivemodel in the framework of dynamical mean field theory [26]. As the AM is extremely wellsuited for the description of QDs, it got an additional boost when the first controlled QDexperiments were carried out.Within this thesis, the AM is of central relevance since (in contrast to the Kondo model) itallows for the calculation of several experimentally accessible quantities, such as the gatevoltage dependence of the conductance, see Fig. 2.3(left panel).

A theoretical model that captures the physics of QDs has to consist of three parts: (i)a local part (Hd), which describes the isolated QD, (ii) a tunneling part between the QDand its surrounding reservoirs (H`d) and (iii) a part that describes the reservoirs (H`), intotal

HAM = Hd + H`d + H`. (3.1)

We start analyzing Eq. (3.1) by considering Hd: the energetic situation inside the QD,as depicted in Fig. 3.1(a), is determined by the bare energy of the i-th local level εdi

(with typical level spacing δE between neighboring levels),1 the charging energy EC , theexchange energy ES and the Zeeman energy. An isolated multi-level QD therefore takesthe form

Hd =∑iσ

εdid†iσdiσ +

∑i

Uini↑ni↓ +∑

σσ′,i6=j

Uijniσnjσ′ − µBgB∑

i

Szi + J

∑

i6=j

Si · Sj, (3.2)

1The bare level energy εdi is measured relative to the chemical potential of the leads µ.

16 3. Introduction to the Kondo effect

��

��

��

��

��

��

��

��

δE

U

��

��

��

��

eVG

� � � � � � � � � � � � � � � � � � � � � � � �

��

��

��

(a) (b)

εεi+1

i

L R

Γi

Γi+1

L R

Figure 3.1: (a) Relevant parameters for the theoretical description of a QD: the chargingenergy U , the local level position of the i-th level εdi (adjustable by VG) separated from itsneighboring levels by the level spacing δE. Additionally the QD-Hamiltonian Hd, Eq. (3.2),includes a Zeeman and an exchange term. (b) The coupling of the i-th local level to theleads (assumed to be non-interacting) results in a level-broadening of width Γi. In generalneighboring levels are not expected to have equal width [27], Γi 6= Γi+1.

with the replacement EC → U and ES → J . Note that multi-level QDs possess an intra-level Ui and an inter-level charging energy Uij, in contrast to single-level dots that only one

possess an (intra-level) charging energy U . Here diσ (niσ = d†iσdiσ) are the Fermi operatorsfor spin σ electrons in level i of the QD, Sz

i = (ni↑ − ni↓)/2, and Si = 12

∑µν d†iµ σµν diν

(with the Pauli-matrices σµν)2. The fourth term of Eq. (3.2) denotes the Zeeman energy of

the local spin with the Bohr magneton µB and the gyro-magnetic ratio g (a quantity thatcharacterizes the coupling between the magnetic field and the ’QD-material’). Spins thatare aligned parallel to an external magnetic field B are consequently favored. Additionallywe included an exchange interaction J (the first correction to pure Coulomb interaction),a term which allows the dot to lower its energy by maximizing its total spin. This term isespecially important for QDs close to a singlet-triplet transition [28].

The coupling between the QD and its environment (H` is the Hamiltonian of the envi-ronment), see Fig. 3.1(b), is described by the tunneling term H`d,

H`d =∑

kσir

(Vkirc

†kσrdiσ + V ∗

kird†iσckσr

)(3.3)

H` =∑

kσr

εkrc†kσrckσr. (3.4)

Here, the operator c†kσr creates an electron with momentum k (corresponding to an energy

εkr) and spin σ in lead r = L,R. The leads, described by H`, are assumed to be identical,

2 Those are are given by σx =(

0 11 0

), σy =

(0 −ii 0

)and σz =

(1 00 −1

), in standard represen-

tation.

3.1 The Anderson model 17

non-interacting and in equilibrium with a dispersion εkL = εkR = εk. Henceforth we willfrequently use the phrasing conduction band (CB) electrons instead of lead electrons. Weconsider the tunneling matrix elements Vkir in H`d, originating from the overlap of the wavefunctions that participate in the tunneling process (i.e. wave functions that correspond to’impurity’ and ’lead’ electrons, respectively) to be k-independent, Vkir = Vir [29]. Notethat the system lowers its energy when the impurity hybridizes with its environment, seeEq. (3.3), i.e. when impurity electrons become delocalized.

Below, we focus on a single-level AM [30, 31], so the index i will be dropped. Thissimplification avoids tedious mathematics while the general concepts still apply.Due to the coupling between the impurity and the conduction band (Vr 6= 0), the locallevel acquires a width Γ, depicted in Fig. 3.1(b). We neglect the additional temperature-dependent level broadening (∼ kBTexp) here, as we are mostly interested in T = 0 quantitiesin this thesis. One expects, based on second order perturbation theory in the tunneling,that the level width Γ is ∝ |Vr|2 and ∝ ρr(ω), the DoS in lead r [note: ρL(ω) = ρR(ω) =ρ(ω) =

∑k δ (ω − εk)]. More rigorously, this quantity can be computed from the imaginary

part of the non-interacting self-energy = [Σ0], Σ0(iω) =∑

kr|Vr|2iω−εk

. After analytic contin-

uation, i.e. iω → ω + iδ, one immediately3 arrives at = [Σ0] = −π∑

kr |Vr|2 δ (ω − εk).When we assume both leads symmetrically coupled, VL = VR = V , we obtain the totallevel-width Γ = −= [Σ0] [29], or equivalently

Γ(ω) = 2πρ(ω)|V |2. (3.5)

Transport through an interacting single-level QD is therefore characterized by the followingfive relevant energies: U , Γ, εd (all adjustable by external gates), B (an external magneticfield) and Texp (set by the fridge).

Before we really solve the AM [as introduced in Eq. (3.1)], however, we perform aunitary transformation [16], which reveals that a single-level impurity couples effectivelyto one channel only, the symmetric combination of the left and the right lead.

3.1.1 Unitary transformation of the AM

In this Section a rotation in the ’L-R’ basis of lead electrons is introduced. In general asingle-level impurity may couple differently to its left and right leads, VR 6= VL. The basistransformation [16]

(αskσ

αakσ

)=

1√|VL|2 + |VR|2

(VR VL

−VL VR

)(ckσR

ckσL

)(3.6)

defines fermionic operators, αskσ/αakσ, that describe electrons in the symmetric and anti-symmetric channel, respectively. By means of this transformation the single-level Ander-son Hamiltonian HAM is rotated such that it couples to the (symmetric) channel only, i.e.

H`d =∑

kσ

√V 2

L + V 2R

(α†skσdσ + d†σαskσ

)[32] (the antisymmetric channel decouples from

3limη→01

x±iη = P (1x

)∓ iπδ(x) with P denoting the principal value.


the impurity).After this transformation, illustrated in Fig. 3.2, the Hamiltonian takes the more compactform

HAM =∑

σ

εdd†σdσ + Un↑n↓− µBgBSz +

∑

kσ

εkα†skσαskσ +

∑

kσ

V(α†skσdσ + d†σαskσ

)(3.7)

with V =√

V 2L + V 2

R. Note that we have dropped the uncoupled channel∑

kσ εkα†akσαakσ

��

RL

V

��

��V~

S A

V RL

Figure 3.2: The unitary transformation Eq. (3.6) reveals that a single-level impurity coupleseffectively to one channel only (described by the fermionic operator αskσ), the symmetriclinear combination of the left and the right lead [∼ (VL ckσL + VR ckσR)]. The strength ofthis coupling is given by V =

√V 2

L + V 2R.

in Eq. (3.7), which does not influence QD-operators at all.QDs can be tuned via VG such that the dot is on average singly occupied [33]; a scenario

that is, as noted in Section 2.6, essential for Kondo physics. For this particular choice ofVG the topmost occupied level contains a single, unpaired electron (known as the LMR)4,which can be used to mimic a magnetic impurity.As will be shown below, an AM in the LMR can be projected onto an effective model.As charge fluctuations are small in this regime, it suffices to perform this projection viasecond order perturbation theory in the tunneling (where Γ/εd is the small parameter).This projection is known as Schrieffer-Wolff (SW) transformation [30].

3.1.2 Mapping onto an effective model: Schrieffer-Wolff trans-formation

To realize a QD in the LMR we fix the gate voltage εd such that

εd ¿ −|Γ| ¿ µ ¿ |Γ| ¿ εd + U, (3.8)

4 For completeness we mention all possible regimes of a single-level AM here: the mixed valence regime(MV) is characterized by large charge fluctuations, realized for εd ∼ µ or (εd +U) ∼ µ. That regime wherethe local level is either empty (εd À µ) or doubly occupied (εd +U ¿ µ) is called the empty orbital regime(EO).

3.1 The Anderson model 19

ensuring that the QD is singly occupied, 〈n〉 ≈ 1. For this particular choice of εd the emptyand the doubly occupied states in the QD can be disregarded, since they are energeticallyhighly unfavorable. Thus, the impurity is either occupied with an ⇑ or ⇓ electron (for therest of the first part of this thesis we use the following notation: σ =⇑ / ⇓ corresponds toa spin σ electron in the impurity and σ =↑ / ↓ to a spin σ electron in the CB). Thus, theimpurity spin can be described by a spin operator S, S ≡ 1

2

∑µν d†µσµν dν .

Due to virtual excitations, see Fig. 3.3, there is still a small (but finite) probabilityfor the QD to contain zero or two electrons, even though condition (3.8) is fulfilled. For|εd| À Γ, these virtual excitations are appropriately described by second order perturbationtheory in the tunneling between the local level and the CB-electrons. Due to the Pauli-principle, virtual processes are only possible between CB and impurity electrons withanti-parallel spin [34]. As these processes lower the energy of the system by an amount∆E, the effective Hamiltonian - describing the low-energy properties of the system - shouldinclude a term that favors anti-parallel alignment between electron spin in the QD and theCB.There are two types of virtual processes, hole-like (excitations to an unoccupied local level;lower middle panel in Fig. 3.3) and electron-like (excitations to a double occupied locallevel; upper middle panel in Fig. 3.3) processes. A hole (electron)-like process lowers5 theenergy of the system [35] by ∆Eh (∆Ee)

∆Eh(εk) =V 2[1− f(εk)]

εk − εd

(3.9)

∆Ee(εk) =V 2[f(εk)]

U + εd − εk

, (3.10)

with the Fermi function of the leads f(εk) = 1/(1 + e(εk−µ)/kBT ).Fig. 3.3 sketches all relevant virtual processes. The impurity spin might: (i) remain

unchanged (either with σ =⇑ or σ =⇓), (ii) flip its spin from ⇑→⇓ (in two possible ways -corresponding to a transition from the left side to the right side in Fig. 3.3) or (iii) flip itsspin oppositely from ⇓→⇑ (also in two possible ways - corresponding to a transition fromthe right side to the left side in Fig. 3.3). As the total spin is conserved, the CB electronspin has to flip oppositely to the impurity spin [in the processes (ii) and (iii)] or remainunchanged [in process (i)]. A compact representation of all those processes is realized byrewriting them with spin operators. Consequently, a QD in the LMR shall contain a termS · s0 =

[Szsz

0 + 12

(S+s−0 + S−s+

0

)]with the QD spin operator S, the CB spin operator s0,

s0 ≡ 12

∑kk′

∑µν α†skµσµναsk′ν [with αskσ as defined in Eq. (3.6)] and the usual definition

of the ladder operators S+/− and s+/−0 , respectively. A spin-flip event between the left and

the right ground state shown in Fig. 3.3, for example, is described by the operator S−s+0 .

Due to the Fermi functions, only states εk > µ [εk ≤ µ] contribute in Eq. (3.9)[Eq. (3.10)]. As we restricted ourselves to εd ¿ µ ¿ εd + U , see Eq. (3.8), we canapproximate the denominators in Eqs. (3.9) and (3.10) by their smallest possible values,

5Virtual excitations lower the energy by an amount ∆E ≈ V 2

Eexc−Eini.


��

��

��

��

��

��

��

��

��

��

εd

εd+U εd+U

εd � � � � � � � � � � � �

��

��

��

Figure 3.3: Sketch of a QD in the LMR with its two possible ground states: the electronin the QD might either be a spin ⇑ (left side) or a spin ⇓ electron (right side). In themiddle of the figure the possible virtual excitations (obeying the Pauli principle), due tohybridization between impurity and lead electrons, are shown. The lower middle panelshows a hole-like excitation, realized by a CB hole that tunnels into the QD. The upperpanel in the middle shows an electron-like excitation, where a CB electron tunnels into theQD. The energy gain due to these processes is described in Eq. (3.9) and (3.10). As theleft and the right state in the figure are connected by virtual transitions the possibility ofa spin-flip inside the QD exists!

i.e. εk,k>µ − εd ≈ µ− εd and U + εd − εk,k≤µ ≈ U + εd − µ.Thus, virtual processes lower the systems energy by an amount

J =V 2

µ− εd

+V 2

U + εd − µ. (3.11)

Summing up all possible virtual transitions sketched in Fig. 3.3, we arrive at an effectiveHamiltonian for the LMR.

Heff = H` + 2JS · s0, (3.12)

with the local (Heisenberg) coupling J between the impurity spin and the conduction elec-tron spins.This heuristic arguments illustrated the general concept of a Schrieffer-Wolff (SW) transfor-mation. For a detailed, more rigorous discussion, see Ref. [30] or [36]. A SW transformationenables one to project a general Hamiltonian (which is the single-level AM here) into a spe-cial subspace of its full Hilbert space (the LMR). The corresponding effective Hamiltonian,Eq. (3.12), describing the LMR is known as the Kondo Hamiltonian Heff = HK .

3.2 The Kondo effect 21

In this Section we argued, based on a perturbation expansion in the tunneling, howthe impurity can lower its energy by means of virtual transitions. In particular processeswhere the impurity spin flips will become of great interest below, when the Kondo effectwill be discussed.

3.2 The Kondo effect

The non-trivial physics associated with the presence of magnetic impurities in a solid isreferred to as the Kondo effect. The experimental discovery of a shallow minimum inthe resistivity of metals that contain magnetic impurities (at temperatures T ∼ 10K)triggered big interest in this field. Kondo was able to relate this phenomenon to spin-flipscattering events. Kondo could show that this scattering mechanism becomes more andmore dominant when the temperature of the system is lowered successively.

3.2.1 The historical origin - the resistivity minimum in bulk

Back in 1934 de Haas, de Boer and van den Berg were measuring the electrical resistivityρel(T ) of Au as a function of temperature. They observed an unexpected local minimumof the resistivity at temperatures T ∼ 10K, see Fig. 3.4. In several other experiments itwas confirmed that the resistivity of ’pure’ metallic samples (like gold, silver and copper)passes through a minimum when the temperature is gradually decreased. However, as wasfound later, the ’pure’ samples were not really pure but contained a small concentration ofmagnetic impurities.

The resistivity of a metal is determined by different scattering mechanisms: (i) theelectrical resistivity due to the scattering between conduction electrons and lattice distor-tions (phonons) ρel

Phonon ∝ T 5 should clearly die out in the limit T → 0. (ii) The resistivitycontribution stemming from electron-electron scattering, ρel

e−e ∝ T 2 (known from Fermiliquid theory), is also expected to vanish for very small temperatures. (iii) The scatteringbetween electrons and static impurities is temperature independent. Consequently alsothe corresponding resistivity reveals no T -dependence (static impurities are present in aconstant concentration, say cimp).

6 Summing up the contributions stemming from themechanisms (i)-(iii) suggests a monotonic temperature dependence of the electrical resis-tivity ρel(T ) = acimpρ

el0 + bT 2 + cT 5 (a, b and c denoting proportionality constants and

ρel0 a characteristic resistivity). Additionally a saturation is expected in the limit T → 0,

limT→0 ρel(T ) = acimpρel0 . Clearly the mechanisms (i)-(iii) can not explain the above men-

tioned anomaly in the electrical resistivity, sketched in Fig. 3.4(a).It took about thirty years until the puzzle of a minimum in ρel(T ) was solved by J.

Kondo (1964) [38]. He associated the minimum with the presence of magnetic impurities,such as Co, in the measured samples.

Magnetic impurities allow for a novel scattering mechanism, spin-flip scattering, notincluded in the mechanisms (i)-(iii). In particular, one finds that this scattering mechanism

6Both magnetic and non-magnetic impurities participate in this scattering mechanism.


Figure 3.4: (a) Sketch of the temperature dependence of the electrical resistance for puremetals (solid line) and metals that contain a small concentration of magnetic impurities(dashed line). Note the local minimum in the resistivity around T ∼ 10K for samplescontaining magnetic impurities. (b) Picture of Jun Kondo. Figure and picture from [37].

is temperature dependent. The spin-flip scattering introduces a new energy scale in theproblem, the Kondo temperature TK . It turns out, that this scattering mechanism startsto dominate for temperatures T that are comparable to TK . Indeed, the logarithmicincrease in the resistivity for temperatures smaller than ∼ 10K, e.g. observed observedin the experiments of de Haas et al., can be nicely explained by that type of scattering[ρel

Kondo(T ) ∝ ln(TK/T )]. The local minimum of ρel(T ) can naturally be explained by addingup all scattering contributions to the resistivity

ρel(T ) = acimpρel0 + bT 2 + cT 5 + cimpρ

el1 ln(TK/T ), (3.13)

with an additional characteristic resistivity ρel1 .

In Section 3.2.3 we give a derivation for the logarithmic temperature dependence of theelectrical resistivity due to the scattering of electrons from magnetic impurities. For thissake we examine the Kondo model, the ’LMR-limit’ [see Eq. (3.8)] of the more generalAM, carefully.


3.2.2 The Kondo model

In 1964 J. Kondo [38] introduced a model that describes the scattering of conductionelectrons from a localized magnetic impurity, i.e. a localized spin [39]. This model, theKondo model [cf. Eq. (3.12)],

HK = H`+2JS·s0 = H`+∑

kk′J

[Sz

(α†sk↑αsk′↑ − α†sk↓αsk′↓

)+ S+α†sk↓αsk′↑ + S−α†sk↑αsk′↓

],

(3.14)was motivated by experiments carried out in the 1930s, as explained in Section 3.2.1. Asmentioned before, the operators S and s0 in Eq. (3.14) denote the impurity and conduction

electrons spin operators, respectively, with S+ |⇓〉 = |⇑〉 and S− |⇑〉 = |⇓〉 (s+/−0 acts

accordingly on CB-electrons). In contrast to the AM (parametrized by V , εd and U),the Kondo model contains only the parameter J , which is related to the parameters ofthe AM via Eq. (3.11). The parameter J characterizes the coupling strength between thelocalized spin and the CB electrons. As HK has non-vanishing matrix elements between thestates 〈⇓| 〈↑| and |↓〉 |⇑〉, 〈⇓| 〈↑| HK |↓〉 |⇑〉 6= 0, the Kondo Hamiltonian obviously containsspin-flip processes. Here the state |↓〉 |⇑〉, for instance, mimics a state where the impuritycontains a spin ⇑ electron and the CB is represented by a spin ↓ electron.

Kondo calculated the resistivity in a perturbation expansion of HK in J . He found thatthe second order contribution (in J) to the scattering amplitude diverges logarithmicallyfor temperatures smaller than a characteristic temperature, the Kondo temperature TK .Therefore the perturbative approach of Kondo is only valid for temperatures T > TK .For temperatures T ≤ TK the proper (theoretical) understanding of the KM arises from’scaling’ ideas, suggested by P.W. Anderson in the late 1960s. This ideas were used todevelop an accurate theoretical description for the regime T < TK . The Renormalizationgroup (RG) theory (developed by Anderson and Wilson) is the appropriate theory in thisregime.

3.2.3 Kondo’s explanation

In metals conduction electrons can be described by plane waves carrying a crystal mo-mentum, say k. The scattering of electrons from magnetic impurities can be described asfollows: an incoming spin σ electron of momentum k gets scattered into an outgoing spinσ′ electron of momentum k′. Since the total spin is conserved in the scattering event, thelocalized spin has to flip if σ 6= σ′. One finds, interestingly, that the spin-flip scattering istemperature dependent. We are going to derive that T -dependence here.

Our interest in scattering events between a localized magnetic impurity (of spin S) withelectrons of momentum k and spin σ suggests to label the involved conduction electronstates as

|Ψ〉 = |k σ〉 (3.15)

and to write the impurity spin in terms of the impurity spin operator S. After the scatteringevent the electron carries a momentum k′ and a spin σ′. The perturbation expansion of


HK in J can be performed by rewriting Eq. (3.14) as HK = H` + H′, with H′ describingthe interaction of the CB electrons with the impurity spin, H′ = 2JS · s0.

In order to determine the eigenstates of HK , HK |Ψ〉 = ε |Ψ〉, we rewrite the Schrodingerequation as (ε − H`) |Ψ〉 = H′ |Ψ〉. The general solution of this equation is a sum of the’homogeneous’ solution and the ’particular’ solution. We can immediately identify planewaves |Ψ0〉 as eigenstates of H`, H` |Ψ0〉 = ε |Ψ0〉, i.e. as the ’homogeneous’ solution.

The formal solution of the full Schrodinger equation is

|Ψ〉 = |Ψ0〉+1

ε + i0+ − H`

H′ |Ψ〉 , (3.16)

known as the Lippmann-Schwinger equation (see e.g. [40]). Eq. (3.16) can be solved bysubstituting the ’new’ value of |Ψ〉 in the r.h.s. of this equation, resulting in |Ψ〉 = |Ψ0〉+

1

ε+i0+−H`T |Ψ0〉. The hereby defined T -matrix takes the form

T =

T (1)︷︸︸︷H′ +

T (2)︷︸︸︷H′ 1

ε + i0+ − H`

H′ +

T (3)︷︸︸︷H′ 1

ε + i0+ − H`

H′ 1

ε + i0+ − H`

H′ + . . . . (3.17)

Eq. (3.17) shows the first three contributions to the T -matrix of the perturbation expansionin J .To first order in J , the T -matrix consists of six possible scattering events, sketched inFig. 3.5. Whereas the impurity spin is conserved in Fig. 3.5(a) [two possibilities] and (b)[two possibilities], it flips in Fig. 3.5(c) [one possibility] and (d) [one possibility].The first-order contributions to the T -matrix can thus easily be inferred from Fig. 3.5 [34]

〈k′ ↑| T (1) |k ↑〉 = JSz,

〈k′ ↓| T (1) |k ↓〉 = −JSz,

〈k′ ↑| T (1) |k ↓〉 = JS−,

〈k′ ↓| T (1) |k ↑〉 = JS+. (3.18)

Note that we kept the dependence on the impurity spin in terms of the impurity spinoperator here. We continue to use this notation below.The probability to scatter from a plane wave with momentum k to another one withmomentum k′, Wkk′ , is related to the amplitude for this scattering event Tkk′ . In particular,T

(1)kk′ , a matrix element of T (1), contains all possible transitions given in Eq. (3.18). The

second order contribution to the scattering probability has the form

Wkk′ =2πNimp

~

∣∣∣T (1)kk′

∣∣∣2

=1

2|J |2 2πNimp

~

2Sz(Sz)† +

2Sz︷︸︸︷S+(S+)† + S−(S−)†

= |J |2 2πNimp

~S(S + 1). (3.19)


k k’

k’k

Sz

k k’

S+

−S

k’k

(a)

(b)

−Sz

, ,

,,

(c)

(d)

Figure 3.5: Possible processes for the T -matrix to first order in J . The lower (dashed) linerepresents the impurity (with corresponding spin S indicated by thick arrows), whereasthe curved (solid) lines represent the CB electrons [with initial (final) momentum k (k′)and spin σ (σ′)]. The impurity spin S (and correspondingly the CB electron spin σ) isconserved in the scattering processes (a) and (b) and it is flipped in processes (c) and (d).

Note that the factor 1/2 [in the second line of (3.19)] ensures the proper average over theinitial impurity spin configurations. Due to our notation, we replaced the impurity spinoperators with their expectation values in the third line of Eq. (3.19), S = 〈Sz〉. Thequantity Nimp labels the number of magnetic impurities in the sample.To finally compute the resistivity ρel

imp, we need to compute the transport relaxation timeat the Fermi energy τ(kF ),

ρelimp =

m

ne2τ(kF ), (3.20)

with the density of conduction electrons at the Fermi energy n = N/V = k3F /3π2. The

transport relaxation time depends sensitively on Wkk′ .7 To second order in J this relaxation

time for an electron at the Fermi surface, i.e. k = kF , is given as [τ(kF )]−1 =3πJ2S(S+1)cimpn

2εF ~ ,

with the Fermi energy εF = ~2k2F /2m and the impurity concentration cimp. Finally, we ob-

tain the second order contribution (in J) to the resistivity by inserting τ(kF ) in Eq. (3.20),

ρel,(2)imp =

3πmJ2S(S + 1)cimp

2e2εF~. (3.21)

Note that ρel,(2)imp is temperature independent indicating that it can not explain the anomalous

temperature dependence found in the experiments of de Haas et al. .

7 1

τ(~k)=

∑~k′ W~k~k′(1− cos θ′)δ(ε~k − ε~k′) with the angle θ′ between ~k and ~k′.


Now, we will go to the next order in perturbation theory, i.e. to T (2). It will turn outthat in this order a prefactor appears that diverges logarithmically in the limit of smalltemperatures.

When second order processes are considered (T (2) = H′ 1

ε+i0+−H`H′), corresponding

diagrams are shown in Fig. 3.6, intermediate states of momentum ki appear that have tobe summed. Fig. 3.6 contains all possible second order processes for incoming and outgoingCB electrons with spin ↑. For instance, Fig. 3.6(c) describes a virtual process, where firstan incoming electron (with momentum k and spin ↑) scatters from the impurity, therebyflipping both the impurity and its own spin and changing its momentum to the intermediatevalue ki. Afterwards, the virtually excited conduction electron (of momentum ki and spin↓) scatters again with the impurity by flipping both spins again and finally leaving withfinal momentum k′ and spin ↑. This second order process, see Fig. 3.6(c), contributes the

k k’k k’ki

SzSz S+ S−

k k’

ki

ki

k k’ki

Sz Sz S+S−

(a)

(b)

(c)

(d)

, ,

, ,

,

,

Figure 3.6: All possible second order processes for CB electrons entering and leaving thesystem with spin ↑. The intermediate states can either be electron-like [(a) and (c)] or hole-like [(b) and (d)]. Note that the impurity spin (thick arrows) flips virtually in processes[(c) and (d)] in contrast to processes [(a) and (b)].

following term to 〈k′ ↑| T (2) |k ↑〉:∑

ki

J2 〈k′ ↑|α†sk′↑αski↓S− 1

ε + i0+ − H`

S+α†ski↓αsk↑ |k ↑〉 =∑

ki

J2S−S+ [1− f(εki)]

ε− εki+ i0+

,

(3.22)

where the relation⟨αski↓

1

ε+i0+−H`α†ski↓

⟩=

1−f(εki)

ε−εki+i0+ was used. Here f is the Fermi-Dirac

distribution.Another second order contribution to 〈k′ ↑| T (2) |k ↑〉 is shown in Fig. 3.6(d). The diagramdescribes a scattering process, where first the impurity spin is flipped (⇑→⇓) by creatingan electron-hole pair with an outgoing electron of momentum k′ and spin ↑. After this


event, an incoming electron of momentum k and spin ↑ scatters with the impurity a secondtime by flipping its spin back to its original state (⇓→⇑) and thereby destroying the holeof momentum ki. Evaluating this diagram results in

∑

ki

J2 〈k′ ↑|α†ski↓αsk↑S+ 1

ε + i0+ − H`

S−α†sk′↑αski↓ |k ↑〉 = −∑

ki

J2 S+S−f(εki)

ε− εk − εk′ + εki+ i0+

=∑

ki

J2 S+S−f(εki)

ε− εki− i0+

, (3.23)

where we used ε ≈ εk ≈ εk′ ≈ µ since we are interested in having both values k and k′ nearthe Fermi level.

The two remaining diagrams in Fig. 3.6, namely (a) and (b), can be evaluated analo-gously and give the following contributions to 〈k′ ↑| T (2) |k ↑〉,

∑

ki

J2 (Sz)2 [1− f(εki)]

ε− εki+ i0+

and∑

ki

J2 (Sz)2 f(εki)

ε− εki− i0+

, (3.24)

respectively. Note that the sum of the two contributions given in Eq. (3.24) is temperatureindependent, since the contributions that include the Fermi function cancel each other8.The contributions from Eqs. (3.22) and (3.23), however, do not cancel. Therefore we iden-tify the spin-flip scattering with the mechanism that gives rise to a temperature dependentscattering.

To second order in J , we approximate the T -matrix with T ≈ T (1) + T (2). Thesecond order contribution T (2) leads to a correction of the scattering probability Wkk′ ,see Eq. (3.19), which we call δWkk′ . The correction δWkk′ is of third order in J . As

Wkk′ ∼ |Tkk′|2 ∼∣∣∣T (1)

kk′

∣∣∣2

+[T

(1)kk′

(T

(2)kk′

)∗+

(T

(1)kk′

)∗T

(2)kk′

]we obtain the correction δWkk′ to

the transition probability as

~2πNimp

δWkk′ = T(1)kk′

(T

(2)kk′

)∗+

(T

(1)kk′

)∗T

(2)kk′ ∼ J3. (3.25)

Adding all spin contributions to δWkk′ leads to

~2πNimp

δWkk′ ∼ S(S + 1)J3∑

ki

f(εki)− 1

2

ε− εki

. (3.26)

The third order contribution to the resistivity, however, requires the tricky determination ofthe corresponding transport relaxation time τ(k). To do so, the sum over the intermediatestates has to be performed

∑

ki

f(εki)− 1

2

ε− εki

∼∫

dεiρ(εi)f(εi)− 1

2

ε− εi

=1

2ρ(0)

∫ D

−D

dεitanh(εi/2kT )

ε− εi

. (3.27)

8The summation over ki is taken as a principal value integration.


Here the energy-dependent DoS ρ(εi) has been replaced by its value at the Fermi energyρ(µ), µ = 0. Moreover, the high-energy divergence has been cut off by restricting theconduction electron states to a band of finite width 2D (D is usually referred to as thebandwidth of the conduction band).We will now consider two limiting cases to obtain an interpolation formula for the inte-gral given in Eq. (3.27): (i) for ε ¿ kT , we approximate the integral with 2 ln(D/kT ),whereas (ii) for ε À kT we approximate it with 2 ln(D/|ε|). This leads to the com-

pact - temperature dependent -approximation∑

ki

f(εki)−1/2

ε−εki∼ ln

(D

max(|ε|,kT )

). There-

fore, the third order contribution (in J) to the transport relaxation time, [τ(kF )]−1 =3πJ2S(S+1)cimpn

2εF ~

[1 + 4Jρ(0) ln D

max(|ε|,kT )

], see also [34], introduces a temperature dependence

in the resistivity.Remarkably T (2) reveals that the existence of magnetic impurities leads to an enhanced

scattering rate for electrons in the close vicinity of the Fermi energy for sufficiently lowtemperatures. This enhanced scattering, known as the Kondo resonance, explains thetemperature dependence of the resistivity observed in bulk materials. Surprisingly, theenergy scale that is related to this phenomenon, the Kondo temperature TK , is not anintrinsic energy scale of the problem such as εd, U or V (in case of the AM). The value ofTK results from an interplay between the impurity spin and all its surrounding conductionelectrons, it is a many-body phenomenon.

Including this order, the resistivity, see Eq. (3.21), becomes temperature dependent. Inparticular, when we assume kT > ε the resistivity takes the form

ρel,(3)imp =

3πmJ2S(S + 1)cimp

2e2εF~[1− 4Jρ(0) ln (kT/D)] . (3.28)

The temperature dependent contribution to the resistivity indeed agrees very well with theresistance minimum found in the experiments of de Haas et al., mentioned in Section 3.2.1.Note that Eq. (3.28) reproduces the logarithmic divergence that was experimentally found,see Eq. (3.13). We are now able to clarify the physical meaning of all contributions to theresistivity, ρel(T ) = acimpρ

el0 + bT 2 + cT 5 + cimpρ

el1 ln(TK/T ). Whereas ρel

0 is a specificresistivity that includes magnetic and non-magnetic impurities, ρel

1 includes only magneticimpurities. At this point we do not want to specify the dependence of TK on the parametersof the system further. This will be done in the next Section.

At the end of this Section, we want to allude a new problem, which arises from the per-turbation expansion performed above: what happens for small temperatures (in particularin the limit T → 0)?Obviously, the behavior suggested in Eq. (3.28) has to be unphysical for T → 0 since itimplies a divergent resistivity in this limit. In fact, the perturbative expansion for theresistivity can not be trusted any more, once the third order contribution becomes of theorder of the second order contribution, i.e. for |4Jρ(0) ln (kT/D)| ∼ O(1) [see Eq. (3.28)].Clearly this happens for sufficiently small temperatures.

For these temperatures, the perturbative method fails and one needs scaling techniquesfor a proper solution of the Kondo problem. Therefore a lot of conceptual work was


needed to overcome the breakdown of perturbation theory to properly describe the emerg-ing regime. We investigate this regime, the regime of extremely small temperatures, in thenext Section.

3.2.4 General properties of the Kondo effect

It was realized in Section 3.2.3, that the spin-flip scattering introduces a logarithmic tem-perature dependence to the electrical resistivity. Obviously, a degeneracy between the twospin states of the impurity is necessary for spin-flip scattering to appear. A magneticfield B [of strength TK (more precisely µBgB = kBTK), see below] clearly removes thisdegeneracy and thereby destroys spin-flip scattering and consequently the Kondo effect.

Below the temperature where perturbation theory breaks down, known as the Kondotemperature TK , an effective screening of the unpaired spin (i.e. the magnetic impurity)takes place due to coherent virtual transitions between the impurity spin and its surround-ing conduction electrons; the (many-body) ground state [40] of the system is a singlet ofbinding energy TK .9 All conduction electrons are arranged such that the magnetic momentis screened and the singlet can be formed. Thus for T < TK , kBTK ∼ De−1/[2ρ(0)J ] (seee.g. [40]) with ρ(0) the leads DoS at the Fermi energy, the localized magnetic impurityinteracts strongly with its surrounding electrons.Once T becomes less than TK the system is in the strong coupling regime, universal scalingsets in. In absence of a magnetic field no other energy scale is left in the problem [40].This means all measurable quantities, like the conductance G(T ), scale as T/TK or ω/TK

for energies smaller than TK .In the framework of the single-level AM [tuned such that the LMR is realized, see

Eq. (3.8)] the value of TK can be estimated in the framework of poor man’s scaling (asdone by Haldane [41]) or by Bethe-Ansatz [29]

kBTK =1

2

√UΓeπεd(εd+U)/ΓU . (3.29)

Note the exponential dependence of TK on the system parameters εd, U and Γ. Accordingto Eq. (3.29), TK is minimal for εd = −U/2. It increases as εd approaches 0 or −U .

For fixed gate voltage VG (i.e. fixed εd), on the other hand, TK can be exponentiallyenhanced by increasing Γ. This can be achieved by opening the QD.This knowledge is of experimental relevance, as it allows one to tune TK to values that areexperimentally accessible (i.e. Texp < TK). However, a continuous increase of Γ has otherdrawbacks. Once Γ exceeds the level-spacing δE, transport processes involve more thanone level. Consequently a multi-level AM has to be considered. We study such a modelin Chapter 6, for instance, where we focus on the εd-dependence of the occupation of amulti-level AM.

The crossover scale TK , given in Eq. (3.29), can also be extracted from the followingdynamic quantity: the imaginary part of the spin susceptibility χ′′(ω), defined in Eq. (4.9),

9In this regime the impurity binds on average one conduction electron to form this singlet. A magneticfield of strength B & TK destroys this singlet.


shows Curie-Weiss behavior for temperatures T > TK , χ′′(ω) ∼ 1ω+TK

, i.e. χ′′(ω) decays

as ∝ ω−1 (apart from logarithmic corrections) for ω > TK . For T < TK , on the otherhand, the system behaves like a Fermi liquid, χ′′(ω) ∝ ω for ω < TK . Numerically, it isconvenient to extract TK from the (numerically) computed spin susceptibility χ′′(ω) byidentifying it with the maximum of χ′′(ω).10 In case of a single-level AM this (numerical)determination of TK agrees very well with the values for TK obtained from Eq. (3.29). Formore complicated models (such as the two-level AM), however, Eq. (3.29) does not applyany more. Then χ′′(ω) can still be used to extract the value of TK .

The fact that the ground state of the system is a singlet for T < TK results in a sharpresonance in the local DoS A(ω) pinned at the Fermi energy of characteristic width ∼ TK

(see Section 4.1.2). Indeed, a three-peak structure in A(ω), two broadened peaks (of widthΓ) at energies εd and εd +U and the Kondo peak, can nicely be seen in Fig. 4.4(left panel),which was calculated with NRG. The determination of the local DoS is crucial since it isthe key quantity for the computation of transport properties.

To summarize: Kondo correlations are established if (i) there are (at least) two degener-ate states in the system (like spin-degenerate states), (ii) the degenerate states are coupledto a Fermi sea with finite strength Γ and (iii) the average number of electrons in the systemis fixed (U suppresses adding a further electron). Some references to experimental resultsrelated to QDs in the Kondo regime were already given in Chapter 2.6.

3.2.5 Poor man’s scaling (PMS)

In 1970 P.W. Anderson introduced a method, the poor man’s scaling (PMS) method [42], toderive an effective Hamiltonian that captures the low-energy properties of a given system.Anderson’s idea was to obtain an effective Hamiltonian, say at an energy ω∗, by succes-sively integrating out high energy states of the system until the scale ω∗ was reached. Heimposed the physical condition that this scaling procedure (equivalent to a decrease in thebandwidth from D to D, see Fig. 3.7) leaves the scattering between conduction electronsand the impurity invariant. One can satisfy this requirement by continuously adapting thephysical parameters of the Hamiltonian (say the coupling J in case of HK) under study, i.e.to allocate a cutoff-dependent parameter J(D) to the corresponding effective HamiltonianHeff(D). This adjustment results in a ’flow’ of the parameter J(D), the coupling becomescutoff dependent.

PMS for the Kondo model

Here we want to discuss the PMS of the KM. Since this procedure can be easily found inthe standard literature, see e.g. Ref. [40] or [43], we keep this discussion short.A scaling equation for the isotropic KM is obtained, when one successively eliminates theexcitations (say of energy ki) that lie in the band edge D ≤ |ki| ≤ D. When one restrictsall excitations that appear in the second order diagrams discussed in Section 3.2.3 (see

10In Chapter 4 we introduce the NRG method and explain how to compute χ′′(ω) in the framework ofthis method. The right panel of Fig. 4.4 shows χ′′(ω) calculated with NRG.


especially Fig. 3.6), to this interval11 the scaling equation for the Kondo coupling J can bederived. For a flat band (ρ =const.) the scaling equation of the Kondo has the form

dJ

d ln D= −2ρJ2. (3.30)

A detailed derivation of this equation can, for instance, be found in Section 3.3 of [40].The consequence of a decrease in the cutoff D can be understood when one integrates

Eq. (3.30) from D to D. This integration yields

J(D) =J

1 + 2ρJ ln(D/D)(3.31)

with the effective coupling J corresponding to the reduced bandwidth D. Provided we arein the weak coupling regime (ρJ ¿ 1), we can infer from Eq. (3.31), that J is continuouslygrowing since 1 + 2ρJ ln(D/D) < 1 (D < D).

The bandwidth is continuously reduced until it reaches the scale of interest, the tem-perature T , D ' T . Consequently, the effective coupling at a temperature T is given asJ(T ) = J

1+2ρJ ln(T/D). One immediately realizes that J(T ) diverges, for 1+2ρJ ln(T/D) = 0.

The temperature that is related to the divergence in J(T ) is called the Kondo temperatureTK

TK = De−1/(2ρJ). (3.32)

PMS for the Anderson model

Here we apply the PMS method to the AM. Following Haldane [41] we calculate how theenergy levels of the AM are renormalized if high energy states are integrated out.

To illustrate this flow, we consider a single-level AM in the limit U → ∞, i.e. therelevant energy level is either empty |0〉 or singly occupied with a spin σ electron |1, σ〉.The flow of the energy of the empty level ε

(0)d and the singly occupied level ε

(1)dσ is obtained,

as mentioned above, by successively integrating out the ’high energy’ band-states (seeFig. 3.7). By this we mean the following: the presence of the highly excited states, of

energy ∼ D, is absorbed by a change of the energies of the empty δε(0)d and singly occupied

levels δε(1)dσ . Within second order perturbation theory in V we obtain

|0〉 : δε(0)d = ε

(0)d − ε

(0)d =

∑σ

∫ −D

−D

ρσ(ω)|Vσ|2ε(0)d − (ε

(1)dσ + |ω|)

dω (3.33)

|1, σ〉 : δε(1)dσ = ε

(1)dσ − ε

(1)dσ =

∫ D

D

ρσ(ω)|Vσ|2ε(1)dσ − (ε

(0)d + |ω|)

dω

+

∫ −D

−D

ρσ(ω)|Vσ|2ε(1)dσ − U + |ω|

dω

︸︷︷︸→0 f. U→∞

. (3.34)

11This means that all ki’s in expressions like Eq. (3.22) are restricted to the interval D ≤ |ki| ≤ D.


��

��

��

��

−D

DD

µ

ω

States to berenoved

−D

~

~

Figure 3.7: The marked high energy states (of width δ, δ → 0) are those that are integratedout in the course of the scaling procedure. D denotes the full bandwidth and D the neweffective bandwidth after the high energy states have been integrated out. This procedureleads to a ’flow’ of the parameters of the considered model so that the derived Hamiltoniandescribes the low-energy physics of the system properly.

Here the tilde denotes the renormalized local levels and σ =⇑ / ⇓ if σ =⇓ / ⇑. Whenwe assume the leads DoS to be constant and spin independent, ρσ(ω) = ρ, and defineΓσ ≡ 2πρ|Vσ|2, this leads to

δε(0)d = −

∑σ

(Γσ

2π

) ∫ −D

−D

dω

|ω|+ ε(1)dσ − ε

(0)d

(3.35)

δε(1)dσ = −

(Γσ

2π

) ∫ D

D

dω

|ω|+ ε(0)d − ε

(1)dσ

. (3.36)

As long as ε(1)dσ − ε

(0)d ¿ D we can approximate |ω| + ε

(0)d − ε

(1)dσ ≈ |ω| in the integrals

in Eqs. (3.35) and (3.36). Thus the flow of εdσ, most easily seen in differential form

δεdσ = δε(1)dσ − δε

(0)d , is described via

dεdσ

d ln D=

Γσ

2π, (3.37)

equivalent to Eq. (2) of [44] [valid for U → ∞]. Note that the renormalization of εdσ

depends on Γσ, i.e. on the coupling of the opposite spin species. The reason is that, theempty level |0〉 hybridizes with the singly occupied level |1, σ〉 (involving band states ofboth spin directions), so both spin species contribute (see Eq. 3.33), while for the σ-occupied level |1, σ〉 (which hybridizes only with |0〉 for U À |εd|) only spin σ electronscontribute (Eq. 3.34), and after a partial cancellation, only the contribution of σ survives.

Chapter 4

Wilson’s Numerical RenormalizationGroup (NRG)

Naively one might expect that high energy states can be disregarded when one is in-terested in the low-temperature properties of a system. As the Fermi function f issmoothened around the Fermi energy µ with a width ∼ kBT (known from the Sommerfeld-expansion [45]) one is tempted to consider only states of energy |ω−µ| ≤ kBT to be relevantfor transport.1 However, as the intermediate states of momentum ki in the (second order)perturbation expansion (see Section 3.2.3) are not restricted to this interval, one might al-ready anticipate, that in the Kondo problem states of all energies are relevant. To accountfor all energies of the problem, |ω| ≤ D, the idea of the renormalization group (RG) wasintroduced.

Within the PMS-approach, introduced in Section 3.2.5, high energy states are succes-sively integrated out, leading to an effective ’flow’ of the parameters of the model. Aprominent example of such a flow, as discussed in Section 3.2.5, is the flow of the pa-rameter J (in case of the Kondo model), see Fig. 4.1. The PMS-method allows one toidentify relevant (parameters that are flowing to larger values under a decrease of the cut-off), marginal (... not flowing under a decrease of the cutoff) and irrelevant (... flowing tozero ...) parameters. Once one has identified the irrelevant parameters of a system, oneknows, in principle2, which mechanisms govern the low-temperature physics of the system,i.e. one has derived an effective Hamiltonian. However, PMS breaks down once the relevantparameters start to diverge.

That drawback was overcome by another prominent RG-method, the NRG-method. Inthis Chapter we introduce this renormalization scheme.

In the early 1970’s K.G. Wilson succeeded to construct a nonperturbative solution ofthe KM [46] (which was later extended to the AM by Krishna-murthy et al. [31, 47]).Since the “NRG-method does not rely on any assumptions regarding the ground state or

1Typical transport integrals include the derivative of f .2The question whether marginal operators are important has to be investigated separately.

34 4. Wilson’s Numerical Renormalization Group (NRG)

leading order divergent couplings” [28] it is superior to a mean-field description or scalingequations. A detailed description of the procedure is given in Appendix A.

Wilson’s ingenious idea was to discretize the CB (of bandwidth 2D) logarithmically [1]via a discretization parameter Λ (Λ > 1). This discretization enabled him to take allCB energies into account. He reached this goal by dividing the CB into energy intervals[−xN ,−xN+1[ and ]xN+1; xN ] with xN = DΛ−N , N ∈ N0. Wilson showed that this CB-discretization becomes exact if the discretization parameter approaches 1, Λ → 1. Moreoverhe was able to show that the method still works extremely well for a moderately large choiceof Λ (up to Λ ∼ 3).

Within the NRG-scheme the continuous CB operators are expanded in a Fourier serieswith fundamental frequency 2ωN = πΛN/ (1− Λ−1) in each (logarithmic) interval. Defin-ing a single fermionic degree of freedom f0σ [defined in (A.13)] allowed Wilson to rewritethe hybridization H`d between the impurity and all CB electrons in a compact way, seeEq. (A.14). The Lanczos algorithm, ansatz (A.22) in Appendix A.2, then allows one totridiagonalize the remaining CB (represented by constant Fourier components within eachlogarithmic interval). This permits one to solve the problem at hand numerically. Indeedthe transformation of the CB onto a semi-infinite chain, the ’Wilson chain’, with corre-sponding fermionic operators f †Nσ creating a spin σ electron at the N -th site of the Wilsonchain [Eq. (A.24)], allows for an iterative diagonalization and consequently a numericalexact solution of the Kondo problem. Further details of this procedure can be found inAppendix A.

More formally, as can e.g. be found in Ref. [31], the KM is recovered in the limit of aninfinite long Wilson chain HK = limN→∞ 1

2(1 + Λ−1) Λ−(N−1)/2HN , with

HN = Λ(N−1)/2

{ˆH0 +

N−1∑n=0,σ

[Λ−n/2ξK

n

(f †nσfn+1σ + f †n+1σfnσ

)]}. (4.1)

Here ˆH0 = H0

(2

1+Λ−1

)and ξK

n = 1−Λ−n−1√(1−Λ−2n−1)(1−Λ−2n−3)

, cf. Eq. (2.18) of [31]. The ξKN ’s are

the tunneling matrix elements between two sites along the Wilson chain (which decay ∝Λ−N/2 [46]) and H0 = JSf †0µσµνf0ν (with S as defined in Section 3.1.2). The typical energy

scale in the N -th iteration, ωN , is roughly set by ξKN . Wilson showed ωN ∼ Λ−(N−1)/2 [46].

Note that a rescaling factor 12(1 + Λ−1) has been introduced [31] here. This factor ensures

that the recursion relation HN+1 =√

ΛHN +∑

σ

[ξKN

(f †NσfN+1σ + f †N+1σfNσ

)]takes a

particular simple form [see Eq. (A.40) for the general case]. In contrast to the more generalcase, discussed in Appendix A, the leads DoS in Ref. [31] was assumed to be constant andspin independent.

Note that the exponential decrease of ξKN (∝ Λ−N/2) in Eq. (4.1) ensures that the

problem can be solved by iterative diagonalization, as the energy scales of the differentiterations separate nicely. Other models like the Hubbard model - where the hopping matrixelement are constant - can not be solved by NRG and one needs to use other methods suchas the density matrix renormalization group (DMRG) method [48] for tackling problemsof this kind.

35

As expected from PMS the parameter J of the Kondo problem is small at the beginningof the NRG-procedure corresponding to high energies (weak coupling regime). Once thestrong coupling regime is reached, i.e when the NRG-iteration has proceeded far belowthe energy scale TK (see Fig. 4.1), however, J diverges. To capture the crossover regime(at an energy TK) between the weak and the strong-coupling regime well, it is crucial tokeep sufficiently many the lowest lying eigenstates of the system in each iteration. Fig. 4.1shows the crossover of J as a function of decreasing energy (i.e. increasing iteration).

Iteration N

J=

J=0��

��

��

��

SCWC

Figure 4.1: In the course of the iteration, i.e. with decreasing energies, the Kondo couplingJ ’flows’ from a small value [weak coupling (WC) regime] to a very big value [strongcoupling (SC) regime], a behavior that was already indicated in Eq. (3.31). The shadedregions mark these two regimes. Especially the complicated crossover regime (white region)can only reliably be calculated via NRG.

In summary, the scheme provided by Wilson [46] relates effective Hamiltonians on suc-cessive energy scales [40] by a transformation R, HN+1 = R[HN ]. Due to the separationof energy scales Wilson showed that the Kondo problem can be solved by iterative pertur-bation theory [meaning the (N + 1)-th site of the Wilson chain can be viewed as a smallperturbation of HN ].

As an example we shortly show the principle of the numerical procedure for HK : thematrix that describes the initial Hamiltonian of the KM, H0 is a 8×8 matrix which consistsof the impurity and the first site of the Wilson chain. We define the eight basis states of thismatrix as product states of the two possible impurity states |⇑〉 and |⇓〉 and four possibleelectronic states at the first site of the Wilson chain |0〉0, f †0↑ |0〉0 = |↑〉0, f †0↓ |0〉0 = |↓〉0


and f †0↑f†0↓ |0〉0 = |↑↓〉0. An arbitrary state in the zeroth-iteration can thus be written as

|Ψ〉0 = |i〉0 |σ〉, with i ∈ {0, ↑, ↓, ↑↓} and σ = {⇑,⇓}. In this basis spanning the Hilbert

space H0 the Hamiltonian H0 takes the form

H0 =

|0〉 |⇑〉 |↑〉 |⇑〉 |↓〉 |⇑〉 |↑↓〉 |⇑〉 |0〉 |⇓〉 |↑〉 |⇓〉 |↓〉 |⇓〉 |↑↓〉 |⇓〉|0〉 |⇑〉 0 0 0 0 0 0 0 0|↑〉 |⇑〉 0 1

2J 0 0 0 0 0 0

|↓〉 |⇑〉 0 0 −12J 0 0 J 0 0

|↑↓〉 |⇑〉 0 0 0 0 0 0 0 0|0〉 |⇓〉 0 0 0 0 0 0 0 0|↑〉 |⇓〉 0 0 J 0 0 −1

2J 0 0

|↓〉 |⇓〉 0 0 0 0 0 0 12J 0

|↑↓〉 |⇓〉 0 0 0 0 0 0 0 0

.

We start the procedure by diagonalizing H0, H0 → Hd0 . This is achieved by rotating H0 to

a proper basis, UT0 H0U0 = Hd

0 , where U0 is determined numerically. Since we are interestedin the low-temperature properties of HK , the eigenenergies of H0 are sorted with respectto increasing energy and in the course of the iteration only the low-energy states are kept.This sorting is important, as the size of the Hilbert-space grows as ∼ 4N , indicating thatalready after a few iterations the Hilbert space has to be truncated. Obviously all relevantoperators one is interested in, in particular the fermionic operators acting on the zerothsite of the chain f0σ, have to be transformed to this new basis as well and afterwards sortedw.r.t. increasing eigenenergies of H0.

In this basis the matrix corresponding to H1 can easily be obtained by connecting the0-th and 1-st site of the Wilson chain [Eq. (A.40)]

H1 =

|0〉1

|↑〉1

|↓〉1

|↑↓〉1

|0〉1

√ΛHd

0 ξK0 f ∗†0↑ ξK

0 f ∗†0↓ 0

|↑〉1

ξK0 f ∗0↑

√ΛHd

0 0 ξK0 f ∗†0↓

|↓〉1

ξK0 f ∗0↓ 0

√ΛHd

0 −ξK0 f ∗†0↑

|↑↓〉1

0 ξK0 f ∗0↓ −ξK

0 f ∗0↑√

ΛHd0

,

with f ∗0σ = UT0 f0σU0. Note: (i) Hd

0 and f ∗0σ are 8× 8 matrices (from the Hilbert space H0),thus the Hilbert-space H1 corresponding to H1 has dimension 32. (ii) states including thefirst site of the Wilson chain are written as |Ψ〉

1= |i〉

1|Ψ〉

0, with i ∈ {0, ↑, ↓, ↑↓}, or more

general

|Ψ〉N+1

= |i〉N+1

|Ψ〉N. (4.2)

Analogously to above, we can now determine the eigenspectrum of H1 by diagonalizingH1. Before we do so, however, we need to determine f †1σ which ensures that the iterationcan be continued. In the original (undiagonalized) basis of the Hilbert space H1 these

37

operators are trivially given as

f †1σ =

|0〉1

|↑〉1

|↓〉1

|↑↓〉1

|0〉1

0 0 0 0|↑〉

11δσ,↑ 0 0 0

|↓〉1

1δσ,↓ 0 0 0|↑↓〉

10 −1δσ,↓ 1δσ,↑ 0

, (4.3)

where 1 denotes 8× 8 unit-matrices. This steps [diagonalization, sorting w.r.t. increasingeigenenergies, truncating, adding the next site of the Wilson chain, setting up the fermionicoperator that acts on the ’new site’ of the Wilson chain, see Eq. (4.3), diagonalization, . . .]are repeated until the iteration converges.

Convergence can be checked in a flow diagram (see Fig. 4.2), where the lowest lyingeigenstates of the system are plotted as a function of the iteration number. Once the strongcoupling fixed point is reached the method has converged.

5 15 25 35 45 55N

0

0.5

1

1.5

2

2.5

[Ei−

E0]

ΛN

/2

1st

2nd

3rd

degeneracy: 4

degeneracy: 6

degeneracy: 8

Figure 4.2: Eigenenergies (with corresponding degeneracy) of the KM as a function of theiteration N corresponding to an energy ∼ Λ−N/2. The energies of all levels are drawnwith respect to the ground state energy (diamonds on x-axis) and are rescaled by ΛN/2,as they are exponentially decreasing in the course of the iteration. After ∼ 20 iterationsthe system flows to the strong coupling fixed point. The iteration converged after ∼ 35iterations. Parameters: J = 0.25, Λ = 2.

We would like to make a final remark here: as the size of the Hilbert-space grows


exponentially fast it is crucial to make use of the system’s symmetries to speed up thenumerical calculation, see Appendix C.

4.1 Physical quantities computed with NRG

There are two classes of quantities we are interested in: (i) thermodynamic quantities, ob-tained by properly averaging over a given operator, say A, 〈A〉 = 1

Z

∑m e−βEm 〈m| A |m〉.3

A typical example of this class is the impurity occupation A = n (in case of the AM). Aswill be shown below, another thermodynamic quantity,the impurity specific heat Cimp(T ),requires not even the knowledge of matrix elements. (ii) Dynamic quantities, such as thespectral function A(ω, T ), which are harder to compute since, additional to the task of cal-culating the corresponding energy spectrum and matrix elements of the involved operators,appearing δ-functions have to be combined and broadened properly (see Appendix B).

The energy-dependence (or alternatively temperature-dependence) of the quantities ofinterest enters via the iteration number N , as in the N -th step of the NRG-iterationphysical quantities are computed at a typical energy ωN ∼ Λ−(N−1)/2D (or temperaturekBTN ∼ Λ−(N−1)/2D).

In contrast to zero-temperature calculations one has to stop the NRG-iteration at aspecific iteration number in case of finite temperature calculations. Once ωN reaches thescale kBT , to be precise kBT = 1

2(1 + Λ−1) Λ−(N−1)/2, i.e. after NT iterations

NT =

[1− 2

log(

2kBT1+Λ−1

)

log(Λ)

](4.4)

with [. . .] denoting the integer function, the NRG-iteration has to be stopped. The reasonfor this is the following: for N > NT the typical energy scale ωN is smaller than the thermalenergy, i.e. relevant excitations of the system are not included any more since they havealready been truncated.

4.1.1 Thermodynamic quantities

Using two specific examples we give a rough idea how to compute thermodynamic quantitiesin this Section.

The knowledge of the energy spectrum is already sufficient for the calculation of theimpurity specific heat Cimp(T ), C(T ) = −T ∂2

∂T 2 F (T ) (with the total free energy of a canon-ical ensemble F (T ) = −kBT ln Z). Cimp(T ) is obtained by subtracting the free electronspecific heat (corresponding to J = 0, i.e. the specific heat in absence of the impurity)from the specific heat of the whole system [47]. This difference can be rewritten as [46]

Cimp(T ) =d

dT

[T 2 d

dTlim

N→∞

(ln tr(e−βΛ−(N−1)/2HN (J))− ln tr(e−βΛ−(N−1)/2HN (J=0) + ln 2

)]

3Z denotes the partition function Z =∑

m e−βEm with β = (kBT )−1.

4.1 Physical quantities computed with NRG 39

=d

dT

[T 2

(d ln Fimp(T )

d ln T

)].

For the calculation of other thermodynamic quantities, such as the impurity occupationn0 =

∑σ d†σdσ, one has to implement the initial matrix elements of n0 as well. In case of

a single-level AM n0 has the form

n0 =

|0〉 |⇑〉 |⇓〉 |⇑⇓〉|0〉 0 0 0 0|⇑〉 0 1 · 1 0 0|⇓〉 0 0 1 · 1 0|⇑⇓〉 0 0 0 2 · 1

where 1 denotes 4× 4 unit matrices. Obviously n has to be transformed in the same man-ner as the Hamiltonian in the course of the iteration. To finally obtain the expectationvalue 〈n〉, one has to average over the eigenstates of the system, i.e. a knowledge of theeigenspectrum of HN and of the matrix elements of the operator n (in the N -th iteration)is required.Thermodynamic quantities can be computed rather accurately (more accurate than dy-namic quantities, see Section 4.1.2), since they are obtained from the low-energy part ofthe spectrum (where NRG has, by construction, its highest resolution).

Indeed, a careful comparison of the gate voltage dependence of the zero-temperatureoccupation between the NRG and the Bethe-Ansatz solution (an analytic exact solutionof the problem) allows for an accuracy check of the numerics. In Fig. 4.3(left panel) weplot the local level occupation n ≡ 〈n〉 (for T = 0) for a single-level AM as a function ofεd once computed via Bethe-Ansatz [49, 50] and once computed via NRG (see also [33]).We find perfect agreement between both methods.In contrast to Bethe-Ansatz, however, NRG allows us to study the QD occupation asa function of decreasing temperature as well. Fig. 4.3(right panel) shows how the QDoccupation evolves upon lowering the temperature. As already mentioned, in the limit ofsmall temperatures three regimes can be identified (EO: |n − 1| ∼ 1, MV: |n − 1| ∼ 0.5,LMR: |n− 1| ∼ 0).

For T = 0 the impurity occupation can be used to compute the phase shift ∆φσ a spinσ electron experiences when being scattered off the magnetic impurity, ∆φσ = nσπ, knownas the Friedel-sum rule [21]. For T = 0 a relation between ∆φσ and the conductance holds[Gσ ∼ sin2(∆φσ)] which is based on Fermi liquid theory . This relation allows one todetermine the conductance G from the knowledge of the occupation of the impurity.

4.1.2 Dynamic quantities: A(ω, T ) and χ′′(ω, T )

Dynamic quantities are harder to compute than thermodynamic ones since they are en-ergy and temperature dependent. Nevertheless we are interested in those quantities, sincein most experiments transport properties instead of thermodynamic properties are mea-sured. Transport properties can typically be expressed via a function Lml, m, l ∈ Z, where


−1 −0.5 0εd / U

0

0.5

1

1.5

2

n

Bethe−AnsatzNRG

U/Γ=6T=0

−6 −4 −2 0log[kBT/U]

0

0.5

1

1.5

2

n

U/εd=6

U/Γ=6

U/εd=0

U/εd=−6

U/εd=−3

U/εd=−2

U/εd=−1.5

U/εd=−1.2

U/εd=−1

LM

RM

VE

OE

OM

V

U/εd=−0.86

Figure 4.3: Left: Gate voltage dependence of the local occupation n of a single-level AM.We find excellent agreement between the exact Bethe-Ansatz solutions [49, 50] and NRG-calculations, indicating the high accuracy of the numerics. The right panel shows the threepossible QD regimes: the Local-moment-regime (LMR) (n ≈ 1), where Kondo correlationscan exist (see also the ’Kondo plateau’ in the left panel), the mixed valence (MV) regime(|n− 1| ≈ 1

2) and the empty orbital (EO) regime where the QD is either empty or doubly

occupied. For temperatures smaller ∼ 0.01U the system reaches the strong coupling limit.Note the particle-hole symmetry around εd = −U/2. Parameters: U = 0.12D, Λ = 2.

Lml ∝∫

∂f∂ω

τ lωmdω (f labels the Fermi-function of the leads). Examples are the resistivityρ(T ) = 1

e2 L01 or the thermopower S(T ) = − 1eT

L11

L01.

As such integrals depend crucially on the lifetime τ(ω, T )−1 ∼ A(ω, T ) [51], a good knowl-edge of the system’s spectral function A(ω, T ) is essential to compute transport propertiesof a given system. Another important dynamical quantity is the spin spectral functionχ′′(ω, T ).

Consequently, we introduce this two important dynamical quantities, A(ω, T ) andχ′′(ω, T ) in this Section.

The spectral function A(ω, T ) is defined as

A(ω, T ) = − 1

π= [GR(ω, T )

], (4.5)

where GR(ω, T ) is the Fourier-transformed4 of the retarded Green’s function GR(t, T ) ≡−iθ(t)〈{d(t), d†(0)}〉. Obviously the calculation of A(ω, T ) requires the determination ofthe matrix elements of d. Consequently one has to implement the matrix elements of theoperator d in the zeroth step of the NRG-iteration, d0.

For arbitrary temperatures the retarded Green’s function can be written (in Lehmannrepresentation) as

GR(t, T ) = −iθ(t)∑nm

e−βEn + e−βEm

Z(β)〈n| d(t) |m〉〈m| d† |n〉 , (4.6)

4GR(ω, T ) =∫∞−∞ dt eiωtGR(t, T )


where Z(β) =∑

i e−βEi denotes the partition function and |n〉 an eigenstate with corre-

sponding eigenenergy En of H.When we insert the time evolution5 d(t) = eiHtd(0)e−iHt into Eq. (4.6) and finally performthe Fourier transformation we arrive at the spectral function [51]

A(ω, T ) =∑nm

e−βEn + e−βEm

Z(β)

∣∣〈m| d† |n〉∣∣2 δ(ω − (Em − En)). (4.7)

In particular, the T = 0 limit (in which we are mostly interested in) can easily be obtainedfrom Eq. (4.7),

A(ω, T = 0) =

ω>0︷︸︸︷∑n

∣∣〈n| d† |0〉∣∣2 δ(ω − (En − E0)) +

ω<0︷︸︸︷∑n

∣∣〈0| d† |n〉∣∣2 δ(ω − (E0 − En)),

(4.8)where |0〉 denotes the ground state of the system. As finite temperature spectral functions

A(ω, T ) involve transitions between excited states∣∣〈m| d† |n〉

∣∣2, they are more difficult tocalculate than A(ω, T = 0) (see e.g. Fig. B.1). We already pointed out in Eq. (4.4) that itis crucial to stop the iteration after NT steps when one is interested in a finite temperaturecalculation.

In Fig. 4.4 we plot the (properly renormalized) spectral function A(ω, T = 0) for dif-ferent values of εd for a single-level AM obtained from NRG. The accuracy of the numericscan be checked by comparing the numerically obtained value of A(ω = 0, T = 0) with that

known from the Friedel-sum-rule [21], A(0, 0) = sin2(πn/2)πΓ/4

[with n =∑

σ〈d†σdσ〉; here we

took Γσ = 12Γ and nσ = 1

2n (since B = 0)]. A further check of the numerical accuracy

of the calculation of the spectral function A(ω, T = 0) can be achieved by comparing its

integrated weight∫ 0

−∞ dω A(ω, T = 0) with the thermodynamically calculated occupationn. As high energy features in A(ω, T = 0) are dominantly responsible for the occupa-

tion obtained via∫ 0

−∞ dω A(ω, T = 0), we do not expect a perfect agreement between∫ 0

−∞ dω A(ω, T = 0) and n. Remember that the NRG-method is designed to describelow-energy properties of the system accurately. A trick due to Bulla [52], that involves theself-energy of the AM, can slightly cure this problem.

In the Kondo regime, the spectral function shows a three peak structure: (i) two atomicresonances of width ∼ Γ near εd and εd + U and (ii) the Kondo resonance of width ∼ TK

(pinned at the Fermi energy). It was already pointed out before that a finite magnetic fieldof strength B . TK leads to a suppression and splitting of the Kondo resonance. An evenbigger magnetic field completely destroys the Kondo effect. As shown by Hofstetter [4],the ’traditional’ NRG-procedure has even to be changed for B 6= 0. In this case the“density-matrix NRG” (DM-NRG) method should be used. A detailed description of theDM-NRG-method is given in Appendix D.

5Here the time evolution operator has the form U(t, 0) = e−i~ Ht.


−1 −0.5 0 0.5 1 1.5ω/U

0

0.2

0.4

0.6

0.8

1

πΓA

(ω)/

4

εd/U=1/6εd/U=0εd/U=−1/6εd/U=−1/2

U/Γ=6T=0

−9 −6 −3log[ω/U]

−7

−5

−3

−1

1

log[

Uχ’

’(ω)]

ε/U=1/6ε/U=0ε/U=−1/6ε/U=−1/2

U/Γ=6T=0

Figure 4.4: Left: Spectral function A(ω, T = 0) for different values of εd. The Kondoresonance (of width ∼ TK) can be nicely seen for that values of εd (namely εd = −U/6 andεd = −U/2) where the impurity is roughly singly occupied, i.e. in the LMR. Note that theFriedel-sum rule [21] is fulfilled extremely well (with a deviation less than 3%). Additionalside peaks of width Γ appear in the spectral function which are due to the discrete (atomic)levels at εd and εd + U . Right: Spin spectral function χ′′ for the same values of εd as inthe left panel. The maximum in χ′′(ω, T = 0) appears at a frequency ω ∼ TK (given theKondo effect exists); for frequencies below TK the system behaves like a Fermi liquid, i.e.χ′′(ω, T = 0) ∝ ω.

In some cases, e.g. when one is interested in the transmission phase shift φ(ω, T ),

tan[φ(ω, T )] ==[GR(ω,T )]<[GR(ω,T )]

, it is necessary to calculate both the imaginary part = [GR(ω, T )]

and the real part < [GR(ω, T )]

of GR. For causal functions, such as the retarded Green’sfunction, this can be achieved by a Kramers-Kronig transformation [7], see Appendix E.

The spin susceptibility χ(t) = χ′(t)+ iχ′′(t), a second dynamic quantity we briefly wantto explain here, describes the response of a system, which is a magnetization ~m(t) at a

time t, after a magnetic field ~h(0) was applied at t = 0

~m(t) = χ(t, 0)~h(0). (4.9)

It was pointed out before, that it is convenient to define TK as the maximum of the spinsusceptibility χ′′(ω) [29], see Fig. 4.4.

In linear response theory the spin susceptibility is given by χ(t) ≡ iθ(t)〈[Szimp(t), S

zimp(0)]〉.

For a single-level AM Szimp has the following matrix elements in the zeroth iteration

(Szimp)0 =

|0〉 |⇑〉 |⇓〉 |⇑⇓〉|0〉 0 0 0 0|⇑〉 0 1

2· 1 0 0

|⇓〉 0 0 −12· 1 0

|⇑⇓〉 0 1 0 0

.


When we write χ(t) in Lehmann representation, achieved by inserting a complete set ofstates, we obtain (for t > 0 and T = 0):

χ(t) = i∑

n

〈0| eiHtSzimpe

−iHt |n〉〈n| Szimp |0〉 = i

∑n

∣∣〈n|Szimp |0〉

∣∣2 ei(E0−En)t.

Its Fourier-transformed then takes the form

χ(ω) = i∑

n


∣∣2 limη→0+

∫ ∞

0

ei[ω−(En−E0)+iη]tdt =

= i2∑

n


∣∣2[P

(1

ω − (En − E0)

)− iπδ(ω − (En − E0))

].

Thus the T = 0 spin-susceptibility χ′′(ω) has the form

χ′′(ω) = π∑

n


∣∣2 δ(ω − (En − E0)). (4.10)

To summarize: additional to the knowledge of the eigenstates |n〉 (with correspondingeigenenergies En) and matrix elements (e.g. 〈n|Sz

imp |0〉) of a system, the computation ofa dynamical quantity involves a (proper) broadening of appearing δ-functions. In Ap-pendix B we focus on this issue.

44

Part II

NRG-studies of Quantum ImpurityModels

46

Chapter 5

QDs coupled to ferromagnetic leads

In Chapter 2 we stressed that lateral semiconductor QDs, commonly used in transportexperiments, are defined by depleting a 2DEG (via negatively charged gates). An exampleof such a structure is shown in Fig. 5.1(left panel). In absence of an external magneticfield the local interacting region, the source and the drain region (all these regions consistof the same material) do not have a preferred spin direction, i.e. they are unpolarized.Consequently, for B = 0 the electrons of the leads screen a localized spin in the QD forT < TK . As explained in Chapter 3 this results in an enhanced linear conductance (dueto the Kondo resonance in the local DoS).

Leads that consist of one spin species only (half-metallic leads), are obviously notable to screen a local moment, no Kondo correlations can develop in this scenario. Thisway of thought brings one to an interesting theoretical question, namely whether Kondocorrelations do exist for a local moment which is coupled to a reservoir with a finite spinpolarization (partially polarized leads). In this scenario it is not obvious what kind ofconsequences the leads’ polarization on the Kondo effect has.

Experimentally the idea of contacting a QD to a lead that consists of a different ma-terial than the QD itself came into reach when Liang et al. [53] managed to contact adivanadium-molecule (serving as QD) to gold electrodes, see Fig. 5.1(right panel). Usuallythe contacting between the QD and its surrounding leads is rather difficult. In conductanceexperiments through this device, however, the Kondo effect was observed, indicating thatLiang et al. managed to realize a sufficiently good contact between the molecule and itsleads. If one replaced the gold electrodes by manganese, the model we are studying in thischapter would be realized.

In the following papers [15, 54] we discuss the effect of a finite spin polarization in theleads on the Kondo resonance. In Section 5.1 we study the consequence of a finite leadspolarization P 6= 0 on Kondo physics (with fixed gate voltage εd). It turns out that a finitespin polarization P 6= 0 results in a splitting and a suppression of the Kondo resonance.

In Section 5.2 we address the question how the splitting of the Kondo resonance variesupon changing εd for QDs coupled to leads with a ’realistic’ DoS. We find a strong gatevoltage dependence of the splitting of the Kondo resonance that depends crucially on theparticular band-structure in the leads. Surprisingly the suppression and splitting of the

48 5. QDs coupled to ferromagnetic leads

Figure 5.1: Left: ’Traditional’ way of realizing a QD, coupled to leads which consist of thesame material as the QD itself (Photo from: http://www.unibas.ch/phys-meso/). Right:Liang et al. [53] managed to contact a divanadium-molecule, depicted above the image, togold contacts. This opens up the exciting possibility to contact QDs to arbitrary leads.

Kondo resonance, which usually appears in case of a QD contacted to polarized leads, canbe compensated by an appropriately tuned gate voltage εd.

5.1 Effect of a finite spin polarization in the leads on

the Kondo resonance

In this Section we study the effect of a finite spin polarization P in the leads on the Kondoresonance. To ensure that the QD is in the local moment regime, i.e. it is roughly singlyoccupied, we fix the gate voltage at εd = −U/3 throughout this study.

We assume the DoS in the leads ρσ to be flat and normalized∑

σ

∫ D0

−D0ρσdω = 1 but

spin-dependent, as shown in Fig. 5.2(left panel).1 Correspondingly, the spin polarization

P of the leads is determined by the DoS at the Fermi energy P =ρ↑(EF )−ρ↓(EF )

ρ↑(EF )+ρ↓(EF ). For flat

bands, as considered here, the full information about spin asymmetry in the leads can beparameterized by a spin-dependent hybridization function Γσ ≡ 1

2Γ(1 ± P ).2 Based on

second order perturbation theory in the tunneling one expects a spin-dependent shift and

1Here we disregard an additional spin splitting at the edge of the bands (Stoner-splitting), which leadsto a logarithmically suppressed effective magnetic field.

2σ =↑ (↓) corresponds to +(−).

5.1 Effect of a finite spin polarization in the leads on the Kondo resonance 49

��

��

��

��

ω/D

σρ (ω)

��

��

��

��

1

−1

µ

εd

εd

��

��

��

��µ

Γ

Γ

Figure 5.2: Left: Spin-resolved leads DoS ρσ(ω) (ω ∈ [−D; D]) for a flat band with P > 0.The imbalance of available ↑- and ↓-electron states determines the spin polarization P ofthe leads. Right: The cartoon illustrates the consequence of a finite spin polarization inthe leads, namely a spin splitting of the local level εdσ (here: P > 0 and εd > −U/2).

consequently a spin splitting of the local level3 for a finite spin polarization of the leads(P 6= 0), see Fig. 5.2(right panel). Consequently one expects the Kondo resonance of a QDcontacted to spin polarized leads to be suppressed and split, similar to a QD in presenceof a magnetic field.

Below we present a careful NRG-study of the problem outlined above. We find that thefull Kondo resonance of a QD that is coupled to spin polarized leads can be recovered (witha reduced value of TK) given an appropriately tuned external magnetic field is applied.

3For εd 6= −U/2 and Γ↑ 6= Γ↓ one expects a spin-dependent shift of the local level δεdσ, δεd↑ ∼Γ↓

εd+U − Γ↑εd6= Γ↑

εd+U − Γ↓εd∼ δεd↓.


Kondo Effect in the Presence of Itinerant-Electron Ferromagnetism Studiedwith the Numerical Renormalization Group Method

J. Martinek,1,4 M. Sindel,2 L. Borda,2,3 J. Barnas,4,5 J. Konig,1,6 G. Schon,1 and J. von Delft2

1Institut fur Theoretische Festkorperphysik, Universitat Karlsruhe, 76128 Karlsruhe, Germany2Sektion Physik and Center for Nanoscience, LMU Munchen, Theresienstrasse 37, 80333 Munchen, Germany

3Hungarian Academy of Sciences, Institute of Physics, TU Budapest, H-1521, Hungary4Institute of Molecular Physics, Polish Academy of Sciences, 60-179 Poznan, Poland

5Department of Physics, Adam Mickiewicz University, 61-614 Poznan, Poland6Institut fur Theoretische Physik III, Ruhr-Universitat Bochum, 44780 Bochum, Germany

(Received 15 April 2003; published 10 December 2003)

The Kondo effect in quantum dots (QDs)—artificial magnetic impurities—attached to ferromag-netic leads is studied with the numerical renormalization group method. It is shown that the QD level isspin split due to the presence of ferromagnetic electrodes, leading to a suppression of the Kondo effect.We find that the Kondo effect can be restored by compensating this splitting with a magnetic field.Although the resulting Kondo resonance then has an unusual spin asymmetry with a reduced Kondotemperature, the ground state is still a locally screened state, describable by Fermi liquid theory and ageneralized Friedel sum rule, and transport at zero temperature is spin independent.

DOI: 10.1103/PhysRevLett.91.247202 PACS numbers: 75.20.Hr, 72.15.Qm, 72.25.–b, 73.23.Hk

The prediction [1] and experimental observation of theKondo effect in artificial magnetic impurities—semi-conducting quantum dots (QDs) [2,3]—renewed interestin the Kondo effect and opened new opportunities ofresearch. The successful observation of the Kondo effectin molecular QDs such as carbon nanotubes [4,5] andsingle molecules [6] attached to metallic electrodesopened the possibility to study the influence of many-body correlations in the leads (superconductivity [7] orferromagnetism) on the Kondo effect. Recently, the ques-tion arose whether the Kondo effect in a QD attached toferromagnetic leads can occur or not. Several authors havepredicted [8–11] that the Kondo effect should occur.However, it was shown recently [12] that the QD levelwill be spin split due to the presence of ferromagneticelectrodes leading to a suppression of the Kondo effect,and that the Kondo resonance can be restored only byapplying an external magnetic field. The analyses ofRefs. [8–12] were all based on approximate methods.

In this Letter, we resolve the controversy by adaptingthe numerical renormalization group (NRG) technique[13,14], one of the most accurate methods available tostudy strongly correlated systems in the Kondo regime, tothe case of a QD coupled to ferromagnetic leads withparallel magnetization directions.We find that, in general,the Kondo resonance is split, similar to the usualmagnetic-field-induced splitting [15,16]. However, wefind that, by appropriately tuning an external magneticfield, this splitting can be fully compensated and theKondo effect can be restored [17](confirming Ref. [12]).We point out that precisely at this field the occupancy ofthe local level is the same for spin-up and -down, n" � n#,a fact that follows from the Friedel sum rule [19]. More-over, we show that the Kondo effect then has unusualproperties such as a strong spin polarization of the Kondo

resonance and, just as for ferromagnetic materials, for thedensity of states (DOS). Nevertheless, despite the spinasymmetries in the DOS of the QD and the leads, thesymmetry in the occupancy n" � n# implies that thesystem’s ground state can be tuned to have a fully com-pensated local spin, in which case the QD conductance isfound to be the same for each spin channel, G" � G#.

The model.—For ferromagnetic leads, electron-electron interactions in the leads give rise to magneticorder and spin-dependent DOS �r"�!� � �r#�!�, (r �L;R). Magnetic order of typical band ferromagnets suchas Fe, Co, and Ni is mainly related to electron correlationeffects in the relatively narrow 3d subbands, which onlyweakly hybridize with 4s and 4p bands [20]. We canassume that, due to a strong spatial confinement of delectron orbitals, the contribution of electrons from dsubbands to transport across the tunnel barrier can beneglected [21]. In such a situation, the system can bemodeled by noninteracting [22] s electrons, which arespin polarized due to the exchange interaction withuncompensated magnetic moments of the completelylocalized d electrons. In the mean-field approximation,one can model this exchange interaction as an effectivemolecular field, which removes spin degeneracy in thesystem of noninteracting conducting electrons, leading toa spin-dependent DOS. The Anderson model (AM) for aQD with a single energy level d, which is coupled toferromagnetic leads, is given by

~HH �X

rk

rk cyrk crk � d

X

nn � Unn"nn#

�X

rk

�Vr;kdy cr;k � V�

r;kcyr;k d � g�BBSz: (1)

Here crk and d (nn � dy d ) are the Fermi operators for

P H Y S I C A L R E V I E W L E T T E R S week ending12 DECEMBER 2003VOLUME 91, NUMBER 24

247202-1 0031-9007=03=91(24)=247202(4)$20.00 2003 The American Physical Society 247202-1


electrons with momentum k and spin in the leadsand, in the QD, Vrk is the tunneling amplitude, Sz ��nn" nn#�=2, and the last term is the Zeeman energyof the dot. All information about spin asymmetryin the leads can be modeled by the spin-dependenthybridization function r �!� � �

P

k��! k �V2r;k �

��r �!�V2r , where Vr;k Vr and �r �!� is the spin-

dependent DOS.In order to understand the Kondo physics of a QD

attached to two identical ferromagnetic electrodes withparallel configurations, it suffices to study, instead of theabove general model [Eq. (1)], a simpler one, whichcaptures the same essential physics, namely, the factthat r"�!� � r#�!� will generate an effective local mag-netic field, which lifts the degeneracy of the local level(even for B � 0). A simple (but not unique) way ofmodeling this effect is to take the DOS in the leads tobe constant and spin independent, �r �!� �, the band-widths to be equal D" � D#, and lump all spin dependenceinto the spin-dependent hybridization function, r �!�,which we take to be ! independent, r �!� r . Bymeans of a unitary transformation [1], the AM [Eq. (1)]can be mapped onto a model in which the correlatedQD level couples only to one electron reservoir describedby Fermi operators �k with strength �

P

rr :

H �X

k

k�yk �k � d

X

nn � Unn"nn#

�X

k

��

=��q

�dy �k � �y

k d � g�BBSz: (2)

Finally, we parametrize the spin dependence of interms of a spin-polarization parameter P �" #�=, by writing "�#�

12�1� P�, where " � #.

In the model of Eq. (2), we allowed for " � # but notfor D" � D#, as would be appropriate for real ferromag-nets, whose spin-up and -down bands always have aStoner splitting D D" D#, with typical values D=D" & 20% (for Ni, Co, and Fe). However, no essen-tial physics is thereby lost, since the consequence oftaking D" � D# is the same as that of taking " � #,namely, to generate an effective local magnetic field [25].

The occurrence of the Kondo effect requires spin fluc-tuations in the dot as well as zero-energy spin-flip ex-citations in the leads. Indeed, a Stoner ferromagnetwithout full spin polarization 1 < P < 1 provideszero-energy Stoner excitations [26], even in the presenceof an external magnetic field.

Method.—The model [Eq. (2)] can be treated byWilson’s NRG method. This method, with recent im-provements related to high-energy features and finitemagnetic field [15,16], is a well-established methodto study the Kondo impurity (QD) physics. It allowsone to calculate the level occupation n hnn i (a staticproperty), the QD spin spectral function, Im�z

s�!� �F fi��t�h�Sz�t�; Sz�0��ig, where F denotes the Fouriertransform, and the spin-resolved single-particle spectral

density A �!; T; B; P� � 1�ImGR

d; �!� for arbitrarytemperature T, magnetic field B, and polariza-tion P [where GR

d; �!� denotes a retarded Green function].From this we can find the spin-resolved conduc-tance G � �e2= �h��2%�=�% � 1�2

R

11 d! A �!� �

f@f�!�=@!g, where f�!� is the Fermi function and% � L =R denotes the coupling asymmetry. Wechoose % � 1 below.

Generation and restoration of spin splitting.—In thisLetter, we focus exclusively on the properties of thesystem at T � 0 in the local moment regime, where thetotal occupancy of the QD, n �

P

n � 1. The occur-rence of charge fluctuations broadens and shifts the posi-tion of the QD levels (for both spin-up and -down), and,hence, changes their occupation. For P � 0, the chargefluctuations and, hence, level shifts and level occupationsbecome spin dependent, causing the QD level to split [12]and the dot magnetization n" n# to be finite (Fig. 1). As aresult, the Kondo resonance is also spin split [28,29] andweakened (Fig. 2), similarly to the effect of an appliedmagnetic field [15,16]. This means that Kondo correla-tions are reduced or even suppressed in the presence offerromagnetic leads. However, for any fixed P, it is pos-sible to compensate the splitting of the Kondo resonance[Figs. 2(c) and 3(c)] by fine-tuning the magnetic field toan appropriate value, Bcomp�P�, defined as the field whichmaximizes the height of the Kondo resonance. This fieldis found to depend linearly [27] on P [Fig. 1(c)] (aspredicted in [12] for U ! 1). Remarkably, we alsofind (throughout the local moment regime) that at Bcomp

the local occupancies satisfy n" � n# [Fig. 1(b)]. The factthat this occurs simultaneously with the disappearanceof the Kondo resonance splitting suggests that the localspin is fully screened at Bcomp.

Spectral functions.—We henceforth fix the magneticfield at B � Bcomp�P�. To learn more about the propertiesof the corresponding ground state, we computed the spinspectral function Imf�z

s�!�g for several values of P atT � 0 (Fig. 3). Its behavior is characteristic for theformation of a local Kondo singlet: As a function ofdecreasing frequency, the spin spectral function shows a

−1 0 1P

0

0.5

1

−1 0 1P

−0.2

0

0.2

Bcom

p /

Γ

−1 0P

B / Γ=−0.1

a cb

n↓n↑n↓n↑

n↑+n↓n↑+n↓

B = 0

1

FIG. 1 (color online). Spin-dependent occupation of the dotlevel at (a) B � 0 and (b) B � 0:1, as a function of spinpolarization P. (a) For B � 0, the condition n" � n# holds onlyat P � 0. (b) For finite P, it can be satisfied if a finite, fine-tuned magnetic field, Bcomp�P�, is applied, whose dependenceon P is shown in (c). As expected, it is approximately linear[12,27]. Here U � 0:12D, d � U=3, � U=6, and T � 0.


247202-2 247202-2


maximum at a frequency !max which we associate withthe Kondo temperature [i.e., kBTK �h!max at B �Bcomp�P�], and then decreases linearly with !, indicatingthe formation of the Fermi liquid state [14]. By determin-ing TK�P� (from !max) for different P values, we find thatTK decreases with increasing P [Fig. 3(b)]. For metalssuch as Ni, Co, and Fe, where P � 0:24, 0.35, and 0.40,respectively, the decrease of TK is weak, so the Kondoeffect should be experimentally accessible. Remarkably,both Imf�z

s�!�g and the A �!� collapse rather well onto auniversal curve if plotted in appropriate units [Figs. 3(a)and 3(c)]. This indicates that an applied magnetic fieldBcomp restores the universal behavior characteristic for theisotropic Kondo effect, in spite of the presence of spin-dependent coupling to the leads. Figure 4(a) shows thatthe amplitude of the Kondo resonance is now stronglyspin dependent, which is unusual and unique. The natureof this asymmetry is related to the asymmetry of theDOS in the leads, and its value is exactly proportional to�1= . As a result, the total conductance G is not spindependent [Fig. 4(b)]. This indicates the robustness of theKondo effect in this system: If the external magnetic fieldhas been tuned appropriately, it is able to compensate thepresence of a spin asymmetry in the leads by creating aproper spin asymmetry in the dot spectral density,thereby conserving a fully compensated local spin andachieving perfect transmission.

Friedel sum rule.—Further insights can be gainedfrom the Friedel sum rule, an exact T � 0 relation [19]that holds for arbitrary values of P and B [30]. Theinteracting Green’s function can be expressed as [14]GR

d; �!� � �! d� � i � �!��1, with spin de-

pendent d� and ; the former due to Zeeman splitting(d� � d 1=2 g�BB) and the latter due to the ferro-magnetic leads. Here � �!� denotes the spin-dependentself-energy. Now, the Friedel sum rule [19] implies that,at T � 0, the following relations hold:

n � ( =� �1

2

1

�tan1

�

d� F � �R �F�

�

; (3)

A �F� �sin2��n �

�

; (4)

where �R �!� Re � �!�, and ( �!� is the phase of

GRd; �!�. Since �R

" �F� � �R# �F�, an equal spin occupa-

tion, n" � n#, is possible only for �d" F � �R" �F��=

" � �d# F � �R# �F��=#, which can be obtained only

for an appropriate external magnetic field B � Bcomp. Forthe latter, in the local moment regime (n � 1), we haven" � n# � 0:5, so that (" � (# � �=2, which impliesthat the peaks of A" and A# are aligned. Thus, theFriedel sum rule clarifies why the magnetic field Bcomp,at which the splitting of the Kondo resonance disappears,coincides with that for which n" � n#. For B � Bcomp, thespin-dependent amplitude A �F� of Eq. (4) and the con-ductance G � A �F� agree well with the above-mentioned NRG results [Figs. 4(a) and 4(b)].

In conclusion, we showed that the Kondo effect in a QDattached to ferromagnetic leads is in general suppressed,because the latter induce a spin splitting of the QD level,which leads to an asymmetry in the occupancy n" � n#.Remarkably, the Kondo effect may nevertheless be re-stored by applying an external magnetic field Bcomp, tuned

10−4

10−2

100

102

104

ω/TK(P)

10−8

10−6

10−4

10−2

100

Im χ

S

z(ω

)/[π

/ TK(P

)]

P=0.0

P=0.2

P=0.4

P=0.6

P=0.8

0 0.2 0.4 0.6 0.8P

0

0.5

1

TK(P

)/T

K(0

)

−4 −2 0 2 4ω/T

K(P)

0

1

π Σ

σ A

σ(ω

)Γσ/2

a

b

c

B=Bcomp

(P)

FIG. 3 (color online). (a) The spin spectral functionImf�z

s�!�g. (b) Dependence of TK on spin polarization P. Thedotted line shows the prediction from Ref. [12]—namely,TK�P�=TK�0� � exp�C arctanh�P�=P�, where the best fit is ob-tained for C � 5:98. Equation (6) from Ref. [12] with ��" ��#�J0 � �=��U=�jdj�U � d�� would lead to C � 4:19. (c)QD spectral function for several values of P. Parameters U, d,, and T are as in Fig. 1, B � Bcomp�P�.

−1 1 3ω / | ε

d |

0

1

πΓA

(ω)/

4

P=0.0

P=0.2

P=0.4

P=0.6−0.125 0 0.1250

1

πΓA

(ω)/

4

−0.125 0 0.125ω / | ε

d |

0

1

πΓA

σ(ω

)/2

−0.05 0 0.05ω / Γ

0

1

πΓA

(ω)/

4

B / Γ0.09 0.70 0.22

0.10 0.47 0.47

0.11 0.27 0.71

a

b

P=0.5

cP=0.4, B=0

Γ=U/6ε

d=U/3

T=0, B=0

FIG. 2 (color online). (a) QD spectral function A�!� �P

A �!� for several values of spin polarization P; inset:expanded scale of A�!� around F. (b) The spin-resolvedspectral function for fixed P. For P ! P, we have A !A . (c) Compensation of the spin splitting by fine-tuning anexternal magnetic field. Parameters U, d, , and T as in Fig. 1.


247202-3 247202-3


such that the splitting of the Kondo resonance is compen-sated and the condition n" � n# is fulfilled. Although theKondo resonance is strongly spin polarized, it then fea-tures a locally screened state, a spin-independent con-ductance, and a Kondo temperature which decreases withincreasing spin asymmetry.

We thank R. Bulla, T. Costi, L. Glazman,W. Hofstetter,H. Imamura, B. Jones, S. Maekawa, A. Rosch, M. Vojta,and Y. Utsumi for discussions. This work was supportedby the DFG under the CFN, ‘‘Spintronics’’ NetworkRTN2-2001-00440, Project No. PBZ/KBN/044/P03/2001, OTKA T034243, and the Emmy-Noether program.

Note added.—After submission of our paper, a preprint[31] studying a similar problem using the NRG techniqueappeared, with conclusions consistent with ours.

[1] L. I. Glazman and M. E. Raikh, JETP Lett. 47, 452(1988); T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61,1768 (1988).

[2] D. Goldhaber-Gordon et al., Nature (London) 391, 156(1998).

[3] S. M. Cronenwett et al., Science 281, 540 (1998);F. Simmel et al., Phys. Rev. Lett. 83, 804 (1999);J. Schmid et al., Phys. Rev. Lett. 84, 5824 (2000); W. G.van der Wiel et al., Science 289, 2105 (2000).

[4] J. Nygard et al., Nature (London) 408, 342 (2000).[5] M. R. Buitelaar et al., Phys. Rev. Lett. 88, 156801 (2002).[6] J. Park et al., Nature (London) 417, 722 (2002); W. Liang

et al., Nature (London) 417, 725 (2002).[7] M. R. Buitelaar et al., Phys. Rev. Lett. 89, 256801 (2002).[8] N. Sergueev et al., Phys. Rev. B 65, 165303 (2002).[9] P. Zhang et al., Phys. Rev. Lett. 89, 286803 (2002).

[10] B. R. Bułka et al., Phys. Rev. B 67, 024404 (2003).[11] R. Lopez et al., Phys. Rev. Lett. 90, 116602 (2003).[12] J. Martinek et al., Phys. Rev. Lett. 91, 127203 (2003).[13] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975); T. A. Costi

et al., J. Phys. Condens. Matter 6, 2519 (1994).[14] A. C. Hewson, The Kondo Problem to Heavy Fermions

(Cambridge University Press, Cambridge, England,1993).

[15] W. Hofstetter, Phys. Rev. Lett. 85, 1508 (2000).[16] T. A. Costi, Phys. Rev. Lett. 85, 1504 (2000); Phys. Rev. B

64, 241310 (2001).[17] The remarkable full recovery of the Kondo effect should

be contrasted to the competition between Kondo physicsand magnetic RKKY interactions in heavy-fermion sys-tems [18]. There, depending on the relation between therelevant energy scales TRKKY and TK, either the local spinis quenched and no magnetic order occurs (for TK >TRKKY), or the local molecular field removes spin degen-eracy and suppresses the Kondo effect (for TK < TRKKY).

[18] A. H. Castro Neto and B. A. Jones, Phys. Rev. B 62,14 975 (2000), and references therein.

[19] D. C. Langreth, Phys. Rev. 150, 516 (1966).[20] W. Nolting et al., Z. Phys. B 96, 357 (1995).[21] E.Yu. Tsymbal and D. G. Pettifor, J. Phys. Condens.

Matter 9, L411 (1997).[22] We expect that a finite s-electron interaction, Us > 0,

will lead only to a slight renormalization of TK.Recently, an Anderson impurity embedded in a corre-lated host (without magnetic order) was studied [23,24],and an enhancement of TK was found for weak s-electroninteraction, Us=D < 1, whereas, for intermediate andstrong interactions, TK was found to be reduced.

[23] T. Schork and P. Fulde, Phys. Rev. B 50, 1345 (1994);G. Khaliullin and P. Fulde, Phys. Rev. B 52, 9514 (1995).

[24] W. Hofstetter et al., Phys. Rev. Lett. 84, 4417 (2000).[25] A lowest-order perturbation calculation shows that for

jU � dj; jdj � D the spin splitting of the bottom ofthe band D (for s bands we consider no spin splittingfor the band top) generates an effective local magneticfield BStoner � �=2 ln�1 D=D"�. To account for thisin our model of Eq. (2), one could renormalize P (or elseuse a shifted compensation field Bcomp below).

[26] K. Yosida, Theory of Magnetism (Springer, New York,1996).

[27] A poor man’s scaling analysis [12] yields a renormaliza-tion �d� of the level d� proportional to , i.e., Bcomp /��d" �d#� / �# "� � P.

[28] A similar splitting was predicted [D. Boese, et al., Phys.Rev. B 64, 125309 (2001); W. Hofstetter, cond-mat/0104297] for a two-level QD system which, in a specialsituation, can be mapped on our model.

[29] For certain special (nongeneric) cases, the spin-up and-down level positions and, hence, also the Kondo reso-nance do not split up in two, even for P � 0: This occurs,for example, at the symmetry point d � U=2 of theAnderson model, provided the bands are also symmetricaround "F (but this is unrealistic for ferromagnets).

[30] Our NRG confirms Eq. (4) for arbitrary P and B.[31] M. S. Choi et al., cond-mat/0305107.

−1 −0.5 0 0.5 1

P

0

2

4

6

8

πΓA

σ(ω

=0)/

2

0.95

1

1.05

Gσ [e

2/h

]

up

down

1/(1±P) see Eq.(4)

a

b

FIG. 4 (color online). (a) Spin resolved QD spectral functionamplitude A �! � 0� at the Fermi level, and (b) the QDconductance G , as functions of the spin polarization P, forB � Bcomp�P� and symmetric couplings (L � R ), with U,d, , and T as in Fig. 1, implying n � 0:5. The dashed line in(a) is 1=�1� P� [Eq. (4) with n � 0:5]. As expected, we findG � e2=h, with a numerical error less than 1%.


247202-4 247202-4


5.1.1 Quantitative determination of the compensation field Bcomp

After finishing the work presented above we became particularly interested (motivated byquestions of D. Ralph) in (i) a quantitative determination of the compensation field Bcomp

and (ii) why the condition n↑ = n↓ implies an unsplit Kondo resonance.We use poor man’s scaling to derive the total shift of the local level δεdσ, which is

due to electrons at the band edges, i.e. at energies ω ∼ ±D. For the case of finite U thescaling starts at U and breaks down when εd is reached, thus Eq. (3.37) has to be adjustedcorrespondingly. The shift of the level εdσ is given as

δεdσ =ln

(U|εd|

)

2πΓσ. (5.1)

Note that Eq. (5.1) holds only for εd > −U/2 (with finite εd), as Eq. (3.37) was derivedin the limit U → ∞. The case εd < −U/2 produces opposite signs, as one expects fromparticle-hole symmetry. For P > 0, i.e. Γ↑ > Γ↓, and in absence of an additional appliedmagnetic field one obtains from Eq. (5.1) εd↑ < εd↓, so n↑ > n↓ [see Fig. 5.2(right panel)].Obviously, one can apply an additional local magnetic field, which decreases n↑ somewhatfaster than n↓ is increased (since the former is broader than the latter). It turns out that acompensation field Bcomp, tuned such that the condition n↑ = n↓ is fulfilled, realizes the casewhere the QD-spin is perfectly screened even though the leads have a finite polarization.Note that the condition n↑ = n↓ implies φ↑ = φ↓ (due to the Friedel sum rule [21]) whichresults in nσ ∼ 1

2or φσ ∼ π/2 for σ =↑ / ↓ as εd is fixed such that n ∼ 1. At T = 0, in

strong coupling (the opposite limit to poor man’s scaling), the interacting Green’s functioncan be written as Gσ(ω) = 1

ω−εdσ+iΓσ−Σσ(ω)= |Gσ(ω)|eiφσ . Therefore the condition n↑ = n↓

implies that both spin-resolved spectral functions are on resonance - the Kondo resonanceis not split [see question (ii) above]. To have a resonance at the Fermi energy the realpart of the self-energy has to fulfill ΣR

σ (0) = −(εdσ − εF ), which means that the self-energyterm Σ is not small, indicating the importance of correlation effects. One can get anapproximation about the dependence of Bcomp by means of PMS, Eq. (5.1), via

Bcomp ≈ δεd↑ − δεd↓ = −ln

(U|εd|

)

2πΓP. (5.2)

That Bcomp should be linear in P and Γ can be understood from poor man’s scaling (comingfrom weak coupling) and is confirmed by the numerics presented before [see question (i)above].

5.1.2 Experimental relevance

The theoretical study presented in this Section was recently confirmed in an experimentof Pasupathy et al., “The Kondo Effect in the Presence of Ferromagnetism”, [55]. It wasconfirmed in this experiment that C60 molecules (serving as QDs) can be strongly coupledto nickel electrodes (serving as ferromagnetic leads) so as to exhibit the Kondo effect.


The shapes of the nickel electrodes were designed such that the two electrodes havedifferent magnetic anisotropies. Consequently, these electrodes undergo a magnetic reversalat different values of the magnetic field. This design has the fundamental advantage thatthe relative orientation of the magnetic moments in the two electrodes can be controlled(via an external magnetic field) between parallel and antiparallel alignment.

Pasupathy et al. found, that ferromagnetism suppresses Kondo-assisted tunneling, butKondo correlations are still present in QDs contacted to ferromagnetic leads. In agreementwith our prediction they observed a split conductance, see Fig. 5.3, for the case of theparallel aligned nickel electrodes. For antiparallel aligned electrodes, on the other hand,the strong-coupling Kondo effect was recovered, in agreement with a prediction of Martineket al. [44] (see Fig. 5.3 for B ≈ 15mT).

Figure 5.3: Conductance curves for a Ni-C60-Ni device. An external magnetic field al-lows to switch the magnetic orientation of the electrodes between parallel and antiparallelalignment. For B ≈ 15mT (see right panel) the alignment between the electrodes is ap-proximately antiparallel, the strong-coupling Kondo effect (with an enhanced conductanceat V = 0 as shown in the right panel) is recovered even in presence of ferromagnetic elec-trodes. The peak in the conductance at V = 0mV, shown in the left panel, corresponds tothis particular magnetic field. For a different magnetic field (a magnetic field B ≈ 175mTis marked in the right panel) the electrodes are essentially parallel oriented and the con-ductance is split and suppressed (the split conductance in the left panel corresponds tothis magnetic field). This observation was predicted in the study presented in this Section.Figure taken from 5.3.

Currently the group of D. Ralph [56], Cornell University, is also trying to contact Mn-atoms (serving as QDs) to ferromagnetic leads, made of Mn as well. Unfortunately, thecontacting between the Mn-atoms and the Mn-leads turns out to be extremely difficult.


Another possibility to realize QDs contacted to polarized leads is to use QDs basedon single wall carbon nanotubes (SWNT), instead of semiconductor QDs. This systemsallow for various studies of Kondo effects in devices with magnetic or superconductingelectrodes [57].

SWNT are usually grown on iron catalysts. If this (ferromagnetic) catalyst is by ac-cident close to the contact between a non-magnetic lead and the SWNT, it can evidentlyappear that the tunneling between the SWNT and the leads becomes spin-dependent [58],a scenario that directly applies to the study presented in Section 5.1.

In a recent study Nygaard et al. [59] presented experimental data on a SWNT basedQD, where the Kondo resonance was split at zero magnetic field, see Fig. 5.4. This splittingwas interpreted, in agreement with the findings presented in this Section, as evidence forthe coupling of the dot to a ferromagnetic impurity.

Figure 5.4: Differential conductance as a function of V of a nanotube QD [59]. The splittingof the zero bias anomaly can nicely be explained by a QD coupled to ferromagnetic leads.The splitting disappears when the temperature T is increased. We could qualitativelyreproduce this behavior in an (unpublished) NRG-calculation.

5.2 Gate controlled spin splitting in the Kondo regime 57

5.2 Gate controlled spin splitting in the Kondo regime

In the above mentioned experiment of Nygaard et al. [59] also the gate voltage dependenceof the splitting of the Kondo resonance was studied, see Fig. 5.5. Three typical classes ofgate voltage dependence of this splitting were found, which the numerical study presentedin this Section confirms.

Figure 5.5: The gate voltage dependent spin splitting of the Kondo resonance as observedin the experiment of Nygaard et al. [59].

In the following paper (submitted to Phys. Rev. Lett.) we neither assume the leadsDoS to be flat nor we disregard the Stoner splitting. The leads DoS of our choice can befound in Fig. 1 of the paper presented below. In contrast to the previous work [15] wereally take the energy-dependence of the DoS into account. The chosen DoS corresponds toa dispersion as found for free electrons. Form a methodological point a view this approachis more sophisticated than the iteration scheme introduced by Wilson [1]. Here, Wilson’siteration scheme [1] has to be extended to handle leads with an arbitrary DoS. The wayhow this can be achieved is described in Appendix A. Note that the spin-dependent DoSin the leads we use here (which is, by the way, not particle-hole symmetric) complicatesthe NRG-iteration dramatically, as spin-dependent hopping matrix elements and spin-dependent onsite energies along the Wilson chain have to be determined numerically. Thisis achieved by solving the equations introduced in Appendix A. As these matrix elementsare decaying exponentially fast, their numerical determination requires rather advancednumerical techniques (for further information see Appendix A.2).

In this study we find that it is not sufficient to consider only the leads DoS at the Fermienergy ρσ(εF ) for a proper description of the resulting splitting of the Kondo resonance. Itturns out, instead, that all lead energies are necessary to describe the resulting splittingof the Kondo resonance properly - even though high energy states are logarithmicallysuppressed. For all choices of ρσ(ω) we find that one can tune the splitting of the Kondoresonance by means of an external gate voltage. Moreover we find that the Kondo resonanceis fully recovered for a special choice of εd for a particular form of ρσ(ω) [see Fig. 1(c) of the


paper below] . This finding suggests the interesting possibility to mimic a local magneticfield inside a QD by means of an external gate voltage.


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0.0 1.0 2.0

G [e2/h]

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0.02 0.04-0.02-0.02 0.0 0.040.02

0.0 0.04-0.04

0.0

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B/U

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c) Q=0.3

a) Q=0.0 b) Q=0.1

0.0

-0.50 -0.45 -0.40

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0.0

0.5

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- nd)

-1.0 -0.5 0.00

1

2

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Q=0.3B=0 B/U=0.0083

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Figure 5.6: An alternative representation of the spectral function A(ω, εd) of a QD con-nected to leads with a spin- and energy-dependent density of states (as shown in the inset).In contrast to a QD coupled to leads with a constant and spin-independent density of states,where a Kondo-plateau in A(ω = 0, εd) is observed for −U . εd . 0, a three peak structurein A(ω = 0, εd) is found here. For that particular density of states, the splitting of theKondo resonance, its full recovery and the resulting opposite splitting upon a variation ofεd can be inferred. (Parameters are equivalent to Fig. 1(f) of Ref. [54]).


5.2.1 Comprehensive study dealing with QDs coupled to ferro-magnetic leads

Currently we are working on a paper that summarizes the effects of a particular leads DoSon the Kondo resonance. Our goal is to study both the gate voltage and the polarizationdependence of the Kondo resonance for the three classes of DoS presented below. In Fig. 5.7these three classes of leads DoS are sketched.The DoS sketched in (a) was studied in Section 5.1 without, however, analyzing the gatevoltage dependence of the spin splitting. Even though this issue was already addressedin [60], we reexamine it here again, as its interpretation was in parts misleading.The DoS corresponding to Fig. 5.7(b) has, so far not been studied at all. Our numericsconfirms the expectation, based on poor man’s scaling, that such a Stoner splitting leadsto an logarithmically suppressed effective magnetic field.The most general case is studied in case (c). Even though this case has already intensivelybeen studied in Section 5.2 we will incorporate it in this comprehensive study.

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Figure 5.7: In a project we are currently working on we are studying the effect on theKondo resonance for these three DoS classes. This study summarizes the studies presentedabove. Case (a) was investigated in [15] without investigating the gate voltage dependenceof the spin splitting. The scenario with a finite Stoner-splitting, case (b), has so far notbeen investigated with NRG. Case (c) corresponds to a DoS as used in [54].

Chapter 6

Non-monotonic occupation intwo-level QDs

In Chapter 3.2.4 we pointed out that an increase in Γ, i.e. an opening of the QD, resultsin a significant enhancement of TK , see Eq. (3.29). This adjustment is sometimes requiredto reach the goal TK . Texp (where Texp is set by the fridge). However, when Γ becomesof the order of the level spacing ∆, Γ ∼ ∆, inter-orbital processes inside the QD mightbecome relevant [61], i.e. a single-level AM does not properly describe the QD any more.

Indeed, recent experiments on a ’Kondo-QD’ [8, 62]1, observed an anomaly which cannot be explained by means of a single-level AM. To be more specific: in Refs. [8, 62] Yanget al. observed an increase in the transmission phase φ of a QD in the Kondo valley byalmost 2π (for detailed information see [8, 62]). This observation can not be explainedin the framework of a single-level AM, as theoretically shown by Gerland et al. [33] (inRef. [33] the transmission phase of a single-level AM was predicted to increase by π insidea Kondo valley). However, since the QD in [8, 62] is rather open (with a ratio ∆/Γ ≈ 1-3)it is very likely that a simple single-level AM is not capable of all processes involved inthese experiments.

Motivated by these experiments [8, 62] and a theoretical study of Silva et al. [61] weinitiated a study on a two-level AM. In contrast to Ref. [61], however, we studied a two-levelQD including interactions.

In this studies we find, somehow surprisingly, a non-monotonic filling of the local levelsupon lowering them w.r.t. the chemical potential of the leads. Naively one might expect,based on standard Coulomb blockade, that the QD levels should fill one after the other.It is the interplay between the relevant energy scales ∆, Γu (the coupling strength of theupper level), Γ` (the coupling strength of the lower level) and U that can lead to a non-monotonic filling of the two local levels, as presented below. Thus, the appropriate fillingscheme of a more complicated QD depends strongly on the above mentioned parameters.Of course, in the limit ∆ À Γu, Γ` standard Coulomb blockade is recovered.

1The parameters of the QDs in these experiments were: U ≈ 1.5meV, ∆ ≈ 0.1-0.5meV, U/Γ ≈ 1-10,TK ≈ 500mK and Texp = 50mK.

66 6. Non-monotonic occupation in two-level QDs

As the observed non-monotonicity is due to interaction and not to spin effects, wefirst study this phenomenon with spinless electrons. The results of this study are shownbelow [27] (submitted to Phys. Rev. Lett.). A generalization of the findings presentedbelow to the spinfull case is currently in preparation.

67

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1

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(b)

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�

0.5

nl,u

nlnu

−1.5 0.5ε/U

−3

3

ε iH/Γ

i

0.5

1

−1.5 0.5ε/U

0

0.5

1

−1.5 0.5ε/U

s=+1 s=−1

Γl /∆=1.0

u

l

γ=0.1

γ=4.0∆/U=0.2

∆/U=0.2

(a) (b)

(c)

(d)

(e) (f)

γ=30∆/U=0.1Γl /∆=0.1

NRGscHA

Γl /∆=1.0

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Chapter 7

Frequency-dependent transport inthe Kondo regime

In this study (submitted to Phys. Rev. Lett.) we focus on the linear AC conductanceG(ω) = G′(ω) + iG′′(ω) (usually a complex quantity) of a single level QD (i.e. ∆/Γ À 1).In typical experiments where Kondo correlations are important usually the linear DCconductance is measured. Therefore recent studies have mostly focused on this quantity.However, a measurement of the AC conductance has recently come into experimentalreach [63].

Therefore, we generalize previous conductance studies in the Kondo regime to theAC case. For this sake we have to determine the complex quantity G(ω). Note thatthe DC conductance G(ω = 0) is purely real, G(ω = 0) = G′(ω = 0). We use the Kuboformalism (see appendix F) to obtain a reliable, nonperturbative calculation of G(ω) whichis reliable at all frequencies ω. This is because for sufficiently large values of ω, ω À U ,the charge on the island is fluctuating, so the used formalism has explicitly to allow forcharge fluctuations.

We derive an approximate formula, a first order perturbation expansion of the result ofJauho et al. [64], for the linear AC conductance in the Kondo regime valid for frequenciesω much smaller than the charge-excitation energy ∆c = min{|εd|, |U + εd|}. The validityof this formula is demonstrated by comparing it with the conductance calculated withinthe framework of the Kubo formalism.

The numerics reveals that G′(ω) has the same power laws as the spectral function A(ω)for frequencies ω ≤ TK . This fact suggests that A(ω) and G′(ω) are related to each other.Indeed, the derived analytical formula (from [64]) allows us to relate both quantities inthe limit of |ω| ¿ ∆c. This establishes a useful relation between the frequency-dependentconductance G(ω) and the spectral function A(ω).

Moreover, it was demonstrated in recent experiments of Deblock et al. [65] that it isfeasible to measure the current fluctuations of a mesoscopic device at frequencies up to90GHz. Motivated by this experiments we also calculate the frequency-dependent equilib-rium current fluctuations C(ω). The fluctuation dissipation theorem (FDT) establishes arelation between the conductance G(ω) and the current fluctuations C(ω), which enables us

72 7. Frequency-dependent transport in the Kondo regime

to establish a relation between the frequency-dependent noise C(ω) with the symmetrizedspectral function As(ω)1 (which by itself is related to G(ω)), for |ω| ¿ ∆c. This rela-tion is of experimental relevance, as it allows one to extract the symmetrized equilibriumspectral function As(ω) (for |ω| < ∆c, i.e. the Kondo peak) from a measurement of thefrequency-dependent noise.

In our calculations we assume a time-dependent chemical potential in the leads. Sincethe leads are capacitively coupled to the local dot level one expects also an oscillating locallevel - an effect which we disregard as this capacitive coupling is a rather weak one. Westudy the effect of an alternating chemical potential of the left and right lead which isrealized by applying an AC bias voltage (of frequency ωAC). Experimentally the frequencyof the (time-dependent) AC bias voltage has to be chosen such, that the Kondo scale can bedetected, i.e. ~ωAC ∼ kBTK . For a typical value of TK , TK = 1K, the frequency-dependentconductance G(ω) has thus to be determined for frequencies ωAC up to ∼ 20GHz.

1For |ω| ¿ ∆c this quantity is identical to A(ω) to a very good approximation.

73

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Chapter 8

Kondo correlations in opticalexperiments

Recent photoluminescence (PL) experiments [2] revealed that it is feasible to measure theabsorption and emission spectrum of a single self-assembled QD (an InAs QD embeddedin a GaAs semiconductor). It was found that an illumination of the semiconducting QDwith photons (of energy hν) triggers transitions inside the QD where an electron fromthe valence band (VB) level is elevated to the conduction band (CB) (or vice versa).Hereby an excited state, an exciton (electron-hole pair), is created (or destroyed). In anabsorption (emission) experiment the QD absorbs (emits) a photon of energy hν by creating(destroying) an excitonic state, say X.

The spatial confinement of the QD results in a discrete level structure inside the QD.An external gate voltage can rigidly shift the internal level structure leading to a filling(emptying) of the QD. The gate voltage dependence of optical transitions, as found inRef. [2] [see Fig. 8.1(top panel)], can be understood, by identifying a particular valueof the gate value with a particular excitonic state. Here a single exciton with n extraelectrons in the CB, for instance, is named Xn−. The lower panel of Fig. 8.1 shows thediscrete PL signals found in [2] which stem from the excitons X, X1−, X2−, . . . realizedupon successively lowering the gate voltage.

Indeed, the transitions observed in the PL experiment (PL counts vs. the gate voltage),shown in the top panel of Fig. 8.1, can be nicely explained by an atom-like electronic levelstructure inside the QD. The lower right panel of Fig. 8.1 nicely illustrates the involvedstates. The filling scheme suggested there quantitatively describes all observed PL-signals.For instance, the two appearing lines for the decay of the X2− state can be understood verywell when one identifies the two possible decay channels of this state: after the exciton hasdecayed, two electrons are left in the CB - one electron in the s- and another one in thep-level. These two electrons have either (i) parallel or (ii) anti-parallel spins. In case (ii)the excited state can immediately relax to the ground state of the system (the s level beingdoubly occupied). In case (i), however, a spin-flip event of the p-electron is required beforethe system can relax into its ground state. Therefore one expects a long-living state, i.e.a sharp signal, in case (i) and a short living one (correspondingly a broad signal) for case

78 8. Kondo correlations in optical experiments

(ii).The expectation that case (ii) has a smaller amplitude and is broader w.r.t. case (i)

can be clearly seen in the experiment. In summary: PL-experiments reveal an atom-likeelectronic level structure of the QD.

The possibility of tuning the QDs filling by an external gate voltage enables one toaddress interesting optical transitions within this experiment. We are particularly inter-ested in a scenario when the wetting layer surrounding the dot is filled with electrons andhybridizes with the local level in the dot. In this case an interesting optical transition be-tween a strongly correlated ’Kondo’ and an uncorrelated ’non-Kondo’ state is possible. Tostudy this effect we investigate the simplest transition that fulfills this requirement, namelythe absorption experiment where an initially singly occupied CB (left panel of Fig. 8.2)becomes doubly occupied under creation of an exciton (right panel of Fig. 8.2). For thissake we study an extended AM, consisting of a local VB level and a local CB level which iscoupled to a lead, as sketched in Fig. 8.2. Additionally, we introduce a Coulomb attractionbetween VB holes and CB electrons in the QD which accounts for the exciton bindingenergy Uexc. In the unphysical limit Uexc = 0, where the VB is completely decoupled fromthe CB, we find that the absorption spectrum is determined by the local density of statesof the QD. Even though this limiting case is unphysical, it provides a good check of theaccuracy of the involved numerics.

For finite values of Uexc we observe two rather dramatic new features: firstly, the thresh-old energy below which no photons are absorbed, say ω0, shows a marked, monotonic shiftas a function of Uexc. Secondly, as Uexc is increased, the absorption spectrum shows atremendous increase in peak height. In fact, the absorption spectrum diverges at thethreshold energy ω0, in close analogy to the well-known X-ray edge absorption problem.Exploiting analogies to the latter, we propose and numerically verify an analytical ex-pression for the exponent that governs this divergence, in terms of the absorption-inducedchange in the average occupation of the local conduction band level.

The work presented here (to be submitted to Phys. Rev. B) has already partly beenpublished in the Diploma thesis of Rolf Helmes, a diploma-student of Prof. Jan von Delft.

79

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Figure 8.1: Top: The PL energy vs. gate voltage characteristics shows sharp distinctPL signals that can be nicely explained by an atom-like electronic level structure in thesemiconductor QD. An external gate voltage allows for a controlled filling of the CB ofthe QD one by one. An excitonic state with n excess charges is named Xn−. [Photo:courtesy of A. Hoegele (LMU)]. Bottom left: PL vs. energy plots for the distinct gatevoltage regions found in the top panel (e.g. the lowest plot corresponds to a gate voltage−0.8V. VG . −0.6V). Bottom right: The filling scheme shown on the right explains thesharp observed PL signals very well. Thus, the PL experiment serves as direct evidencefor the atom-like electronic level structure inside the QD. Figure taken from [2].


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Figure 8.2: Sketch of an absorption process. The left panel shows the initial state (asingly occupied CB). An incoming photon of energy hν excites an electron from the VBinto the CB. The resulting (final) state, X1−, is shown in the right panel. Here we considera scenario where the wetting layer (surrounding the dot) is filled and hybridizes withthe electrons inside the dot. This example marks a transition from a strongly correlated(’Kondo’) into an uncorrelated (’non-Kondo’) state, as marked in the figure. Since themass of the holes in the VB is significantly larger than the electron mass in the CB wedisregard the coupling between the VB holes and the lead electrons.

81

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� � u � v � ��x | v�s ��t�� u � w�~ w � w u � w xkv � u v�� | � w ��w ��w { v ��| { { � ~�� w �$v � w��&w v v � � �� u � w ~�� ~ ~ | � � x � � ��v � w��t"} w { | ��w �� w xa�� v �5w � w { v ~ | � � ��} w { u � � w$v � w$~ w � � � v � � �� w ~ �� w u�{ u �`� � } ~ � x � � w�� v �av � wk� | { u �w � w { v ~ | �� w � w � �&| ��v � w�x | v ��$| v � � u v w x�} ��v � w � w�w � ��w ~ � ��w � v � � �&w�� v � x ��u ��w � v w � x w x�� x w ~ � | ��| x w � �� { �5x w � { ~ � }�w ��u�� | { u ��{ | � x � { v � | �a} u � x5� w � w ��{ | � � � w xav |�u�� w ~ �� w u ��} � v�u � � |�� { � � x w ��u�� | { u �� u � w � { w�} u � x$� w � w � �� w�u ~ w�� v w ~ w � v w x� � �$� u ~ v � { � � u ~ � | �� | ��u � � � }�| x ��{ | ~ ~ w � u v � | � ��~ w � � � v � � �� ~ | �v � w�� ~ w � w � { w�| �v � w�� w ~ �� w u�u ��w { v&v � w�w �� | ��u � x�u } � | ~ � v � | �� w { v ~ u ��¡�� `� � � | �¢ �&£�� w ~ � { u �� w � | ~ ��u � � � u v � | ��¤�~ | � �� w�{ u � { � � u v w�v � w�� w ~ | � v w �� w ~ u v � ~ w�w �� | �$s u } � | ~ � v � | � �� w { v ~ � �B| �u��t�� { �$w � x �� $� ��s � v u ~ v �&� ~ | ��u�� v ~ | � � � �${ | ~ ~ w � u v w xk¥�| � x |�� ~ | � � x$� v u v w �� w�� ~ w x � { v�v ��|�� w u v � ~ w � y�� ~ � v � � ��v � w�v � ~ w � � | � xaw � w ~ � ��¦�§��u } | � w�� { �5� |�� | v | �a� ��w �� v v w x`s } w � | �� { �a� |k� � | v | �a� �u } � | ~ } w x�� | ��u��u ~ ¨ w x�&��| � | v | � � {k� � � � v�u ��uk� � � { v � | �5| ��v � w$w � { � v | �a} � � x � � ��w � w ~ � �`©�ª « ¬ � w { | � x � � � ��w�� x$v � u v�v � w�� w { v ~ � �®� � | ��u�� | ��w ~�� u �Bx � � w ~ � w � { w � �� v ��u �kw � � | � w � v�v � u v�{ u ��} w� � x w ~ � v | | x$} �$u � u � | � ��v |�v � w��w � � � ¨ � | ��$¯�� ~ u ��w x � w�u } � | ~ � v � | �$� ~ | } � w �$�° ±&²�³�´ µ ¶�· ¸ ¹ º » ¼�½ ¾ ¿ À�° ±�²�³�½ Á Á ¸ ½ ¹�Â ¸ ¹ ¸

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*�. Í . 8 F"< Î F 7 Í > /�. Ï�Î. = 7 0$. 8 F 1 3 Ð @'Ñ�1 7 8�ÒÓ1 . / E Ô > 1 1 . 0�Õ�/ . 9Ö 8�×�1�Ø Ñ�> 8 F Ñ�0Ù9�< F 1kÚ Û�D�1 Ü 2&. 0�Õ. 9�9�. 957 8`Ý�> ×�1 2&1 Þ�< ß�. 9F Þ�> F�7 F�7 1&E . > 1 7 Õ�/ .�F <�0$. > 1 Ñ�= .�F Þ�.�> Õ�1 < = Î F 7 < 8�> 8�9$. 0$7 1 1 7 < 81 Î. Í F = Ñ�0"< E�>�1 7 8�Ò / .�Û�D�+ Ö 8k. 0$7 1 1 7 < 8k1 Î. Í F = Ñ�0%0$. > 1 Ñ�= . Ô0$. 8 F 1 2�> 8�. / . Í F = < 8 Ô Þ�< / .�Î�> 7 =�Ú . Ï�Í 7 F < 8ÜkÍ = . > F . 9�Õ à�/ > 1 . =. Ï�Í 7 F > F 7 < 8�= . Í < 0�Õ�7 8�. 1�7 81 7 9�.�F Þ�.kÛ�D�2�ß�Þ�. = . Õ à�>�Î�Þ�< F < 87 1$. 0$7 F F . 9�ß�Þ�7 Í Þ�7 1$0$. > 1 Ñ�= . 9�+�Ö 8�> Õ�1 < = Î F 7 < 8`1 Î. Í F = Ñ�00$. > 1 Ñ�= . 0$. 8 F 1 2�< 8kF Þ.�< F Þ�. =�Þ�> 8�9�2�Î�Þ< F < 8�1�> = .�> Õ�1 < = Õ. 97 8�1 7 9�.�F Þ�.�Û�D�Õ àk. Ï Í 7 F < 8�. Ï�Í 7 F > F 7 < 8�+D�Ñ�.�F <�1 Î�> F 7 > /�Í < 8 á8�. 0$. 8 F 2 F Þ�.�Û�D�Î�< 1 1 . 1 1 . 1�>�Í Þ�> = Ò Ô

7 8�Òa. 8�. = Ò àa> 8�9`>a9�7 1 Í = . F .k. 8�. = Ò à5/ . C . /�1 F = Ñ�Í F Ñ�= . 2�ß�Þ�7 Í ÞÍ > 85Õ.k= 7 Ò 7 9�/ àa1 Þ�7 E F . 9`ß�7 F Þ`= . 1 Î. Í F�F <�F Þ�.�â�. = 0$7�. 8�. = Ò àÕ à�C > = à 7 8�Ò`> 8�. Ï F . = 8�> /�Ò > F .�C < / F > Ò .aã�ä +®å�Þ�. = . E < = .aã�ä> / / < ß�1�E < =�> 8k. Ï�Î. = 7 0$. 8 F > /Í < 8 F = < /< E�F Þ�.�8 Ñ�0�Õ. =�< E�. / . Í ÔF = < 8�1�7 8�F Þ�.�Û�D�2�ß�Þ�7 Í Þ�7 8kF Ñ�= 8�9�. F . = 0$7 8�. 1�F Þ�.�. 8�. = Ò à$< EF Þ�.k> Õ�1 < = Õ. 95> 8�95. 0$7 F F . 9`Î�Þ�< F < 8�1 +�Ö 8�9�. . 9�2�F Þ�.k< Î F 7 Í > /9�> F >a1 Þ�< ß">a9�7 1 F 7 8�Í Fkã ä Ô 9�. Î. 8�9�. 8�Í .�> 89kæ Ñ�1 F 7 E à5F Þ�.�> 1 Ô1 Ñ�0$Î F 7 < 8k< E�>�9�7 1 Í = . F .�. 8�. = Ò à$/ . C . /�1 F = Ñ�Í F Ñ�= .�< E�F Þ�.�Û�D 3 +Ö 8�F Þ�.�. Ï�Î. = 7 0$. 8 F > / 1 . F Ô Ñ�Î�2 9�. Î�7 Í F . 9�7 8�â&7 Ò�+ ç 2 F Þ�.�Ö 8�×�1

Û�D�1�> = .�1 Ñ�= = < Ñ�8�9�. 9kÕ àk> 8kÖ 8�×�1�0$< 8�< Ô / > à . = 2 Í > / / . 9aè ß�. F ÔF 7 8�Ò�/ > à . = è Ú ,B:&Ü 2 / 7 é .�7 1 / > 8�9�1�7 8�> 8�< Í . > 8�+�×�Õ< C .�>�Í . = F > 7 8C > / Ñ�.�< E�ã ä 2 F Þ�.�Í < 8�9�Ñ�Í F 7 < 8�Õ�> 8�9�< E�9�. / < Í > / 7 ê . 9�1 F > F . 1�< EF Þ�7 1$,�:BÕ. Ò 7 8�1�F <aÕ.$á/ / . 9�2�E < = 0$7 8�Òa>�F ß�< Ô 9�7 0$. 8�1 7 < 8�> /â�. = 0$7�1 . >�< E�9�. / < Í > / 7 ê . 9$. / . Í F = < 8�1 2 7 + . + >�F ß�< Ô 9�7 0$. 8�1 7 < 8�> /. / . Í F = < 85Ò > 1kÚ ë D�ì�Ý�Ü +�å�Þ�.kë D�ì�Ý%Þ à Õ�= 7 9�7 ê . 1�ß�7 F Þ`/ < Í > / Ô7 ê . 9�1 F > F . 1�< E�F Þ�.`Û�D�2�/ . > 9�7 8Ò`F <�> 8�< 0$> / < Ñ�1�. 0$7 1 1 7 < 81 Î. Í F = >�ß�Þ�7 Í Þ�Í < Ñ�/ 9$8�< F�Õ.�. Ï�Î�/ > 7 8�. 9�Õ à�< 8�/ à�Í < 8�1 7 9�. = 7 8�ÒF Þ�.�9�7 1 Í = . F .�/ . C . /�1 F = Ñ�Í F Ñ�= .�< E&F Þ�.�Û�D 3 +4a< F 7 C > F . 9�Õ à5F Þ�. 1 .�. Ï�Î. = 7 0$. 8 F 1 2�Þ�. = .�ß�.�7 8 C . 1 F 7 Ò > F .

F Þ�.�< Î F 7 Í > /Î�= < Î. = F 7 . 1�< E�>$Û�DBÍ < Ñ�Î�/ . 9$F <�>�â. = 0k71 . >�2 > FF . 0$Î. = > F Ñ�= . 1�1 Ñ íkÍ 7 . 8 F / à`1 0$> / /�F Þ�> F$î�< 8�9�<aÍ < = = . / > F 7 < 8�1Í > 8k< Í Í Ñ�=�Ú å�ï�ð Ü +�å�Þ�.�î�< 8�9�<�. ñ�. Í F�7 8�>$Û�D�Þ�> 1�> / = . > 9�àÕ. . 8k9�. F . Í F . 9$7 8�F = > 8�1 Î�< = F�. Ï�Î. = 7 0$. 8 F 1 ò Ð ó 2 ß�Þ�. = .�7 F�/ . > 9�1F <`> 8�. 8�Þ�> 8�Í . 9�/ 7 8�. > =$Í < 8�9Ñ�Í F > 8�Í . +�6 <aE > =$F Þ�.�î�< 8�9�<

. ñ�. Í F�7 8kÛ�D�1�Þ�> 1�Õ. . 8$1 F Ñ�9�7 . 9$> / 0$< 1 F�. Ï�Í / Ñ�1 7 C . / à�7 8$= . / > ÔF 7 < 8$F <�F = > 8�1 Î< = F�Î�= < Î. = F 7 . 1 +�å�Þ�.�. Ï�Î. = 7 0k. 8 F 1�< E�*�. E 1 + 3 Ð @< Î. 8�F Þ�.�. Ï�Í 7 F 7 8�Ò�Î< 1 1 7 Õ�7 / 7 F à�F <�1 F Ñ�9à�F Þ�.�î�< 8�9�<�. ñ�. Í F�7 8< Î F 7 Í > /�. Ï�Î. = 7 0$. 8 F 1 +Ö 8�< Î F 7 Í 1 2 F Þ�.�î�< 8�9�<�. ñ�. Í F�Þ�> 1�F <�F Þ�.�Õ. 1 F&< E�< Ñ�=�é 8�< ß�/ Ô

. 9�Ò .aÕ. . 8B9�7 1 Í Ñ�1 1 . 9�F Þ�. < = . F 7 Í > / / à�< 8�/ à�ß�7 F Þ�= . 1 Î. Í F�F <8�< 8 Ô / 7 8�. > =$> 8�9�1 Þ�> é . Ô Ñ�Î`Î= < Í . 1 1 . 1k7 8�>`Û�D�ô Ð õ +�Ö 8�F Þ�7 1Î�> Î. =�ß�.�7 8 C . 1 F 7 Ò > F .�F Þ�.�> Õ�1 < = Î F 7 < 8�> 8�9$. 0$7 1 1 7 < 8�1 Î. Í F = >< E�>5Û�D�+�,5.k> = .$. 1 Î. Í 7 > / / àa7 8 F . = . 1 F . 9`7 8`< Î F 7 Í > /&F = > 8�1 7 ÔF 7 < 8�1�Ú . Ï�> 0$Î�/ . 1�> = .�1 Þ�< ß�8$7 8�â&7 Ò�+ ö�Õ. / < ß�Ü&E < =�ß�Þ�7 Í Þ$F Þ�.Û�D"1 F > = F 1�7 8a< =�. 8�9�1�Ñ�Îa7 8a>�1 F = < 8�Ò / à�Í < = = . / > F . 9�î�< 8�9�<Ò = < Ñ�8�9�1 F > F . 2�> 89�ß�7 / /�7 8 C . 1 F 7 Ò > F .�Þ�< ß"F Þ�.�î�< 8�9�<5Í < = Ô= . / > F 7 < 8�1$> ñ�. Í FkF Þ�.a< Õ�1 . = C . 9�/ 7 8�.a1 Þ�> Î. 1 +�Ö 8�*�. E + ÷�F Þ�.. 0$7 1 1 7 < 8�1 Î. Í F = Ñ�0®7 8�F Þ�.�î�< 8�9�<�= . Ò 7 0$.�Þ�> 1�> / = . > 9�à�Õ. . 81 F Ñ�9�7 . 9�2 Þ�< ß�. C . =�ß�7 F Þ$0$. F Þ�< 9�1�ß�Þ7 Í Þ$< 8�/ à�Î�= < 9�Ñ�Í .�Ø Ñ�> / Ô7 F > F 7 C .�= . 1 Ñ�/ F 1 +å�Þ�.�Î�> Î. =$7 1$< = Ò > 8�7 ê . 9`> 1�E < / / < ß�1 økÖ 8�6 . Í F 7 < 8�Ö Ö 2�ß�.

. Ï F . 8�9kF Þ�.�1 F > 8�9�> = 9�×�8�9�. = 1 < 8k0$< 9�. / ù�Õ à$7 8�Í / Ñ�9�7 8�Ò$>�/ < ÔÍ > /�C > / . 8�Í .kÕ�> 8�95/ . C . /�Ú : ú�;�:&Ü�Í < 8 F > 7 8�7 8�Ò�F Þ�.kÞ�< / . 1 +�Ö 8Í < 8 F = > 1 F�F <5*�. E 1 + 3 Ð @ ß�.�Í < 8�1 7 9�. =$< 8�/ à5< 8�.k/ < Í > /�Í < 8�9�Ñ�Í ÔF 7 < 85Õ�> 8�9a/ . C . /�Ú :&û�;�:&Ü�F <�1 7 0$Î�/ 7 E à�F Þ�.$Í > / Í Ñ�/ > F 7 < 8�1 +�Ö 86 . Í F 7 < 8�Ö Ö Ö 2 ß�.�. Ï�Î�/ > 7 8�Þ�< ß`,�7 / 1 < 8�è 1�8 Ñ�0$. = 7 Í > / = . 8�< = 0$> / Ô7 ê > F 7 < 8�Ò = < Ñ�Î`Ú ü�*�Ý�Ü�0$. F Þ�< 9�ý$Í > 8aÕ.�> 9�> Î F . 9�F <�Í > / Í Ñ Ô/ > F .�F Þ�.�. 0$7 1 1 7 < 8k> 8�9$> Õ�1 < = Î F 7 < 8k1 Î. Í F = Ñ�0"< E�F Þ�.�Û�D�+ Ö 86 . Í F 7 < 8$Ö ú�2 ß�.�Î�= . 1 . 8 F�F Þ�.�= . 1 Ñ�/ F 1�< E�< Ñ�=&Í > / Í Ñ�/ > F 7 < 8�1&> 8�9Î�= . 9�7 Í F$F ß�<a= > F Þ�. =$9�= > 0$> F 7 Í�8�. ß%E . > F Ñ�= . 1 +�â&7 = 1 F / à 2�F Þ�.F Þ�= . 1 Þ�< / 9�. 8�. = Ò à`> Õ< C .�ß�Þ�7 Í Þ�8�<`Î�Þ�< F < 8�7 1k. 0$7 F F . 9�< =Õ. / < ßBß�Þ�7 Í Þk8�<�Î�Þ�< F < 8k7 1�> Õ�1 < = Õ. 9�2�= . 1 Î. Í F 7 C . / à 2 1 > à�þ&ÿ 21 Þ�< ß�1�>�0k> = é . 9�2 0$< 8�< F < 8�7 Í�1 Þ�7 E F�> 1�>�E Ñ�8�Í F 7 < 8�< E�F Þ�.�. Ï ÔÍ 7 F < 8kÕ�7 8�9�7 8�Ò�. 8�. = Ò à�� ß�.�Ò 7 C .�>�Ø Ñ�> / 7 F > F 7 C .�. Ï�Î�/ > 8�> ÔF 7 < 8�< E F Þ�7 1&Õ. Þ�> C 7 < Ñ�=�Õ à�Í < 8�1 7 9�. = 7 8�Ò�F Þ�.�7 8 F . = Î�/ > à�< EC > = 7 Ô< Ñ�1&= . / . C > 8 F&. 8�. = Ò à�1 Í > / . 1 +&6�. Í < 8�9�/ à 2 > 1�� 7 1&7 8�Í = . > 1 . 9�2F Þ�.�. 0$7 1 1 7 < 8�> 8�9a> Õ�1 < = Î F 7 < 8�1 Î. Í F = >k1 Þ�< ß�>$F = . 0$. 8�9�< Ñ�17 8�Í = . > 1 .�7 8$Î. > é�Þ�. 7 Ò Þ F +�Ö 8�E > Í F 2 F Þ�.�> Õ�1 < = Î F 7 < 8�1 Î. Í F = Ñ�01 Þ�< ß�1&>�Î< ß�. =&/ > ß`9�7 C . = Ò . 8�Í .�> F&F Þ�.�F Þ�= . 1 Þ�< / 9�. 8�. = Ò à�þ&ÿ 27 8�Í / < 1 .�> 8�> / < Ò à�F <�F Þ�.�ß�. / / Ô é 8�< ß�8��Ô = > à�. 9�Ò .�> Õ�1 < = Î F 7 < 8

82 8. Kondo correlations in optical experiments�

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2DEG

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2DEG

Kondo state

2DEG

QD

photon

EF

EF

QD

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photon

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b)

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-1.4 -0.7 0 0.7(ω - εv) / Uc

100

10000

α(ω

)

0.00.30.50.7

-1.4 -0.7 0 0.7(ω + εv) / Uc

-1 0 1 2 3 4 5(ω - ω0) / TK

Emission EmissionAbsorption

Uexc / Uc = εc / Uc = -0.5Uv / Uc = 1.0Γ / Uc = 0.15

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0.001 0.01 0.1(ω-ω0) / TK

1000

10000

α G(ω

)

0.00.20.40.60.81.0

Uexc/Uc=εc =-0.5 UcUv = 1.0 UcΓ = 0.3 Uc

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0 0.2 0.4 0.6 0.8 1∆n

0

0.1

0.2

0.3

0.4

0.5

β y = x - x2/2

εc =-0.5 Ucεc =-0.3 Ucεc =-0.1 Ucεc = 0.1 Ucεc = 0.3 Ucεc = 0.5 Uc

Uv = 1.0 UcΓ = 0.3 Uc

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Uexc

Uexc

Uexc

UcUc

Uexc

N = −1 n/2∆

=G��

ε

photon

c2DEG i

FE<n>

∆n/2

n/2∆

f i + n∆∆n<n> = <n>

N = − n/2∆n/2∆

+ + ...

small, but finite weight

+ + ...

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Chapter 9

Summary and outlook

In this thesis we have been focusing on zero-temperature properties of QD-systems. Thedifferent QD-systems that we analyzed were modeled by the Anderson impurity model,a model that is extremely well suited to describe QD-physics. In the limit of small tem-peratures correlation effects are crucial. Consequently, we employed Wilson’s numericalrenormalization group method, which is essentially a numerical exact method for the deter-mination of the low-temperature properties of the various models we studied. We expectour predictions to be observed in (future) experiments.1

Table 9.1 shows four different classes of impurity models we were solving in this thesis.One can categorize these classes by the number of (spinfull) levels inside the impurityand the number of (spinfull) channels the impurity couples to. In particular, the numberof channels the (complex) impurity couples to sets a serious restriction for the numericalsolution of a particular model.2 Within each of those classes particular symmetries can beexploited. As stressed in Appendix C the use of symmetries is essential for an accuratedetermination of the quantities one is interested in. As this symmetries depend on theparticular model under study we do not refine Table 9.1 w.r.t. particular symmetries here.

Whereas in this thesis, we only considered QDs coupled to fermionic baths,3 the NRG-method is not restricted to fermionic baths. The ’bosonic’ NRG, developed by Bulla etal. [66], overcomes this restriction. This generalization of the NRG-method is a new pathone should definitely follow. It opens the exciting possibility to study models, such as thespin boson model (see [66]), which are, for instance, relevant in the context of quantum-computation [22].

The code that was programmed during my PhD-studies is rather flexible. Therefore itallows for the solution of various problems (dealing with fermionic baths) without changing

1The spin splitting of the Kondo resonance for a QD contacted to ferromagnetic leads has already, inagreement with our predictions, been observed [55] (see also [59]).

2It is the number of coupled channels that governs the growth-rate of the Hilbert space in the courseof the NRG-iteration.

3In Chapter 6 and 7 we solved models up to four different fermionic baths, (two different channels eachcarrying a spin index).

92 9. Summary and outlook

1 channel 2 channel

1 level Chapter 5, 8 Chapter 72 level Chapter 6 Chapter 6

Table 9.1: The table classifies the models that have been solved in Part II of this thesis.Models that fall into one of these classes can, in principle, be solved with the NRG-codethat was produced within my PhD-studies. Since the symmetries one should exploit dependcrucially on the model one is studying we do not refine this table by including possiblesymmetries.

the structure of it.In a recent experiment on single wall carbon nanotubes [67], for instance, a fascinatingfeature of a sixfold split Kondo resonance (in presence of a finite magnetic field) has beenobserved. In this experiment, carried out at TU Delft (Netherlands), the nanotube wastuned such that it behaves, in absence of a magnetic field, as a two-fold degenerate QD4

coupled to two distinct channels. Once the relevant model of this experiment is identified, itis rather straightforward to use the developed code to compute the (level and spin resolved)spectral functions of the system, which enables one to gain a deeper understanding of theexperimental results.

4It turns out that a magnetic field couples differently to the spin and the orbital degree of freedom,leading to a difference between the Zeeman and the orbital splitting.

Part III

Appendix

94

Appendix A

Derivation of the NRG-equations

In the early 1970’s K.G. Wilson succeeded to construct a nonperturbative solution of theKondo Model (KM) [1]. The crucial step in Wilson’s procedure was the mapping of thefree conduction electrons (that surround the impurity) onto a chain, the Wilson chain.Krishna-murthy et al. [31], [47] extended this procedure to the Anderson Model (AM) [25]in the early 1980’s.

In contrast to Wilson and Krishna-murthy et al., who assumed a flat conduction band(CB),1 we consider here a CB with a spin- and energy-dependent density of states (DoS)ρσ(ε) [the energy dispersion in the CB εkσ is related to ρσ(ε) via ρσ(ε) =

∑k δ(ω−εkσ)]. For

this sake the iteration scheme used in [1] has to be generalized for a CB with an arbitraryDoS ρσ(ε). To establish this generalization we follow Bulla [68], [69] and Hofstetter [70]:2

For simplicity the mapping of the single level AM,

HAM = Hd +∑

kσ

εkσc†kσckσ

︸︷︷︸HCB

+∑

kσ

Vkσ

(c†kσdσ + d†σckσ

)

︸︷︷︸H`d

, (A.1)

on the Wilson chain will be described below. As explained in Eq. (3.7), the AM consists ofthree parts, the CB HCB, the tunneling part H`d and the impurity part Hd which does notneed to be specified here. The operators dσ and ckσ denote the impurity and CB operators[actually the symmetric linear combination of the left and right lead (called αskσ in partI, see Section 3.1)] which obey Fermi statistics (with σ describing the z-component of spinorientation), respectively. The corresponding mapping of an impurity that couples to Nchannels3 - the case N = 1 is shown here - can easily be obtained via a generalization ofthe procedure outlined below. In Chapter 6, for instance, the mapping for N = 2 had tobe performed.

1 The bandwidth D of the CB is the natural energy scale; for a flat and normalized CB the DoS isgiven by: ρ(ε) =

∑σ ρσ(ε) = 1

2D , since∑

σ

∫ D

−Ddερσ(ε) = 1.

2An impurity that couples to an arbitrary DoS is of particular importance in DMFT-applications [26].3ckσ → ckiσ, i ε {1, · · · , N}. Note, however, that a single level impurity couples effectively to one

channel only (as a unitary transformation [16] reveals).

96 A. Derivation of the NRG-equations

A.1 Manipulation of the CB

A continuous representation of HCB and H`d is more convenient for the derivation of theNRG-mapping [69]. To achieve this, we replace the k-sums in Eq. (A.1) by a continuousrepresentation

HCB =∑

σ

∫ 1

−1

dε gσ(ε) a†εσaεσ, (A.2)

H`d =∑

σ

∫ 1

−1

dε hσ(ε)(d†σaεσ + h.c.

). (A.3)

In Eq. (A.2) an isotropic energy representation of the CB (with band edge D = 1, i.e.|ε| ≤ 1) with a generalized dispersion gσ(ε) has been introduced. Accordingly, a generalizedhybridization function hσ(ε) was inserted in Eq. (A.3). The continuous CB-operators aεσ

obviously have to obey Fermi statistics {aεσ, a†ε′σ′} = δ(ε− ε′)δσ,σ′ . The equivalence of the

discrete and continuous version of HAM [i.e. between Eq. (A.1) and Eqs. (A.2), (A.3)] isestablished if the effective action on the impurity degree of freedom is identical in bothrepresentations. Given that the functions gσ(ε) and hσ(ε) obey the relation

∂g−1σ (ε)

∂ε

[hσ

(g−1

σ (ε))]2

= ρσ(ε) [Vσ(ε)]2 , (A.4)

where g−1σ (ε) is the inverse of gσ(ε), this requirement is fulfilled and the discrete and the

continuous representation of the AM are identical (see Eq.(10) of [69]).Obviously, there are many possibilities to satisfy Eq. (A.4). One possibility is to choose thedispersion gσ(ε) = g−1

σ (ε) = ε and the generalized hybridization hσ(ε) =√

∆σ(ε)/π, with

ε-dependent hybridization ∆σ(ε) ≡ πρσ(ε) [Vσ(ε)]2 (other possibilities can, for instance,be found in [69]). For reasons that will become clear below, this choice is well suitedfor a constant (or at least sufficiently smooth) hybridization ∆σ(ε). For a hybridizationwith a strong energy-dependence [71], however, the functions gσ(ε) and hσ(ε) are chosendifferently [still satisfying Eq. (A.4)]. An example of the latter case is given in Fig. A.1(a)[the corresponding functions gσ(ε) and hσ(ε) are sketched in Fig. A.1(b)].

Eq. (A.3) reveals that the impurity couples to all energy scales of the problem, i.e.infinitely many degrees of freedom. Wilson’s original idea, a logarithmic discretization ofthe CB, allowed him to transform H`d into a tractable form. The logarithmic discretizationresolves low-energy states with a high accuracy, but does not simply disregard high energystates of the CB, as shown in Fig. A.1(a). By introducing a discretization parameter Λ(Λ > 1) Wilson divided the CB into energy intervals ranging between

[−Λ−n;−Λ−(n+1)[

and]Λ−(n+1); Λ−n

], n ε N0. In each logarithmic interval a Fourier Series

Ψ±np(ε) =

{1√dn

e±iωnpε if Λ−(n+1) < ±ε ≤ Λ−n; p ε Z0 else

(A.5)

is defined where the superscript ± labels wave functions defined for positive/negative en-ergies. The subscripts n and p mark the interval and the harmonic index of the Fourier

A.1 Manipulation of the CB 97

expansion. Note that the width of the n-th interval dn = Λ−n (1− Λ−1) determines thefundamental Fourier frequency ωn = 2π/dn of that interval. The expansion (A.5) now

��

��

+=

Vσ

ε/D

σ(ε)ρ

ε/D

σ(ε)∆

��

��

−n

−2

−Λ0

−Λ−1−Λ−2

−nΛΛ−1

Λ0

−ΛΛ

b)

]2

ε/D

(ε)σ

Imp

��

��

��

��

� � ��

��

��

��

��

��

ε/D

��

��

��

Imp

1

0

−1

a)

n[h σg

Figure A.1: (a) Since the impurity couples to all energies, the CB is logarithmically dis-cretized (with discretization parameter Λ). (b) For an energy-dependent hybridization∆σ(ε) (left) one chooses the generalized hybridization hσ(ε) = h±nσ to be constant in eachlogarithmic interval; for this choice only the (p = 0)-component of the CB couples to theimpurity (right panel). The corresponding dispersion gσ(ε), sketched in the middle panel(solid line), follows from solving Eq. (A.4). For reference the linear dispersion gσ(ε) = ε(dashed line) is plotted as well.

allows one to replace the continuous operators aεσ in HCB and H`d [cf. (A.2) and (A.3)] bydiscrete ones,

aεσ =∑np

[anpσΨ+

np(ε) + bnpσΨ−np(ε)

]. (A.6)

A discrete set of anti-commuting operators anpσ and bnpσ [note: {anpσ, a†n′p′σ′} = δnn′δpp′δσσ′

and analogously for bnpσ] with harmonic index p and spin σ,

anpσ =

∫ 1

−1

dε[Ψ+np(ε)]

∗aεσ, (A.7)

bnpσ =

∫ 1

−1

dε[Ψ−np(ε)]

∗aεσ, (A.8)

that act on the n-th positive or negative logarithmic interval is defined hereby.


A.1.1 Transformation of H`d

We start the derivation of the NRG-equations by a transformation of the tunneling partH`d of the AM. Inserting the expansion (A.6) into Eq. (A.3) yields

H`d =∑npσ

{[d†σ

(anpσ

∫ Λ−n

Λ−(n+1)

dε hσ(ε)Ψ+np(ε) + bnpσ

∫ −Λ−(n+1)

−Λ−n

dε hσ(ε)Ψ−np(ε)

)]+ h.c.

}.

(A.9)Henceforth, we take the generalized hybridization function hσ(ε) to be constant [69] inall logarithmic intervals, hσ(ε) → h±nσ, see Fig. A.1(b), which ensures that the impuritycouples to s-waves only, i.e. to the (p = 0)-contributions (Riemann-Lebesgue Lemma) [70].Consequently the harmonic index p can be dropped in Eq. (A.9). This particular choice ofthe generalized hybridization function, the averaged hybridization function

h+nσ ≡

{ √1dn

∫ Λ−n

Λ−(n+1) dερσ(ε)[Vσ(ε)]2 if Λ−(n+1) < ε ≤ Λ−n

0 else, (A.10)

h−nσ ≡{ √

1dn

∫ −Λ−(n+1)

−Λ−n dερσ(ε)[Vσ(ε)]2 if −Λ−n < ε ≤ −Λ−(n+1)

0 else, (A.11)

forces us to adjust the generalized dispersion gσ(ε) such that Eq. (A.4) still holds. Ingeneral, as indicated in Fig. A.1(b), this results in a nonlinear generalized dispersion gσ(ε).Inserting Eqs. (A.10) and (A.11) into Eq. (A.9) simplifies H`d considerably

H`d =

√1

π

∑σ

d†σ

∼f0σ︷︸︸︷∑n

(γ+

nσanσ + γ−nσbnσ

)+h.c.

, (A.12)

with γ+nσ ≡

√∫ Λ−n

Λ−(n+1) dε ∆σ(ε) and γ−nσ ≡√∫ −Λ−(n+1)

−Λ−n dε ∆σ(ε), respectively. As indicated

in Eq. (A.12), it is convenient to define a fermionic operator f0σ (with {f0σ, f†0σ′} = δσσ′),

which has the form

f0σ ≡ 1√ϑ0σ

∑n

(γ+

nσanσ + γ−nσbnσ

), (A.13)

with ϑ0σ =∑

n

[(γ+

nσ)2+ (γ−nσ)

2]

=∫ 1

−1∆σ(ε)dε. The operator f †0σ creates a spin σ electron

in the maximally localized state (which is essentially the conduction electron field on theimpurity site), equivalent to the zeroth site of the Wilson chain, see Fig. A.2. In the limitof constant hybridization, ∆σ(ε) = ∆σ, Eq. (A.13) reproduces the well-known result [31],f0σ =

∑∞n=0

√dn/2 (anσ + bnσ) =

√(1− Λ−1)/2

∑∞n=0 Λ−n/2(anσ + bnσ).

We insert the maximally localized state into Eq. (A.12) and finally obtain

H`d =∑

σ

[√ϑ0σ

π

(d†σf0σ + h.c.

)]

. (A.14)

A.1 Manipulation of the CB 99

Eq. (A.14) nicely illustrates that the impurity couples to a single fermionic degree offreedom only, the zeroth site of the Wilson chain. To fully map HAM on a linear chain,however, still HCB needs to be transformed properly. We focus on this issue in the nextSection.

A.1.2 Transformation of HCB

When we insert the expansion (A.6) into Eq. (A.2) the CB-Hamiltonian HCB reads

HCB =∑

σ

∫ 1

−1

dε gσ(ε)

{∑

npp′

[a†npσ

[Ψ+

np(ε)]∗

anp′σΨ+np′(ε) + b†npσ

[Ψ−

np(ε)]∗

bnp′σΨ−np′(ε)

]}

.

(A.15)Though only (p = 0)-contributions couple directly to the impurity [remember that we choseconstant hybridization h±nσ, see Eqs. (A.10) and (A.11)], (p 6= 0)-contributions might be-come important via an indirect coupling to the impurity [via the (p = 0)-mode].Therefore two contributions remain in the pp′-sum of Eq. (A.15):

∑p=0,p′ [. . .] = [. . .]p=0,p′=0+∑

p′ 6=0 [. . .]p=0.

The first term of this sum (p = p′ = 0), which we label H(1)CB, reduces Eq. (A.15) to

H(1)CB =

∑nσ

[ζ+nσa

†nσanσ + ζ−nσb

†nσbnσ

], (A.16)

with the general expression ζ+nσ = (1/dn)

∫ Λ−n

Λ−(n+1) dε gσ(ε) and ζ−nσ = (1/dn)∫ −Λ−(n+1)

−Λ−n dε gσ(ε).Note the following: since the single-particle energies of the CB electrons ζ±nσ depend onlyon the integral over the logarithmic intervals an exact knowledge of gσ(ε) [which can beinferred from Eq. (A.4)] is not required.Bulla et al. [69] showed that for the particular choice taken here [hσ(ε) → h±nσ in eachlogarithmic interval], the single-particle energies ζ±nσ are given by

ζ+nσ ≡

∫ Λ−n

Λ−(n+1) dε ε ∆σ(ε)∫ Λ−n

Λ−(n+1) dε ∆σ(ε), (A.17)

ζ−nσ ≡∫ −Λ−(n+1)

−Λ−n dε ε ∆σ(ε)∫ −Λ−(n+1)

−Λ−n dε ∆σ(ε). (A.18)

A comparison between the general expression for ζ+nσ and Eq. (A.17) reveals that the dis-

persion is indeed nonlinear here, gσ(ε) = ε{

[dn∆σ(ε)]/[∫ Λ−n

Λ−(n+1) dε∆σ(ε)]}

.

For constant hybridization ∆σ(ε) = ∆σ the linear dispersion [gσ(ε) = ε] is recovered withinthis more general framework. Also the result for HCB of Krishna-murthy et al. [31] [who as-

sumed ∆σ(ε) = ∆; see Eq. (2.12) of [31]], H(1)CB = 1

2(1 + Λ−1)

∑nσ Λ−n

(a†nσanσ − b†nσbnσ

),

i.e. ζ±nσ = ±12(1 + Λ−1) Λ−n, is recovered in this more general scheme.


The second contribution to the (pp′)-sum,∑

p′ 6=0 [. . .]p=0, couples off-diagonal angular

momenta. This term, labeled by H(2)CB, takes the form

H(2)CB =

∑nσ

∑

p′ 6=0

a†np=0σanp′σ

I1︷︸︸︷(∫ Λ−n

Λ−(n+1)

dε gσ(ε)[Ψ+

np=0(ε)]∗

Ψ+np′(ε)

)+

+ b†np=0σbnp′σ

(∫ −Λ−(n+1)

−Λ−n

dε gσ(ε)[Ψ+

np=0(ε)]∗

Ψ+np′(ε)

)

︸︷︷︸I2

.(A.19)

Note that the mapping of the AM on the Wilson chain is still exact at this stage (HCB =

H(1)CB + H(2)

CB), with hσ(ε) and gσ(ε) satisfying Eq. (A.4). In particular, Eq. (A.19) stressesthat a complete description of the CB includes all possible angular momenta p.We analyze the importance of H(2)

CB by computing the integral I1 in Eq. (A.19) for theparticular case of a linear dispersion (which results in: I2 = −I1):

4 inserting Eq. (A.5) intoI1 results in

I1 =1

dn

∫ Λ−n

Λ−(n+1)

dε ε exp (iωnp′ε) =1

dn

{exp (iωnp

′ε)[− (iωnp

′)−2+ ε (iωnp′)

]}∣∣∣Λ−n

Λ−(n+1)

which finally reduces to

I1 =1− Λ−1

2πip′Λ−n exp [2πip′/(1− Λ−1)] (A.20)

[see also Eq. (2.10) of [31]].

We conclude from Eq. (A.20) that H(2)CB becomes less and less important for Λ ap-

proaching 1. Wilson showed [1] that the p 6= 0 contributions, i.e. H(2)CB, can be dropped to

a very good approximation for Λ = 2, which was mostly used in this thesis.5

From now on we will disregard H(2)CB, which is like dropping a small perturbation in

HCB. The first approximation we use is therefore

HCB ≈ H(1)CB, (A.21)

i.e. we replace Eq. (A.2) by Eq. (A.16).

4The reasoning also holds for an arbitrary dispersion, see [69].5The approximation works still pretty good for bigger values of Λ, e.g. Λ = 3.

A.2 Mapping of the CB onto a semi-infinite chain 101

A.2 Mapping of the CB onto a semi-infinite chain

To solve HAM numerically, it is convenient to bring HCB, Eq. (A.16), into tridiagonal from,see Fig. A.2, i.e. to map HCB onto a semi-infinite chain (the Wilson chain HWC),

∞∑n=0σ

(ζ+nσ a†nσanσ + ζ−nσ b†nσbnσ

)

︸︷︷︸HCB

=∞∑

n=0σ

[εnσf

†nσfnσ + tnσ


)]

︸︷︷︸HWC

. (A.22)

For this mapping (see e.g. [72]) we use the tridiagonalization procedure developed byLanczos [73] with diagonal matrix elements εnσ and off-diagonal ones tnσ [70].The tridiagonalization procedure of Lanczos allows us to determine HCB |Ψnσ〉, where|Ψnσ〉 = f †nσ |0〉 denotes a one-particle state (with Fock vacuum |0〉, n ε N0),

6

HCB |Ψnσ〉 =

εnσ︷︸︸︷〈Ψnσ| HCB |Ψnσ〉 |Ψnσ〉 +

tn−1σ︷︸︸︷〈Ψn−1σ| HCB |Ψnσ〉 |Ψn−1σ〉

+

tnσ︷︸︸︷〈Ψn+1σ| HCB |Ψnσ〉 |Ψn+1σ〉 . (A.23)

We can immediately identify the matrix elements εnσ, tn−1σ and tnσ by comparing Eq. (A.23)with the ansatz (A.22). Eq. (A.23) nicely illustrates that the n-th site of the Wilson chainis only connected to its two neighboring sites, the (n− 1)-th and the (n + 1)-th site of thechain. It can also nicely be seen that the recursive determination of the fermionic oper-ators fnσ involves the calculation of the onsite energies and the hopping matrix elementsalong the Wilson chain. Below, the derivation of the corresponding recursion relations issummarized.

The single-particle operator fnσ that acts on the n-th site of the Wilson chain is givenby the ansatz

fnσ =∞∑

m=0

(unmσamσ + vnmσbmσ) , (A.24)

with coefficients unmσ and vnmσ that need to be determined recursively.The initial values u0mσ and v0mσ are immediately found by comparing the ansatz (A.24)with f0σ as given in (A.13),

u0mσ =γ+

mσ√ϑ0σ

, v0mσ =γ−mσ√ϑ0σ

. (A.25)

An inversion of the ansatz (A.24) leads to anσ =∑∞

m=0 umnσfmσ and bnσ =∑∞

m=0 vmnσfmσ.When we insert anσ and bnσ in the l.h.s. of Eq. (A.22) and compare corresponding fnσ

operators on both sides of this equation we obtain

∞∑m=0

(ζ+msunmσa

†mσ + ζ−msvnmσb

†mσ

)= εnσf

†nσ + tnσf

†n+1σ + tn−1σf

†n−1σ. (A.26)

6 f0σ was already given in Eq. (A.13); see Eq. (A.24) for the general definition of fnσ.


In particular, the (n = 0)-comparison in Eq. (A.26) (which reflects the comparison of thef0σ operators) yields

∞∑m=0

(ζ+mσγ

+mσ√

ϑ0σ

a†mσ +ζ−mσγ

−mσ√

ϑ0σ

b†mσ

)= ε0σf

†0σ + t0σf

†1σ, (A.27)

where Eq. (A.25) has already been inserted.Since the operators fnσ obey Fermi-statistics, {fnσ, f

†n′σ′} = δnn′δσσ′ , the anticommutator

of the r.h.s. of Eq. (A.27) with f0σ yields {ε0σf†0σ + t0σf

†1σ, f0σ} = ε0σ. The corresponding

anticommutator of the l.h.s. of Eq. (A.27) with f0σ as given in (A.13),7 finally leads to therelation

ε0σ =1

ϑ0σ

∑m

[ζ+mσ

(γ+

mσ

)2+ ζ−mσ

(γ−mσ

)2]. (A.28)

The initial hopping matrix element t0σ can now easily be determined from Eq. (A.27)by computing the anticommutator {ε0σf

†0σ + t0σf

†1σ, ε0σf0σ + t0σf1σ} = (ε0σ)2 + (t0σ)2 =

∑m

{(ζ+mσγ+

mσ√ϑ0σ

a†mσ + ζ−mσγ−mσ√ϑ0σ

b†mσ

),(

ζ+mσγ+

mσ√ϑ0σ

a†mσ + ζ−mσγ−mσ√ϑ0σ

b†mσ

)†}. As ε0σ is known we obtain

t0σ as

(t0σ)2 =1

ϑ0σ

{∑m

[(ζ+mσ

)2 (γ+

mσ

)2+

(ζ−mσ

)2 (γ−mσ

)2]−

∑m

[ζ+mσ

(γ+

mσ

)2+ ζ−mσ

(γ−mσ

)2]}

.

(A.29)When we insert Eqs. (A.13), (A.28) and (A.29) into Eq. (A.27) and compare this with theansatz for f †1σ, Eq. (A.24), we obtain

u1mσ =γ+

mσ√ϑ0σt0σ

(ζ+mσ − ε0σ

), (A.30)

v1mσ =γ−mσ√ϑ0σt0σ

(ζ−mσ − ε0σ

). (A.31)

Following the above argumentation one finally obtains the spin-dependent onsite energiesεnσ and hopping matrix elements tnσ at the n-th site of the Wilson chain as

εnσ =∑m

[(unmσ)2ζ+

mσ + (vnmσ)2ζ−mσ

], (A.32)

(tnσ)2 =∑m

[(unmσ)2 (

ζ+mσ

)2+ (vnmσ)2 (

ζ−mσ

)2]− (tn−1σ)2 − (εnσ)2 , (A.33)

with the coefficients of the single-particle operator fn+1σ, defined in Eq. (A.24),

un+1mσ =1

tnσ

[(ζ+mσ − εnσ

)unmσ − tn−1σun−1mσ

], (A.34)

vn+1mσ =1

tnσ

[(ζ−mσ − εnσ

)vnmσ − tn−1σvn−1mσ

]. (A.35)

7Note that the discrete operators anσ and bnσ are anticommuting as well.

A.3 Iterative numerical diagonalization 103

Even though, only the matrix elements along the Wilson chain (εnσ and tnσ) are finallyrequired, it is crucial to determine the coefficients un+1mσ and vn+1mσ as well, since εnσ andtnσ depend on these coefficients, see Eqs. (A.32) and (A.33).

Note: the mapping of the CB onto the Wilson chain does not change electron-holesymmetry (as it is a unitary transformation). The Wilson chain, given in Eq. (A.22),consists only of hopping matrix elements tnσ and onsite energies εnσ. Thus an electron-hole symmetric hybridization ∆σ(ε) implies that all onsite energies along the Wilson chainvanish, εnσ = 0 ∀n ∈ N0. More rigorously the reasoning goes as follows: for an electron-hole symmetric hybridization [∆σ(ε) = ∆σ(−ε)] the relations ζ+

nσ = −ζ−nσ and γ+nσ = γ−nσ

are valid, thus ε0σ = 0 [Eq. (A.28)]. This results in u1mσ = −v1mσ what leads to ε1σ = 0[Eq. (A.32)]. From Eqs. (A.34) and (A.35) one realizes that this leads to u2mσ = −v2mσ

which results in ε2σ = 0 etc. .We shall make a final remark here about the numerical solution of the above mentioned

equations: since the band energies are exponentially decaying in the course of the iterationone has to use reliable numerical routines (i.e. arbitrary-precision Fortran routines [69]) tosolve for the coefficients (unmσ, vnmσ) and matrix elements (tnσ, εnσ).

A.3 Iterative numerical diagonalization

When we rename the impurity operator dσ ≡ f−1σ the AM takes a particular compactform

HAM = Hd +

HCB+H`d︷︸︸︷∞∑

n=−1,σ

[tnσ


)+ εn+1σf

†n+1σfn+1σ

], (A.36)

with the coupling between the impurity and the CB t−1σ ≡√

ϑ0σ

π[see Eq. (A.14)], the

kinetic energy (∝ tnσ) [see Eq. (A.33)] and the onsite energy (∝ εnσ) [see Eq. (A.32)] ofthe CB, written in form of the Wilson chain. Note that there is no possibility of a spin-flipevent along the chain. The Hamiltonian (A.36) is valid for an arbitrary energy- and spin-dependent hybridization function ∆σ(ε), it is therefore a generalization of the one obtainedby Krishna-murthy et al. [see Eq. (2.14) of [31]]. For the particular case considered byKrishna-murthy et al. [31] (∆σ(ε) = ∆, i.e. flat and spin-independent bands), their resultis recovered with

tnσ =1

2(1 + Λ−1)Λ−n/2

ξKn︷︸︸︷

1− Λ−n−1

√(1− Λ−2n−1)(1− Λ−2n−3)

(A.37)

and εnσ = 0 ∀n ∈ N0.The obtained ’chain-Hamiltonian’, depicted in Fig. A.2, can be solved by iterative

diagonalization. In the limit of an infinite long Wilson chain the AM is recovered,


t0 t1 tN−1 tNt−1

ε0 ε1 ε2 εN

H0

H1

HN

IterationImp 0. 1. 2. N.

energyresolution

∼∆1/2 ∼Λ0 ∼Λ−1/2 ∼Λ−(N−1)/2

t2

Figure A.2: The Wilson chain can be iteratively diagonalized, since only neighboring sitesare coupled with each other. One proceeds by first diagonalizing HN , the Hamiltonian thatdescribes the system consisting of the impurity and the first N sites of the Wilson chain,then setting up HN+1 [via Eq. (A.40)] by coupling HN to the (N+1)-th site of the chain andrestarting the diagonalization procedure. Along the chain the energy resolution increasessince smaller and smaller energies (∼ Λ−(N−1)/2 in the N -th iteration) are coupled to HN .The tN ’s are the exponentially decaying tunneling matrix elements and the εN the onsiteenergies along the chain. For symmetric hybridization, ∆(ε) = ∆(−ε), εN = 0 ∀ N ∈ N0.

HAM = limN→∞

Λ−(N−1)/2HN (A.38)

with the Hamiltonian HN that contains the first N sites of the Wilson chain

HN = Λ(N−1)/2

{Hd +

N−1∑n=−1,σ

[tnσ


)+ εn+1σf

†n+1σfn+1σ

]}.

(A.39)Indeed, Eq.(A.39) coincides with the formula found by Krishna-murthy et al. [Eq.(2.18)of [31]] for ∆σ(ε) = ∆.

Eq. (A.39) allows us to set up the NRG-iteration, heavily used in this thesis

HN+1 =√

ΛHN + ΛN/2∑

σ

[tNσ

(f †NσfN+1σ + f †N+1σfNσ

)+ εN+1σf

†N+1σfN+1σ

]. (A.40)

The latter equation defines a transformation R that (i) relates effective Hamiltonians onsuccessive lower energy scales with each other and (ii) couples only neighboring sites along

A.4 Results of the iterative diagonalization 105

the (Wilson) chain (HN+1 = R[HN ]).8

Note that the energy resolution in the N -th iteration is of order ∼ Λ−(N−1)/2, see alsoFig. A.2. The rescaling factor Λ(N−1)/2 in Eq. (A.39) ensures that HN contains numbersthat are ∼ O(1), i.e. numerically “good” numbers. HN is an effective Hamiltonian that cap-tures the physics at a temperature kBT/D ∼ Λ−(N−1)/2. Lower temperatures are reachedwhen one proceeds along the Wilson chain.In the course of the iteration one solves the Schrodinger equation iteratively, i.e. one com-putes the eigenstates and corresponding eigenenergies of HN . The exponential decay ofthe matrix elements tNσ and εNσ is crucial for the success of the iteration scheme (A.40).

The iteration scheme (A.40) becomes particularly transparent in matrix-notation,

HN+1 =

|0〉N+1

|↑〉N+1

|↓〉N+1

|↑↓〉N+1

|0〉N+1

√ΛHd

N tN↑f†N↑ tN↓f

†N↓ 0

|↑〉N+1

tN↑fN↑√

ΛHdN + εN+1↑ 0 tN↓f

†N↓

|↓〉N+1

tN↓fN↓ 0√

ΛHdN + εN+1↓ −tN↑f

†N↑

|↑↓〉N+1

0 tN↓fN↓ −tN↑fN↑√

ΛHdN +

∑σ εN+1σ

.

Here each entry represents a (truncated) matrix which is written in the basis where HN isdiagonal, Hd

N .Since the Hilbert space is growing exponentially fast when one proceeds along the chain atruncation is necessary. Thus, one only keeps the lowest lying eigenstates of HN (typically∼ 1000) in each iteration. This is achieved by diagonalizing HN , HN → Hd

N , sorting itw.r.t. increasing eigenenergies and finally truncating it.

The NRG-iteration for the KM is identical to the one for the AM (with identical εnσ

and tnσ). The only thing that needs to be adjusted is the initial Hamiltonian H0. For theKM the coupling between the impurity and the zeroth site of the Wilson chain is ∼ J(the Kondo coupling) (and not ∼ √

∆ as for the AM).

A.4 Results of the iterative diagonalization

As explained above, before the Hamiltonian HN is coupled to the next site of the chain,it is diagonalized, Hd

N |Ψn〉N = ENn |Ψn〉N . The knowledge of the n-th eigenstate |Ψn〉N (in

the N -th iteration) and its corresponding eigenenergy ENn enables us to calculate physical

quantities at a scale ωN ∼ Λ−(N−1)/2D. Additionally, the calculation of an expectationvalue of an operator O requires the knowledge of the corresponding matrix elements (inthe basis of Hd

N ; see Section 4.1) of this operator.

The calculation of dynamic quantities (see also Section 4.1), such as the spectralfunction Aσ(ω), defined as Aσ(ω) = − 1

π= [GR

σ (ω)], with the retarded Green’s function

GRσ (t) = −iθ(t)

⟨{dσ(t), d†σ(0)

}⟩, requires even more tricks. The spectral function obtained

8For constant hybridization one obtains from Eq. (A.37): tNσ → Λ−N/2 for N →∞.


in the N -th iteration ANσ (ω) takes the following form in Lehmann representation [74]

ANσ (ω, T ) =

∑nm

[e−βEN

m + e−βENn

ZN(T )

]∣∣

N〈Ψn| d†σ |Ψm〉N

∣∣2 δ(ω − (

ENn − EN

m

)). (A.41)

Two problems arise from the way Eq. (A.41) is written: (i) what is the ’optimal’ way ofreplacing the δ-functions in Eq. (A.41)? An important question since one is interested incomputing the continuous spectral function Aσ(ω, T ) (see Appendix B.1). (ii) How shouldone mix the matrix elements in the spectral function AN

σ (ω, T ) to obtain the best possiblespectral function of the system Aσ(ω, T )? We comment on this issue in Appendix B.2.

Appendix B

How to obtain continuous dynamicfunctions

It was pointed out in Section 4.1 that dynamic quantities, such as the spectral functionA(ω, T ) [see Eq. (4.5)], can be written as a sum of δ-functions; the spectral function corre-sponding to the N -th NRG-iteration, for instance, can be found in Eq. (A.41). Figure B.1shows the eigenspectrum of the N -th iteration (horizontal lines) with possible transitions(indicated by the arrows), i.e. non zero matrix elements

N〈Ψn| d†σ |Ψm〉N , for (a) T = 0 and

(b) T 6= 0. Note that for T 6= 0 transitions between excited states are possible.Since the typical energy scale in the N -th iteration is ωN , AN(ω, T ) contains mostly

transitions at this energy scale, i.e. AN(ω, T ) ∼ A(ωN , T ). Information about bigger(smaller) energies in A(ω, T ) are obtained at an earlier (later) stage of the iteration pro-cedure.

Our final goal, namely to obtain a continuous spectral function A(ω, T ) (ω ∈ [−D; D]),is achieved when the following two problems are resolved: (i) δ-functions appearing inAN(ω, T ), see Eq. (A.41), have been broadened properly and (ii) spectral functions AN ofdifferent iterations have been combined appropriately as explained above [i.e.

∑N AN ∼

A(ω)].Task (i) will be resolved in Section B.1, task (ii) in Section B.2.

B.1 The broadening of the δ-functions

The broadening of the δ-functions by gaussians or lorentzians suggests itself, as the iden-tities δ(c − ε) = limσ→0

1σ√

2πe−(c−ε)2/(2σ2) or δ(c − ε) = limσ→0

1π

σ(c−ε)2+σ2 [76] hold, re-

spectively. It turns out, however, that dynamic quantities (such as the spectral function)are very sensitive to the properties of the corresponding broadening-functions. Based onsome properties of the spectral function, such as (i) the height of the spectral functionat the Fermi energy A(ω = 0, T = 0) (Friedel-sum-rule), (ii) the accuracy of the relation

n(T = 0) =∫ 0

−∞ dωA(ω, T = 0) [the (thermodynamically computed) occupation n(T = 0)is known with high precision] and (iii) the width of the atomic resonances (which is Γ),

108 B. How to obtain continuous dynamic functions

bE E

0 T

T

0

a

Figure B.1: Possible transitions that contribute to AN(ω, T = 0) (a) and AN(ω, T 6= 0)(b). Here all energies are measured w.r.t. the ground state energy of the N -th iteration.Note the following: (i) for finite temperatures the possibility of transitions between excitedstates exists and (ii) an additional energy scale T comes into play. Therefore one shouldstop the NRG-iteration once the energy resolution ∼ Λ−(N−1)/2 is comparable to T . Thedashed lines in (a) and (b) mark the cutoff in the energy spectrum which arises from thetruncation of highly excited states. Figure taken from [75].

one can decide which broadening-function is preferable.It turns out that it is favorable to replace the appearing δ-functions by gaussians rather

than lorentzians. This is because gaussians, in contrast to lorentzians, decay exponentiallyfast. Sakai et al. [77] showed that a gaussian adopted to the logarithmic grid used in theNRG-scheme, i.e. a logarithmic gaussian, has even better properties than the ’usual’ gaus-sian has. Fig. B.2 shows a comparison between both functions: the logarithmic gaussianis not symmetric around its center, in contrast to the ’usual’ gaussian. It can be inferredfrom Fig. B.2 that logarithmic gaussians suppress low energies even more than ’usual’gaussians do. Since these energies are resolved at later iterations this suppression makesthem preferable w.r.t. ’usual’ gaussians.

Consequently appearing δ-functions are broadened by a logarithmic gaussian [75]

δ(c− ε) → e−σ2/4

σε√

πe−

(ln(c)−ln(ε))2

σ2 , (B.1)

with positive energies ε and c [for negative energies Eq. (B.1) has to be adjusted corre-spondingly]. In typical NRG-calculations we use σ ≈ 0.6.To improve our intuition of the gaussian and the logarithmic gaussian we compute theirwidth here: at energies c ± σ

√2 the ’usual’ gaussian has decayed to 1/e of its maximal

B.2 How to combine information from different iterations 109

0 2 4 6ε

0

0.2

0.4

0.6 log. gaussiangaussian

σ=0.6

Figure B.2: Logarithmic gaussian [solid; see Eq. (B.1)] vs. ’usual’ gaussian (dashed) forc = 2 and σ = 0.6. The logarithmic gaussian is not symmetric around its center c; it hascomparably more weight at higher energies than a ’usual’ gaussian does. Note that thewidth of the logarithmic gaussian (see text) scales with its center, whereas the width ofthe gaussian is determined by σ only.

value, its width is thus ∼ σ. A logarithmic gaussians, however, decays to 1/e of its maximalvalue at energies c± δc, with δc = c (e±σ − 1). This implies that its widths scales with thevalue of its center c.

B.2 How to combine information from different iter-

ations

To illustrate problem (ii) we use the spectral function A(ω) as a showcase. A good summaryabout this topic is given in an article of Bulla et al. [75].

One way to obtain a continuous spectral function A(ω) is to analyze the sum of the N -

th NRG-iteration, AN(ω, T = 0) =∑

n

∣∣N〈n| d† |0〉

N

∣∣2 δ(ω− (ENn −EN

0 )) at a frequency ωN

which is typical for the N -th iteration, say ωN = c ωN (typically c ≈ 2; ωN = Λ−(N−1)/2).In this scheme, the spectral function of the N -th iteration is replaced by the averaged valueat frequency ωN , AN(ω) → AN(ωN). One chooses c such that it corresponds to a statewith an energy that lies roughly in the middle of the N -th cluster (by cluster we mean theset of all possible transitions included in the N -th iteration). Rather big (small) values ofc are better described at an earlier (later) stage of the iteration. The continuous spectralfunction A(ω) is finally obtained by connecting the ’averaged’ points (ωN ; AN(ωN)). In thisapproach one does not have to worry about the issue of over-counting the matrix elementsof the spectral function since consecutive intervals are evaluated separately. This issue isrelevant for the procedure outlined below.

The second possibility to evaluate A(ω, T ) is to first combine information of the relevant


matrix elements from neighboring clusters, say the N -th and (N + 2)-th cluster (to avoideven/odd oscillations), and finally to broaden the combined spectrum. This procedure wassuggested by Bulla et al. [75]. As shown in Fig. B.3, one combines the matrix elementsof the N -th (which contains all relevant matrix elements of previous clusters) and the(N + 2)-th cluster such that one finally obtains a mixed cluster that includes informationof both original clusters. To do so one embarks the following strategy: those energies ofthe N -th cluster that are not contained in the (N + 2)-th cluster are just copied in themixed cluster. Correspondingly energies of the (N +2)-th cluster that are not contained inthe N -th cluster are copied in the mixed cluster. For this sake relevant energies ωN+2

min , ωNmin

and ωN+2max (see Fig. B.3) have to be identified. Formally, matrix elements with energies

ε > ωN+2max or ε < ωN

min are copied in the mixed cluster.The energy region where the N -th and (N+2)-th cluster overlap is combined with appropri-ate weighting of the contributions stemming from the two initial regions. Double-countingof matrix elements is avoided by introducing a weighting function whose weight increases(or decreases) linearly between 0 and 1 (or 1 and 0) for ωN

min ≤ ε ≤ ωN+2max , as shown in

Fig. B.3. As the (N + 2)-th cluster describes smaller energies better than the N -th clusterdoes, it is appropriate that the weighting function of the (N+2)-th cluster has its maximumweight at ωN

min (the contrary holds for the N -th cluster). After completing this procedurea set of all relevant (properly weighted) matrix elements (still at discrete energies) for the(N + 2)-th iteration is obtained.The spectral function A(ω, T ) is finally obtained by broadening all δ-functions of the finalcluster. In this approach, every δ-peak is broadened which has the substantial advantagew.r.t. the first procedure that the parameter c does not have to be chosen in this scheme.This approach, introduced for finite temperature calculations first, works very well in theT = 0 limit, too. To summarize: in contrast to the previous approach, here one broadensthe full (combined) spectrum and one does not average over a cluster.

Below we want to comment on some technical details of the broadening procedure.As the logarithmic gaussian is restricted to positive (or negative) energies (cf. Fig. E.1)it is important to introduce a broadening scheme that interpolates between negative andpositive energies. In the finite temperature broadening scheme suggested by Bulla et al. [75]a lorentzian is used for broadening the δ-functions at energies |ε| < 4T . For |ε| > 4T theδ-functions are broadened by the logarithmic gaussians introduced in Eq. (B.1).1 Thereforethe δ-functions are approximated by

δ(c− ε) →{ 1

πσ

(c−ε)2+σ2 , |ε| < 4T, σ = 0.6T,

e−σ2/4

σ|ε|√πe−

(ln(|c|)−ln(|ε|))2σ2 , |ε| > 4T, σ = 0.3.

Finally we want to show a plot of the matrix elements of neighboring clusters. In the his-togram Fig. B.4 matrix elements of the N -th and (N + 2)-th cluster are plotted versus thetypical energy of the N -th cluster Λ−(N−1)/2D. It can be nicely inferred from this figure thatthe (N + 2)-th cluster contains excitations at smaller energies than the N -th cluster. For

1Note that the logarithmic gaussians are restricted to positive (negative) frequencies for c > 0 (c < 0).

B.2 How to combine information from different iterations 111

ωN ωN+2ωN+2

up to N+2

N+2

up to N

0

0 ω

ω

0 ωminmin max

Figure B.3: The initial cluster contains information about all (even) iterations up to theN -th iteration. The combination of this cluster with the (N + 2)-th cluster, schematicallydepicted, results in a cluster that contains all relevant information up to the (N + 2)-thiteration. The vertical lines correspond to relevant matrix elements (obtained via NRG)that one needs to combine to achieve a continuous spectral function. Relevant energies inthis scheme are: the biggest energy of a non-vanishing matrix element of the (N + 2)-thiteration ωN+2

max and the corresponding smallest energies of the N -th and (N +2)-th iterationωN

min and ωN+2min . The linear weighting functions ensure that the obtained spectral function

does not suffer from double counting of involved matrix elements.

this particular example, Fig. B.4 reveals that there are just a few nonzero matrix elementswithin each cluster. Additional to this, one realizes that the absolute value of the matrixelements decreases in the course of the iteration. This histogram is an important tool tofind out which of the above mentioned two procedures of combing the matrix elements isbetter suited to obtain a smooth dynamical function.Given the individual clusters contain many nonzero matrix elements, the procedure sug-gested by Bulla at al. [75] is preferable. For only a few matrix elements within eachcluster (as shown in Fig. B.4), however, the first suggested method turns out to be moresuccessful [78].


0 4 80

0.02

0.04

0 4 8ε Λ(Ν−1)/2

/D

0

0.02

0.04

N+2

N

Figure B.4: Matrix elements of the Kubo operator (see Appendix F) in the case of acalculation done for one level coupled to two leads (two channels) (as used in [7]). Theodd channel decouples in this particular example. Therefore there are only a few nonzeromatrix elements of this operator left. Note that the procedure suggested in Fig. B.3 is notsuccessful here, since the different clusters contain only a few nonzero matrix elements.Here, the evaluation of a continuous function according to the first described procedure ismore successful.

Appendix C

Symmetries in the NRG-scheme

The most time-consuming steps of the NRG-iteration are the iterative numerical diago-nalizations of (truncated) Hamiltonians HN of size (NN ×NN) and the following unitarytransformations of all relevant operators. Since both processes scale like N 3

N it is crucial toidentify symmetries of the problem which enables us to divide the full Hilbert space intosmaller subspaces. This ensures that the model one is interested in can be solved with suf-ficient precision (set by the value of NN) while keeping the computational time tolerable.Especially when one studies more complicated models, such as two channel models, theuse of symmetries is essential.

An additional reason why one should use the symmetries of a model is the following:as mentioned in Appendix A, the Hilbert space along the Wilson chain is growing expo-nentially fast. To circumvent this problem a truncation scheme (a pure energy criterion) isintroduced where only the lowest lying eigenstates of the system are kept. This truncationscheme, however, might introduce ’artificial perturbations’ in the NRG-iteration. For in-stance, if one does not use the full spin symmetry (present in absence of a magnetic field;for details see below), one can introduce an ’artificial magnetic field’ during the truncationprocess by dividing spin multiplets. One can get rid of these ’artificial perturbations’ byimplementing the corresponding symmetry.

Neither the impurity part Hd nor the chain part of the NRG-iteration (A.40) createsor annihilates an electron. The total charge of a system that contains the impurity andthe first N sites of the Wilson chain, QN (normalized to half-filling),

QN =N∑

σ,n=0

(f †nσfnσ − 1

)+

(d†σdσ − 1

), (C.1)

is therefore a conserved quantity within the NRG-procedure, formally[QN , HN

]= 0

∀N ∈ N0. From[QN , HN

]= 0 follows that HN has to be diagonal in QN ; therefore the

full Hilbert space of the N -th iteration can be decomposed into subspaces sorted w.r.t. allpossible charge quantum-numbers QN . Thus it is convenient to add a charge label to an

114 C. Symmetries in the NRG-scheme

arbitrary state of the N -th iteration |Ψ〉N

= |Ψ; QN〉N .Even though one is tempted to introduce also a total spin symmetry S, we do not

implement this symmetry, as it is broken in presence of a finite magnetic field B (whichis often used in our studies). If one would, however, want to use this symmetry, one hasto compute Clebsch-Gordan coefficients to be able to use reduced matrix elements.1 Inpresence of a finite magnetic field in z-direction, however, the z-component of the spin Sz

N ,defined as

SzN =

1

2

N∑n=0

(f †n↑fn↑ − f †n↓fn↓ + d†↑d↑ − d†↓d↓

), (C.2)

is still conserved,[Sz

N , HN

]= 0. Note that this symmetry is broken if the possibility of a

spin-flip event along the chain exists or if an additional magnetic field in x or y direction ispresent. Given this symmetry exists, the Hilbert space of the N -th iteration can be furtherdecomposed. In this case the states are labeled by their total charge and the z-componentof the total spin |Ψ〉

N= |Ψ; QN , Sz

N〉N .In case of two-level impurities, as studied e.g. in [27], one might introduce additional

symmetries such as a ’symmetric’ and an ’antisymmetric’ charge quantum number, QsN and

QasN , respectively, given the system can be separated into a symmetric and an antisymmetric

subsystem. When these conditions are met, it is convenient to write an arbitrary state as|Ψ〉

N= |Ψ; Qs

N , QasN , Sz

N〉N .

1 For B = 0 the eigenstates of HN are eigenstates of the operators QN , (S2N ) and Sz

N . In absence of amagnetic field one can label the eigenstates of HN with |Ψ; QN , SN , ms

N 〉N .

Appendix D

The density matrix NumericalRenormalization Group (DM-NRG)

The traditional NRG-scheme inherits only one energy scale (corresponding to the effectivetemperature of the N -th iteration ∼ Λ−N/2). In cases where different energy scales areinvolved serious problems arise, as outlined in Ref. [4]. If a finite magnetic field is present,for instance, the energy scale ωmagn ∼ gµBB is a relevant energy scale as well. A cure of thisproblem was invented by Hofstetter [4], who proposed a generalization of the traditionalNRG-method.

As a magnetic field (with characteristic energy ωmagn) affects the spectral functionA(ω, T ) on all frequencies ω, it was shown in Ref. [4] that one has to distinguish carefullybetween the two scales ω and ωmagn. For B 6= 0 the expectation value of an operator cannot be calculated by ’just’ combining the matrix elements as suggested in Appendix B.The scheme one should use instead is Hofstetter’s ’DM-NRG’ procedure [4].The fundamental new idea within this approach is that any expectation value is calculatedby considering the density matrix ρ which contains information of the full system.Indeed, as shown in [4], this more sophisticated treatment has only consequences (in par-ticular on ’high-energy’ features) when more than one energy scale is relevant - else the’DM-NRG’-method coincides with the traditional method on all energy scales.

In presence of a magnetic field one would naively expect the following: at high energiesa tiny perturbation (e.g. a small magnetic field) does not affect the spectrum at firstglance. However, this small perturbation might break the spin symmetry of the groundstate leading to a finite polarization of the impurity and therefore result in a remarkableeffect at high energies. As shown in Fig. D.1, a finite magnetic field (due to a finite spin-polarization in the leads) results in a systematic redistribution of spectral weight from thelower to the upper atomic level. The ’traditional’ NRG-scheme (lower panel in Fig. D.1)is not capable of this effect.

Hofstetter [4] realized that in such a scenario it is crucial to start with the correctground state (which is the one in presence of a magnetic field), the real many-body groundstate of the system, to observe the behavior anticipated above. Therefore the DM-NRG-method consists of two stages:

116 D. The density matrix Numerical Renormalization Group (DM-NRG)

First the usual NRG-procedure is run until the strong coupling fixed point is reached, sayat the energy TN∗ . At this energy the density matrix of the system ρ,

ρ =1

Z

∑m

e−EN∗m /(kBTN∗ ) |m〉

N∗〈m| , (D.1)

with Z =∑

m e−EN∗m /(kBTN∗ ), is constructed. Here EN∗

m are the eigenenergies of the eigen-

states |m〉N∗ of HN∗ . ρ fully describes the physical state of the system. In particular, the

(retarded) equilibrium Green’s function takes the form

GRσ (t) = −iθ(t)tr

[ρ{dσ(t), d†σ(0)}] . (D.2)

In the DM-NRG-scheme one uses Eq. (D.2) instead of Eq. (4.7) to compute A(ω, T ) [relatedto GR

σ via Eq. (4.5)], i.e. the spectral function is evaluated by using the correct reduceddensity matrix ρred. As shown in [4] the reduced density matrix is obtained by splittingthe Wilson chain into a smaller cluster of length N (N < N∗) and an environmental partwhich contains the remaining degrees of freedom.

In practice, one has to store all unitary transformations (up to the N∗-th iteration)that lead to the real ground state, evaluate the density matrix at this iteration, and thenevaluate the expectation value given in Eq. (D.2) backwards (i.e. ∀N ∈ N0, N < N∗).In the course of this ’backwards-iteration’ the reduced density matrices are obtained bytracing out the environmental degrees of freedom. Consequently one deals with a triplesum at every iteration step N (N < N∗) and finally obtains the spectral function of theN -th iteration as

ANσ (ω, TN∗) =

∑i,j,m

N〈j| d†σ |m〉N N

〈j| d†σ |i〉N ρredim,Nδ(ω − (EN

j − ENm))

+∑i,j,m

N〈m| d†σ |j〉N N

〈i| d†σ |j〉N ρredim,Nδ(ω − (EN

i − ENj )). (D.3)

Since we only keep the matrix elements of d†, we wrote the last expression in the corre-sponding form. It is worthwhile mentioning that in the DM-NRG-approach transitionsbetween excited states are allowed which are forbidden in ’traditional’ zero-temperatureNRG-calculations. Thus, both terms in Eq. (D.3) contribute at positive and negativefrequencies to AN

σ (ω, TN∗).For the spin correlation function, introduced in Eq. (4.10), one obtains a similar formula

χ′′N(ω) = π∑i,j,m

N〈m|Sz |j〉

N N〈j|Sz |i〉

Nρred

im,Nδ(ω − (ENj − EN

m))

−π∑i,j,m

N〈m|Sz |j〉

N N〈j|Sz |i〉

Nρred

im,Nδ(ω − (ENi − EN

j )), (D.4)

in the framework of the DM-NRG-method.

117

−1 1 3ω / | εd |

0

1πΓ

A(ω

)/4

P=0.0P=0.2P=0.4P=0.6 −0.125 0 0.125

0

1

πΓA

(ω)/

4

DM−NRG

Γ=U/6εd=−U/3T=0, B=0

−1 1 3ω / | εd |

0

1

πΓA

(ω)/

4

P=0.0P=0.2P=0.4P=0.6 −0.125 0 0.125

0

1

πΓA

(ω)/

4

NRG

Γ=U/6εd=−U/3T=0, B=0

Figure D.1: Spectral function A(ω, T = 0) of a QD coupled of polarized leads with varyingleads polarization P [15] once computed with the DM-NRG (top) and for comparison withthe usual NRG-method (bottom). As discussed in [15], finite P has similar effects as anapplied magnetic field, leading to a systematic shift (indicated by the arrow in the upperplot) of spectral weight to the upper atomic level. The spectral functions computed via theDM-NRG nicely show this behavior while those computed with the ’conventional’ NRG-method are not capable of this shift. For small energies, both methods are essentiallyidentical.

118 D. The density matrix Numerical Renormalization Group (DM-NRG)

Appendix E

Kramers-Kronig (KK) relation

The KK-relations [79] establish a relation between the real and the imaginary part of acausal function1 κ(ω) = < [κ(ω)] + i= [κ(ω)], ω ∈ {−∞,∞},

< [κ(ω)] =1

πP

∫ ∞

−∞

= [κ(ω′)]ω′ − ω

dω′, (E.1)

= [κ(ω)] = − 1

πP

∫ ∞

−∞

< [κ(ω′)]ω′ − ω

dω′, (E.2)

where P ∫∞−∞

=(<)[κ(ω′)]ω′−ω

dω′ = limε→0

[∫ −ε

−∞=(<)κ(ω′)

ω′−ωdω′ +

∫∞ε

=(<)[κ(ω′)]ω′−ω

dω′]

denotes the prin-

cipal value of the corresponding integrals. In cases when one is interested in both, realand imaginary part of a causal function, it suffices to compute only one of them. Thecorresponding conjugated part can be obtained from relation (E.1) or (E.2), respectively.However, the computation of the principal value (with high precision) makes the KK-transformation a nontrivial numerical task.

We follow an idea of Bulla [5] for performing the KK-transformation here. In the courseof NRG the imaginary part of an arbitrary causal function, say = [κ(ω)], is given (beforebroadening) as a sum of matrix elements αn at energies pn, = [κ(ω)] =

∑n αnδ(ω − pn).2

Since the integration is a linear operation, there are obviously two possibilities for theKK-transformation of = [κ(ω)]: (i) one first sums up the broadened δ-functions (leadingto a continuous function) and afterwards performs the KK-transformation, < [κ(ω)] =[∑

n αnδ(ω − pn)]KK or (ii) one performs the KK-transformation of all δ-functions before

summing them up, < [κ(ω)] =∑

n αn [δ(ω − pn)]KK .Since typical functions of our interest, like = [GR(ω)

], reveal sharp features around the

Fermi energy (Kondo resonance) it is much harder to obtain the KK-transformed followingapproach (i). For this reasons we follow the second approach.

Let us consider the causal function = [κ1(ω)] = δ(ω − pn), a single delta function at

energy pn. The complex function κ1(z), κ1(z = ω + iη) =(− 1

π

) ∫∞0

dω′ δ(ω′−pn)

z−ω′ , z ∈ C1Such as the retarded Green’s function GR(ω).2Note: = [GR(ω)

] ∼ ∑n

∣∣〈n| d† |0〉∣∣2 δ(ω − pn), i.e. a sum of δ-functions.

120 E. Kramers-Kronig (KK) relation

and η → 0+, is easily obtained by analytic continuation. We know from Appendix B thatthe broadening of the δ-peak works best when one replaces them by logarithmic gaussians

δ(ω − pn) → e−b2/4

bpn√

πexp

[− (ln ω−ln pn)2

b2

], centered at the positive energy pn (analogously for

pn < 0) with b ≈ 0.6. It is convenient to write the logarithmic gaussian as a function f

that only depends on ω/pn, 1pn

e−b2/4

b√

πexp

[− (ln(ω/pn)2

b2

]= 1

pnf

(ωpn

), thus

κ1(z = ω + iδ) =

(− 1

π

) ∫ ∞

0

dω′1pn

f(

ω′pn

)

z − ω′. (E.3)

By means of the definitions x′ ≡ ω′pn

, x ≡ ωpn

and δ′ ≡ ηpn

we can therefore compute both

the real and the imaginary part of κ1(z)

< [κ1(z = ω + iδ)] =

(− 1

π

)1

pn

g(x)︷︸︸︷∫ ∞

0

dx′f (x′)x− x′

δ′2 + (x− x′)2= −g(x)

π pn

, (E.4)

= [κ1(z = ω + iδ)] =

(− 1

π

)1

pn

[−

∫ ∞

0

dx′f (x′)δ′

δ′2 + (x− x′)2

]

︸︷︷︸h(x)

= −h(x)

π pn

, (E.5)

with the function f(x) (defined for x > 0 only)3 given as

f(x) =e−b2/4

b√

πexp

[−(ln x)2

b2

]. (E.6)

The functions that describe the real and the imaginary part of κ1(z), namely g(x) andh(x), cf. Eqs. (E.4) and (E.5) have to be determined for sufficiently small δ′, chosen suchthat the integrals in Eqs. (E.4) and (E.5) have converged (typically δ′ ≈ 0.01). Note thatthe required convergence makes an adjustment of δ′ necessary when the value of b in thelogarithmic gaussian is changed. The big advantage of this treatment is that one has tocompute (numerically) the function g(x) for any given b, with an accordingly adjusted δ′,only once. Consequently we store the function g(x) for the most relevant values of b onthe hard disk.

Finally we remark on an anomaly: although the logarithmic gaussians are confined topositive or negative frequencies, their KK-transformed, obtained from g(x), see Eq. (E.4),are not restricted to those intervals, see Fig. E.1.

3Note: for x ≤ 0 Eq. (E.6) is not defined, thus we define f(x) = 0 ∀x ≤ 0.

121

−3 −1.5 0 1.5 3 4.5x

−3

−2

−1

0

1

2

−5 −3.5 −2 −0.5 1 2.5x

−3

−2

−1

0

1

2pn < 0 pn > 0

f(x)

g(x)g(x)

f(x)

Figure E.1: KK-transformation of the logarithmic gaussian, Eq. (E.6), f(x) (for b = 0.50)for the case of pn > 0 and pn < 0. g(x), defined in Eq. (E.4), is shown for both cases aswell (with δ′ = 0.001). In contrast to f(x), its KK-transformed, g(x), is finite for positiveand negative energies.

122 E. Kramers-Kronig (KK) relation

Appendix F

Kubo formalism

The linear conductance through an interacting region can be calculated for an arbitraryimpurity contacted to a left and a right lead within the Kubo formalism [6]. In contrast tothe formula developed by Meir and Wingreen [80], where the conductance G through aninteracting region is related to =[GR], the condition of proportional coupling (ΓL ∝ ΓR) isnot required for the Kubo approach. In particular, in multi-level QDs with a relative signin the tunneling matrix elements between the different levels, the condition of proportionalcoupling is not met, see e.g. the s = −1 case in [27].

Since the conductance is calculated in a nonlocal way within the Kubo formalism,one really has to perform a ’two-channel’ calculation (which means that the system sizeincreases by a factor of 16 in each step of the NRG-iteration) when one uses this approach.

The Kubo formalism allows one to relate the linear conductance with the current-current correlator, see Izumida et al. [6]. Obviously the operator K∗ = ˆNL − ˆNR =i~ [H, NL − NR], with NL =

∑kσ c†kσLckσL and NR =

∑kσ c†kσRckσR [with Fermi operators

ckσL (ckσR) of the left (right) lead], is of particular interest within this formalism. Since onlythe tunneling part of H does not commute with NL−NR, one obtains K∗ = i

~ [H`d, NL−NR].As shown in Section 3.1, it is often convenient to work in the symmetric/antisymmetricbasis described by the Fermi operators αskσ and αakσ [as defined in Eq. (3.6)]. In this basisK∗ has the form

NL − NR =∑

k,σ

(c†kσLckσL − c†kσRckσR) =∑

kσ

(α†akσαskσ + α†skσαakσ). (F.1)

We illustrate this procedure by looking at the case of a two level, two channel problemwith a relative minus sign in one tunneling matrix element, as studied in Ref. [27]. Inthe symmetric/antisymmetric basis H`d takes the form H`d =

∑k,σ V1(d

†1σαskσ + h.c.) +

V2(d†2σαakσ + h.c.) (diσ are the Fermi operators of the impurity level i, i = {1, 2}, and

Vi =√

2Vi are the corresponding tunneling matrix elements), which finally leads to

ˆNL − ˆNR =i

~∑

kσ

[V1(d†1σαakσ − α†akσd1σ) + V2(d

†2σαskσ − α†skσd2σ)]. (F.2)

124 F. Kubo formalism

Note, even though the operator ˆNL − ˆNR is purely imaginary, we can use real matrices tocompute Eq. (F.2).

F.1 Derivation of the Kubo formula

When a small bias voltage is applied across a QD, a current is driven through the system.Here we compute the linear conductance G through an interacting QD which is connected toleads of chemical potentials µL = eV/2 and µR = −eV/2, respectively (shown in Fig. F.1).Consequently the system consists of an equilibrium part (the equilibrium Anderson model

��

��

��

��

eV/2

−eV/2

L R

DOT

Figure F.1: The leads connected to the QD have different chemical potentials which drivesa current through the QD. An overall potential difference of magnitude V is applied on theQD. Note that V has to be sufficiently small to justify the use of linear response theory.

Heq) and an additional perturbation, Hsys = Heq + H′, H′ = −NLµL − NRµR. Since we

consider the perturbation H′ to be small, we use first order perturbation theory in H ′

(linear response theory) to compute G. The introduction of a linear response tensor σ,〈Nα〉 =

∑β σαβµβ (with α/β = L/R), allows us to write G(ω) as

G(ω) =〈I(ω)〉

V=

e

V

〈NL〉 − 〈NR〉2

=e2

4[σLL(ω) + σRR(ω)− σLR(ω)− σRL(ω)] . (F.3)

To compute σαβ, we first compute 〈Nα(t)〉 for arbitrary times (see [40], Appendix B)1 andthen take the expectation value 〈Nα(t = 0)〉. The steady state conductance is thus relatedto

〈Nα(t = 0)〉 =1

i~tr

{∫ t=0

−∞[−Nβ, ρeq]Nα(−t′)µβdt′

}=

∫ t=0

−∞σαβ(0− t′)µβ, (F.4)

1 In linear response theory, the change ∆A(t) of an observable A due to a (time dependent) perturbationV (t) = BF (t) is given as 〈∆A(t)〉 = 1

i~ tr∫ t

−∞[B, ρeq]A(t − t′)F (t′)dt′. Here we are interested in a time-independent perturbation, an applied bias voltage.

F.1 Derivation of the Kubo formula 125

which implies

σαβ(t′) =1

i~tr

{[−Nβ, ρeq]Nα(t′)

}=−1

i~tr

{[Nβ, ρeq]e

i~ Heqt′Nαe−

i~ Heqt′

}. (F.5)

Here ρeq = 1Ze−βHeq denotes the density matrix of the unperturbed Hamiltonian. The cyclic

property of the trace allows us to write the frequency-dependent conductivity σαβ(ω), theFourier transformed of Eq. (F.5), in the following way

σαβ(ω) = limδ→0+

i

~tr

∫ ∞

0

eiωt−δt e−i~ Heqt[Nβ, ρeq]e

i~ HeqtNα︸︷︷︸

χ(t)

dt

. (F.6)

To simplify notation we will skip the limδ→0+ henceforth. Integrating Eq. (F.6) by partsyields

σαβ(ω) =i

~

{eiωt−δt

iω − δtr(χ(t))|∞0 −

∫ ∞

0

dteiωt−δt

iω − δtr

[(−i

~Heq)e

− i~ Heqt[Nβ, ρeq]e

i~ HeqtNα

+e−i~ Heqt[Nν , ρeq](

i

~Heq)e

i~ HeqtNα

]}

=i

~

{ −1

iω − δtr([Nν , ρeq]Nα)− 1

iω − δ

∫ ∞

0

dteiωt−δt

[tr

((−i

~Heq)[Nβ, ρeq]Nβ(t)

)

+tr

(Nα(t)[Nβ, ρeq](

i

~Heq)

)]}. (F.7)

We rewrite the terms tr{(−i~ Heq)[Nβ, ρeq]Nα(t)} = − i

~tr{ρeqNβ(t)HeqNβ−ρeqHeqNβNα(t)}and tr{Nα(t)[Nβ, ρeq](

i~Heq)} = i

~tr{ρeqNα(t)NβHeq − ρeqNβHeqNβ(t)} in Eq. (F.7) andfinally obtain the frequency-dependent linear conductivity

σαβ(ω) =i

~

{ −1

iω − δtr

([Nβ, ρeq]Nα

)

− 1

iω − δ

∫ ∞

0

dteiωt−δttr

ρeq

Nα︷︸︸︷i

~[Heq, Nβ] Nα(t)− Nα(t)

Nβ︷︸︸︷i

~[Heq, Nβ]

=i

~

[ −1

iω − δtr

([Nβ, ρeq]Nα

)− 1

iω − δ

∫ ∞

0

dteiωt−δt⟨[Nβ, Nα(t)]

⟩eq

]. (F.8)

Analogously to [6] we define Kαβ(ω) ≡ −i~

∫∞0

dteiωt−δt⟨[Nβ(0), Nα(t)]

⟩eq

which enables us

to simplify Eq. (F.8)

σαβ(ω) =1

iω{Kαβ(ω)−Kαβ(0)} , (F.9)


with Kαβ(0) = i~tr

{[Nβ, ρeq]Nα

}. Note that the complex causal function Kαβ(ω) =

K ′αβ(ω)+ iK ′′

αβ(ω) is obtained by averaging over the unperturbed system. When one insertsa complete set of states, it is straightforward to perform the integration2 that is necessaryto obtain Kαβ(ω)

Kαβ(ω) =−i

~

∫ ∞

0

dteiωt−δt∑n,m

(〈n| ρeqNβ |m〉〈m| e i

~ HeqtNαe−i~ Heqt |n〉 (F.10)

− 〈m| ρeqei~ HeqtNαe

−i~ Heqt |n〉〈n| Nβ |m〉

)

=−i

~∑n,m

∫ ∞

0

dteiωt−δtei~ (Em−En)t

[1

Z

(e−βEn − e−βEm

)] 〈n| Nβ |m〉〈m| Nα |n〉

=−i

~1

Z

∑n,m


) 〈n| Nβ |m〉〈m| Nα |n〉∫ ∞

0

dtei~ (~ω−(En−Em))t−δt

=1

Z

∑n,m


) (Em − En + ~ω)− i~δ(Em − En + ~ω)2 + (~δ)2

〈n| Nβ |m〉〈m| Nα |n〉 .

Eq. (F.10) yields K ′αβ(ω) =

∑n,m [ e−βEn−e−βEm

Z] 〈n| Nβ |m〉〈m| Nα |n〉 ~ω−(En−Em)

(~ω−(En−Em))2+(~δ)2

and K ′′αβ(ω) =

∑n,m

[e−βEn−e−βEm

Z

]〈n| Nβ |m〉〈m| Nα |n〉 −~δ

(~ω−(En−Em))2+(~δ)2 (with δ → 0+)

resulting in3

K ′αβ(ω) = −

∑n,m

[e−βεm − e−βεn

Z

]〈n| Nβ |m〉〈m| Nα |n〉 1

~ω − (εn − εm)(F.11)

K ′′αβ(ω) = π

∑n,m

[e−βεm − e−βεn

Z

]〈n| Nβ |m〉〈m| Nα |n〉 δ(~ω − (εn − εm)). (F.12)

Obviously the functions K ′αβ(ω) and K ′′

αβ(ω) have the following symmetries: K ′αβ(ω) =

K ′αβ(−ω) and K ′′

αβ(ω) = −K ′′αβ(−ω)4 implying

K ′αβ(0) 6= 0 (F.13)

K ′′αβ(0) = 0. (F.14)

Even though we can only compute K ′′αβ(ω), but not K ′

αβ(ω) within the framework of NRG,we can access K ′

αβ(ω) by performing a KK-transformation (see Appendix E) of K ′′αβ(ω) [as

2∫∞0

dtei~ (~ω−(En−Em))t−δt = ~ i(Em−En+~ω)+δ~

(Em−En+~ω)2+(~δ)23 πδ(α) = lima→0

aα2+a2

4For T = 0 an integration of K ′′αβ(ω) results in:

∫∞−∞ dωK ′′

αβ(ω) = π 〈0| [Nα, Nβ ] |0〉, with the ground

state of the unperturbed system |0〉. Note that[Nα, Nβ

]= 0 (trivially for α = β; in the case α 6= β the

conservation of charge, N = 0, with N = NL + NR, yields Nα = −Nβ confirming the assertion). Theaccuracy of the numerics can be checked, since

∫G′′(ω)dω should give an exact zero; the numerical value

we obtain is ∼ 10−4, i.e. it is in excellent agreement with this sum-rule.

F.1 Derivation of the Kubo formula 127

Kαβ(ω) is a causal function]

K ′αβ(ω) =

1

πP

∫ ∞

−∞

K ′′αβ(ω′)

ω′ − ωdω′, (F.15)

K ′′αβ(ω) = − 1

πP

∫ ∞

−∞

K ′αβ(ω′)

ω′ − ωdω′, (F.16)

with the principal value P of the integral. Finally Eq. (F.10) simplifies to

σαβ(ω) =1

iω

(K ′

αβ(ω)−K ′αβ(0) + iK ′′

αβ(ω))

(F.17)

with its real-part σ′αβ(ω) (directly obtained from NRG)

σ′αβ(ω) =K ′′

αβ(ω)

ω(F.18)

and its imaginary part [obtained via a KK-transformation of K ′′αβ(ω)]

σ′′αβ(ω) = −K ′αβ(ω)−K ′

αβ(0)

ω. (F.19)

Thus, the frequency dependent conductance [7] is, substituting (F.9) in (F.3), G(ω) ∼K(ω)−K(0)

ω, where K(ω) = KLL(ω) + KRR(ω) − KLR(ω) − KRL(ω). G′(ω) can thus be

written as

G′(ω)2e2

h

=π

4

π∑

n,m

[e−βEm−e−βEn

Z

] ∣∣∣〈n|(−K

)|m〉

∣∣∣2

δ(~ω − (En − Em))

~ω

. (F.20)

The definition

W(ω) ≡ π∑n,m

[e−βEm − e−βEn

Z

] ∣∣∣〈n|(−K

)|m〉

∣∣∣2

δ(~ω − (En − Em)) (F.21)

allows us to write G′′(ω) as the KK-transformed of G′(ω) [see Eq. (F.19)],

G′′(ω)2e2

h

= −π

4

{[W(ω)]KK − [W(0)]KK

~ω

}. (F.22)

The operator K = ~iK∗, used in Eqs. (F.20) and (F.22), was defined in the introduction of

this Appendix (remember K∗ ∼ [H`d, NL − NR]). Phase and absolute value of the linearconductance, φ and |G(ω)|, are given as

φ = arctan

(− [W(ω)]KK − [W(0)]KK

W(ω)

)(F.23)

|G(ω)|2e2

h

=π/4

|~ω|

√[W(ω)]2 +

{[W(ω)]KK − [W(0)]KK

}2

. (F.24)


The (numerical) strategy we use is the following: after we implement the operator K, whichallows us to determine W(ω) [see Eq. (F.20)], we take its KK-transformed, [W(ω)]KK ,which determines G′′(ω), see Eq. (F.22).

When one is only interested in the DC conductance G the delicate limit G = limω→0 G(ω)has to be computed. Due to the properties of the real and the imaginary part of Kαβ [cf.Eqs. (F.13) and (F.14); K ′

αβ is an even whereas K ′′αβ is an odd function] the DC conductance

is purely real.

Part IV

Miscellaneous

130

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[74] R. Bulla. The Numerical Renormalization Group Method for correlated electrons. Adv.Solid State Phys. 40, 169–182 (2000).

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List of Publications

The results presented in this thesis have been published in the following papers:

1. Kondo Effect in the Presence of Itinerant-Electron Ferromagnetism Studied with theNumerical Renormalization Group MethodJ. Martinek, M. Sindel, L. Borda, J. Barnas, J. Konig, G. Schon, and J. von DelftPhys. Rev. Lett. 91, 247202 (2003)

2. Gate-controlled spin-splitting in quantum dots with ferromagnetic leads in the KondoregimeJ. Martinek, M. Sindel, L. Borda, J. Barnas, R. Bulla, J. Konig, G. Schon, S.Maekawa, and J. von Delftsubmitted to Phys. Rev. Lett., cond-mat/0406323

3. Charge oscillations in Quantum DotsM. Sindel, A. Silva, Y. Oreg, and J. von Delftsubmitted to Phys. Rev. Lett., cond-mat/0408096

4. Frequency-dependent transport through a quantum dot in the Kondo regimeM. Sindel, W. Hofstetter, J. von Delft, and M. Kindermannsubmitted to Phys. Rev. Lett., cond-mat/0406312

5. Anderson-Excitons: the effect of a Fermi sea on emission and absorption spectraR.W. Helmes, M. Sindel, L. Borda, and J. von Delftto be submitted to Phys. Rev. B

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Acknowledgements

I am grateful to my adviser, Prof. Jan von Delft, for providing guidance and directionfor my research over the last three years. His critical questions were crucial to gain adeeper understanding of the physics behind the different problems. Thanks for teachingme how to give a good talk and how to write scientific publications. I am very grateful forthe possibility of going to conferences and schools, to work with varying collaborators ondifferent projects and the open atmosphere at our chair.

In particular I want to thank Prof. Walter Hofstetter for co-refereeing this thesis andthe possibility of working on common projects with him.

Deep gratitude goes to my friend Laszlo Borda who tolerated me as an office mate andtaught me the powerful numerical technique I used in my thesis.Without the support, help and ideas of my excellent collaborators (and friends), I foundduring the last years, the thesis wouldn’t have been possible. Thanks to Ralf Bulla, TheoCosti, Silvano De Franceschi, Yuval Gefen, Markus Kindermann, Jurgen Konig, JohannKroha, Florian Marquardt, Jan Martinek, Yuval Oreg, Achim Rosch, Alessandro Silva andGergely Zarand.Moreover I would like to thank Udo Hartmann, Theresa Hecht, Rolf Helmes and MarkusStorcz for critical reading of the thesis.

Thanks to all members of our group during the last years: Andreas Friedrich, RongLu, Frank Wilhelm, Ulrich Zulicke, Uli Schollwock, Alexander Khaetskii, Corinna Kollath,Dominique Gobert, Robert Dahlke, Henryk Gutmann, Hamed Saberi, Andreas Weich-selbaum, Michael Mockel, Johannes Ferber, Konstanze Jahne, Igor Gazuz and StephaneSchoonover with whom I spent a great time in Munich.

I would like to thank my family, in particular my parents, for their continuous supportover all the years. Special thanks goes to my girl friend Angelika who kept supporting me.Thanks!

Financial support through the DFG Program “Festkorperbasierte Quanteninforma-tionsverarbeitung: Physikalische Konzepte und Materialaspekte” (SFB 631) is acknowl-edged.

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Deutsche Zusammenfassung

Die vorliegende Arbeit umfasst eine Reihe theoretischer Studien, die sich mit verschiedenenAspekten von Quantenpunkten beschaftigen.Kleine Quantenpunkte mit grossen Niveauabstanden lassen sich sehr gut mit Hilfe desnach P.W. Anderson benannten Storstellenmodelles beschreiben. Da wir in der vorliegen-den Arbeit in erster Linie dieses Modell benutzen, erwarten wir, dass unsere Vorhersagenexperimentell nachweisbar sind.5 Es sei hier bemerkt, dass sich alle Parameter der obenbeschriebenen Quantenpunkte durch externe Gatterspannungen experimentell einstellenlassen.In besonderer Weise interessieren wir uns fur den Temperaturbereich, unterhalb dessenein lokaler Elektronenspin (im Quantenpunkt) stark mit den ihn umgebenden Leitungse-lektronen (in den Zuleitungen) wechselwirkt. Dieser Temperaturbereich wird durch diesogenannte Kondo Temperatur TK festgelegt (d.h. T < TK). Zur exakten Berechnungverschiedener Transporteigenschaften von Quantenpunkten fur Temperaturen T < TK be-nutzen wir die von Wilson entwickelte numerische Renormierungsgruppenmethode [1].

Der vorliegenden Dissertation stellen wir eine allgemeine Einleitung zur Physik vonQuantenpunkten, mit besonderer Betonung auf den Kondo Effekt in Quantenpunkten,voran. Daruber hinaus beinhaltet der erste Teil eine Einfuhrung in die von uns verwendetenumerische Renormierungsgruppenmethode.

Der zweite Teil dieser Arbeit, der Hauptteil, ist in die folgenden verschiedenen Studienaufgeteilt:(i) Wir analysieren die Eigenschaften eines ’Kondo’ Quantenpunktes, der an spinpolar-isierte Zuleitungen gekoppelt ist. Es stellt sich heraus, dass die Spin-Polarisierung derZuleitungen zu einer Aufspaltung und einer Unterdruckung der Kondo Resonanz fuhrt.Wir erweitern unsere Studien auf den Fall eines Quantenpunktes, der an Zuleitungen miteiner beliebigen Zustandsdichte koppelt. Fur diesen Fall untersuchen wir die Gatterspan-nungsabhangigkeit der Kondo Resonanz.(ii) Wir untersuchen das Fullungsverhalten eines spinlosen Anderson Modells das zweilokale Niveaus besitzt als Funktion der Gatterspannung. Wir identifizeren Parameterbere-iche, in denen sich die beiden betrachteten Niveaus nicht monoton fullen, wenn sie rela-tiv zur Fermi-Energie der Zuleitungen abgesenkt werden. Fur asymmetrisch gekoppelteNiveaus finden wir sogar eine Besetzungsinversion, d.h. fur einen bestimmten Gatterspan-nungsbereich ist das energetisch hoher liegende Niveau starker besetzt als das energetisch

5Die theoretische Studie in Kapitel 5.1 wurde bereits experimentell nachgewiesen, siehe [55].

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niedriger liegende Niveau. Das gefundene Verhalten erklaren wir mit Hilfe eines selbstkon-sistenten Hartree Ansatzes.(iii) Wir berechnen den frequenzabhangigen Leitwert eines Quantenpunktes im KondoRegime unter Benutzung der Kubo Formel. Wir leiten eine analytische Formel ab, die eineBeziehung zwischen dem frequenzabhangigen Leitwert und der (lokalen) Gleichgewichts-Spektralfunktion herstellt. Das Fluktuations-Dissipations Theorem gestattet es uns, eineBeziehung zwischen Stromrauschen und der (lokalen) Gleichgewichts-Spektralfunktion auf-zustellen.(iv) Emissionsexperimente in selbstorganisierten Quantenpunkten [2] motivierten uns, op-tische Ubergange zwischen ’Kondo’ und ’Nicht-Kondo’ Zustanden zu untersuchen. Zudiesem Zwecke verallgemeinern wir das ublicherweise verwendete Anderson Modell. Wirfinden, dass sich sowohl das Emissions- als auch das Absorptionsspektrum durch eineAnalogie mit dem ’X-ray edge’ Problem erklaren lassen.

Der dritte Teil dieser Arbeit enthalt Herleitungen, die fur das Verstandis der vorliegen-den Arbeit von grosser Bedeutung sind.Der vierte Teil dieser Dissertation enthalt sonstige Informationen.

Curriculum vitae

Name: Michael Sindel

born: June 26th, 1974 in Rothenburg ob der Tauber

School: 1980 - 1984: Primary school in Schillingsfurst

1984 - 1993: Reichsstadt-Gymnasium, Rothenburg o.d.T.

06/1993: Abitur

Studies: 10/1994 - 06/1998: Study of Lehramt fur Realschule (subjects: mathematics

and physics) at the University of Wurzburg

06/1998: 1. Staatsexamen fur das Lehramt an Realschulen

04/1998 - 08/2000: Study of physics at the University of Wurzburg

10/1999: Vordiplom in physics (University of Wurzburg)

09/2000 - 10/2001: Study of physics at Rutgers,

The State University of New Jersey, New Brunswick

10/2001: Master of Science (Rutgers, State University of New Jersey)

10/2001 - 12/2004: Ph.D. studies in Theoretical Condensed Matter Physics,

supervised by Prof. Dr. Jan von Delft,

Ludwig-Maximilians-University, Munchen

Theses: “Vertauschung von Grenzprozessen von Funktionen einer Veranderlichen”

supervised by Dr. L. Neckermann at Mathematisches Institut,

University of Wurzburg (Zulassungsarbeit 1998)

“Two Patch Model for Transport Properties of Optimally and Overdoped Cuprates”

supervised by Prof. Dr. G. Kotliar at the Center for Materials Theory,

Rutgers, State University of New Jersey (Master Thesis 2001)

Documents

Numerical Renormalization Group studies of Quantum Impurity Models