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The functional renormalization group for zero-dimensional quantum systems in and out of equilibrium [Elektronische Ressource] / vorgelegt von Christoph Karrasch

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219 pages
The Functional Renormalization Groupfor Zero-Dimensional Quantum Systemsin and out of EquilibriumVon der Fakult¨at fu¨r Mathematik, Informatik und Naturwissenschaftender RWTH Aachen University zur Erlangung des akademischen Gradeseines Doktors der Naturwissenschaften genehmigte Dissertationvorgelegt vonDipl.-Phys. Christoph Karraschaus DuderstadtBerichter: Univ.-Prof. Dr. Volker MedenUniv.-Prof. Dr. Herbert SchoellerTag der mu¨ndlichen Pru¨fung: 2. Juli 2010Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfu¨gbar.Contact Information:Christoph KarraschInstitut fu¨r Theoretische Physik APhysikzentrum, RWTH Aachen52056 Aachenphone: +49 241 8027032email: karrasch (at) physik.rwth-aachen.dehomepage: http://www.theorie.physik.uni-goettingen.de/∼karraschBerichter: Volker MedenHerbert SchoellerTag der mu¨ndlichen Pru¨fung: 2. Juli 2010PrefaceIt is more than a mere obligation to express my gratitude to numerous colleagues for exchangingideas and for various resulting collaborations. This work benefited greatly from all of these.First and most of all, I am indebted to Volker Meden for giving me the opportunity to do my Ph.D.studies under his supervision. To (once more) cut a long story short: I could not have imagined abetter tutor.It were certainly Kurt Sch¨onhammer’s enlightening lectures at the G¨ottingen University which raisedmy interest for condensed matter physics in the first place.
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The Functional Renormalization Group
for Zero-Dimensional Quantum Systems
in and out of Equilibrium
Von der Fakult¨at fu¨r Mathematik, Informatik und Naturwissenschaften
der RWTH Aachen University zur Erlangung des akademischen Grades
eines Doktors der Naturwissenschaften genehmigte Dissertation
vorgelegt von
Dipl.-Phys. Christoph Karrasch
aus Duderstadt
Berichter: Univ.-Prof. Dr. Volker Meden
Univ.-Prof. Dr. Herbert Schoeller
Tag der mu¨ndlichen Pru¨fung: 2. Juli 2010
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfu¨gbar.Contact Information:
Christoph Karrasch
Institut fu¨r Theoretische Physik A
Physikzentrum, RWTH Aachen
52056 Aachen
phone: +49 241 8027032
email: karrasch (at) physik.rwth-aachen.de
homepage: http://www.theorie.physik.uni-goettingen.de/∼karrasch
Berichter: Volker Meden
Herbert Schoeller
Tag der mu¨ndlichen Pru¨fung: 2. Juli 2010Preface
It is more than a mere obligation to express my gratitude to numerous colleagues for exchanging
ideas and for various resulting collaborations. This work benefited greatly from all of these.
First and most of all, I am indebted to Volker Meden for giving me the opportunity to do my Ph.D.
studies under his supervision. To (once more) cut a long story short: I could not have imagined a
better tutor.
It were certainly Kurt Sch¨onhammer’s enlightening lectures at the G¨ottingen University which raised
my interest for condensed matter physics in the first place. Years afterwards, they led to this Thesis,
and I appreciate that Herbert Schoeller is refereeing it.
Our studies of the quantum dot Josephson problem were triggered by discussions with Tom´aˇs
Novotn´y and Jens Paaske when Volker was visiting the Nano-Science Center of the University of
Copenhagen. Part of this work was a collaboration with our experimental colleagues – let me espe-
cially mention H´el`ene Bouchiat and Alexander Eichler – from Basel and Paris. I am glad for having
had this opportunity.
Slava Kashcheyevs, currently affiliated to the University of Riga, proposed to contemplate the charg-
ing of a narrow quantum dot level, and Avi Schiller from the Hebrew University of Jerusalem came
up with a bosonisation treatment. Among many other things, it was always fascinating to listen to
Slava giving talks about this project!
I would like to thank Kurt Sch¨onhammer as well as the Aachen people Sabine Andergassen, Severin
Jakobs, Mikhail Pletyukhov, Herbert Schoeller, and Dirk Schuricht for fruitful discussions about
countless issues. In particular, it was a great pleasure to carry out joint investigations of the inter-
acting resonant level model out of equilibrium.
Robert Peters and Thomas Pruschke from G¨ottingen, Akira Oguri from Osaka, and Theresa Hecht
as well as Andreas Weichselbaum from Munich took care of a variety of numerical renormalization
group calculations. Thank you!
I am grateful to the Federal Republic of Germany for generously financing this work. In particular,
support from the Deutsche Forschungsgemeinschaft (via FOR 723) made it possible to participate
in various conferences all over the world.
3Contents
Preface 3
Introduction 9
1 Rather Short Summary 11
2 Introduction 13
2.1 The Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 FRG for Quantum Impurity Systems: Status Report . . . . . . . . . . . . . . . 13
2.3 Fundamental Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 Single Impurity Anderson Model . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Interacting Resonant Level Model . . . . . . . . . . . . . . . . . . . . . 15
2.3.3 Quantum Dot Josephson Junction . . . . . . . . . . . . . . . . . . . . . 17
2.3.4 Charging of a Narrow Level . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 The Model 23
3.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Something about Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
The Method 27
4 Green Functions 29
4.1 Matsubara Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1.3 What about Convergence Factors? . . . . . . . . . . . . . . . . . . . . 32
4.1.4 Diagrammatics – Dyson Equation . . . . . . . . . . . . . . . . . . . . . 33
4.1.5 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1.6 Zero-Temperature Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Keldysh Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.2 The Steady-State Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.3 What about Initial Correlations? . . . . . . . . . . . . . . . . . . . . . . 36
4.2.4 Diagrammatics – Dyson Equation . . . . . . . . . . . . . . . . . . . . . 37
4.3 Generating Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3.1 Functional Integrals in Many-Particle Physics . . . . . . . . . . . . . . . 39
4.3.2 Matsubara Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3.3 Keldysh Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.4 Generating Functionals of Green Functions . . . . . . . . . . . . . . . . 43
4.4 Non-Interacting Green Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 46
56 Contents
4.4.1 Green Functions of an Isolated System . . . . . . . . . . . . . . . . . . 46
4.4.2 Finite-Matrix Dyson Equation . . . . . . . . . . . . . . . . . . . . . . . 47
4.4.3 Local Green Function of the Isolated Leads . . . . . . . . . . . . . . . . 48
4.5 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5.1 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5.2 Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5.3 Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.5.4 Average Occupation Numbers . . . . . . . . . . . . . . . . . . . . . . . 53
5 Functional Renormalization Group 55
5.1 Flow Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1.1 Flow Equations of Connected Green Functions . . . . . . . . . . . . . . 57
5.1.2 Flow Equations of Vertex Functions . . . . . . . . . . . . . . . . . . . . 58
5.1.3 The Steady-State Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.4 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1.5 Truncation Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Choice of a Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.1 Matsubara Formalism at Finite Temperature . . . . . . . . . . . . . . . 63
5.2.2 Sharp Cutoff Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2.3 Reservoir Cutoff Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.4 Again: Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 Conservation of Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.1 Some Trivialities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.2 Complex Conjugation & Time Reversal . . . . . . . . . . . . . . . . . . 69
5.3.3 Causality & DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6 Alternative Approaches 73
6.1 Connection to Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2 Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.3 Bosonisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3.2 Bose Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3.3 An Exact Solution to the Many-Particle Problem . . . . . . . . . . . . . 77
6.3.4 Bosonisation of the Field Operator . . . . . . . . . . . . . . . . . . . . 77
6.3.5 Real-Space Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.4 Numerical Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . 80
Quantum Impurity Systems 81
7 Single Impurity Anderson Model 83
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.3 Application of the FRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.3.1 Parametrisation of the Two-Particle Vertex . . . . . . . . . . . . . . . . 86
7.3.2 Second-Order Flow Equations . . . . . . . . . . . . . . . . . . . . . . . 86
7.3.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.3.4 Choice of a Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.3.5 Static Functional RG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.3.6 Restricted Hartree-Fock & Perturbation Theory . . . . . . . . . . . . . . 90
7.4 Numerical Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.4.1 Discretisation of the Frequency Axis . . . . . . . . . . . . . . . . . . . . 91
7.4.2 Parametrisation of the Two-Particle Vertex . . . . . . . . . . . . . . . . 92
7.4.3 Modification of the Vertex Flow Equation . . . . . . . . . . . . . . . . . 94
7.4.4 Further Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 94Contents 7
7.4.5 Numerical Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.5 Spectral Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.5.1 Pad´e Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.5.2 Spectral Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.5.3 Spectral Weight at Zero Frequency and Average Occupation . . . . . . . 98
7.6 The Kondo Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.6.1 Effective Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.6.2 Static Spin Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.7 Linear Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.7.1 Fermi Function in Terms of Continued Fractions . . . . . . . . . . . . . 102
7.7.2 Finite-Temperature Conductance . . . . . . . . . . . . . . . . . . . . . 104
7.8 Reservoir Cutoff Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.9 Conclusions & Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8 Interacting Resonant Level Model 109
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
8.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.3 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.3.1 Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.3.2 Flow Equations, Matsubara Formalism . . . . . . . . . . . . . . . . . . 113
8.3.3 Flow Equations, Keldysh Formalism . . . . . . . . . . . . . . . . . . . . 115
8.4 Comparison with DMRG and RTRG . . . . . . . . . . . . . . . . . . . . . . . . 118
8.4.1 Steady-State Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.4.2 Conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.5 Scaling Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.5.1 Analysis of the RG Equations – Linear Response . . . . . . . . . . . . . 122
8.5.2 Analysis of the RG Equations – Non-Equilibrium . . . . . . . . . . . . . 124
8.5.3 Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.5.4 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.6 Second-Order Functional RG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.7 Conclusion & Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9 Quantum Dot Josephson Junction 135
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
9.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.3 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
9.3.1 Nambu Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9.3.2 Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.3.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.3.4 Functional Renormalization Group . . . . . . . . . . . . . . . . . . . . . 144
9.3.5 Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.3.6 The Josephson Current . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.3.7 Atomic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
9.3.8 The Non-Interacting Case . . . . . . . . . . . . . . . . . . . . . . . . . 159
9.4 Ordinary Quantum Dot Josephon Junction . . . . . . . . . . . . . . . . . . . . 160
9.4.1 Integrating the FRG Flow Equations . . . . . . . . . . . . . . . . . . . . 161
9.4.2 Atomic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
9.4.3 Comparison with NRG . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
9.4.4 Gate-Voltage Dependence: Connection to the Experiment . . . . . . . . 166
9.4.5 Comparison with Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . 167
9.4.6 Alternative FRG Schemes . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.4.7 Finite Temperatures: QMC vs. NRG . . . . . . . . . . . . . . . . . . . . 170
9.5 Aharonov-Bohm Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
9.5.1 Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1728 Contents
9.5.2 Josephson Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
9.6 Attractive Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.7 Conclusions & Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
10 Charging of a Narrow Level 181
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
10.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
10.3 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
10.3.1 Transformation to a Generalised IRLM . . . . . . . . . . . . . . . . . . 184
10.3.2 Bosonisation and Mapping to the Kondo Problem . . . . . . . . . . . . 186
10.3.3 Functional RG Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 188
10.4 Results: Inverse Charge Fluctuation Time . . . . . . . . . . . . . . . . . . . . . 189
10.4.1 Analytic Description: Power-Law Scaling . . . . . . . . . . . . . . . . . 189
10.4.2 Comparison with Numerics . . . . . . . . . . . . . . . . . . . . . . . . . 190
10.5 Conclusions & Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Conclusions & Outlook 195
11 Conclusions & Outlook 197
11.1 FRG for Quantum Impurity Systems: Status Report . . . . . . . . . . . . . . . 197
11.2 Summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
11.2.1 Single Impurity Anderson Model . . . . . . . . . . . . . . . . . . . . . . 197
11.2.2 Interacting Resonant Level Model . . . . . . . . . . . . . . . . . . . . . 198
11.2.3 Quantum Dot Josephson Junction . . . . . . . . . . . . . . . . . . . . . 199
11.2.4 Charging of a Narrow Level . . . . . . . . . . . . . . . . . . . . . . . . 200
11.3 Prospects for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Supplements 203
Bibliography 205
List of Publications 215
Curriculum Vitae 217
Thank You! 219Introduction
9