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

Christoph Karrasch

Sujets

• Physik

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2010
• Nombre de lectures: 15
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

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: [phone]
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