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Aspects of strong correlations in low dimensions [Elektronische Ressource] / von Florian Schütz

178 pages
Aspects of strong correlationsin low dimensionsDissertationzu Erlangung des Doktorgradesder Naturwissenschaftenvorgelegt beim Fachbereich Physikder Johann Wolfgang Goethe-Universit¨atin Frankfurt am MainvonFlorian Schu¨tzaus G¨ottingenFrankfurt (2005)(D F 1)vom Fachbereich Physik derJohann Wolfgang Goethe-Universit¨atals Dissertation angenommen.Dekan: Prof. Dr. W. AßmusGutachter: Prof. Dr. P. Kopietz,Prof. Dr. M.-R. ValentiDatum derDisputation: 7. September, 2005Abstract of the thesisThe challenging intricacies of strongly correlated electronic systems necessitatethe use of a variety of complementary theoretical approaches. In this thesis, weanalyze two distinct aspects of strong correlations and develop further or adaptsuitable techniques.First, we discuss magnetization transport in insulating one-dimensional spinrings described by a Heisenberg model in an inhomogeneous magnetic field. Dueto quantum mechanical interference of magnon wave functions, persistent mag-netization currents areshown to exist insuch a geometry in analogyto persistentcharge currents in mesoscopic normal metal rings.The second, longer part is dedicated to a new aspect of the functional renor-malization group technique for fermions. By decoupling the interaction via aHubbard-Stratonovich transformation, we introduce collective bosonic variablesfrom the beginning and analyze the hierarchy of flow equations for the coupledfield theory.
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Aspects of strong correlations
in low dimensions
Dissertation
zu Erlangung des Doktorgrades
der Naturwissenschaften
vorgelegt beim Fachbereich Physik
der Johann Wolfgang Goethe-Universit¨at
in Frankfurt am Main
von
Florian Schu¨tz
aus G¨ottingen
Frankfurt (2005)
(D F 1)vom Fachbereich Physik der
Johann Wolfgang Goethe-Universit¨at
als Dissertation angenommen.
Dekan: Prof. Dr. W. Aßmus
Gutachter: Prof. Dr. P. Kopietz,
Prof. Dr. M.-R. Valenti
Datum der
Disputation: 7. September, 2005Abstract of the thesis
The challenging intricacies of strongly correlated electronic systems necessitate
the use of a variety of complementary theoretical approaches. In this thesis, we
analyze two distinct aspects of strong correlations and develop further or adapt
suitable techniques.
First, we discuss magnetization transport in insulating one-dimensional spin
rings described by a Heisenberg model in an inhomogeneous magnetic field. Due
to quantum mechanical interference of magnon wave functions, persistent mag-
netization currents areshown to exist insuch a geometry in analogyto persistent
charge currents in mesoscopic normal metal rings.
The second, longer part is dedicated to a new aspect of the functional renor-
malization group technique for fermions. By decoupling the interaction via a
Hubbard-Stratonovich transformation, we introduce collective bosonic variables
from the beginning and analyze the hierarchy of flow equations for the coupled
field theory. The possibility of a cutoff in the momentum transfer of the in-
teraction leads to a new flow scheme, which we will refer to as the interaction
cutoffscheme. Withinthisapproach,Wardidentitiesforforwardscatteringprob-
lems are conserved at every instant of the flow leading to an exact solution of
a whole hierarchy of flow equations. This way the known exact result for the
single-particle Green’s function of the Tomonaga-Luttinger model is recovered.
- v -Contents
Abstract of the thesis v
1 Overview 1
1.1 Persistent spin currents . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Functional renormalization group with collective fields . . . . . . 2
I Persistent spin currents 3
2 Introduction 5
2.1 Persistent currents in normal metal rings . . . . . . . . . . . . . . 6
2.2 Analogue in mesoscopic spin rings . . . . . . . . . . . . . . . . . . 9
3 Spin-wave theory 13
3.1 Associated classical problem . . . . . . . . . . . . . . . . . . . . . 14
3.2 Semi-classical expansion . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Linear spin-wave theory . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Spin currents in inhomogeneous magnetic fields 27
4.1 Naive definition of the spin current operator . . . . . . . . . . . . 28
4.2 Effective spin currents with correct classical limit . . . . . . . . . 30
4.3 New definition of the spin current operator . . . . . . . . . . . . . 31
4.4 Electrodynamics of spin currents . . . . . . . . . . . . . . . . . . 33
5 Persistent spin currents in ferromagnetic Heisenberg rings 37
5.1 Classical ground state . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Spin-wave spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Parallel transport and geometric flux . . . . . . . . . . . . . . . . 39
5.4 Evaluation of the persistent magnetization current I . . . . . . . 42m
5.5 Estimated experimental parameters . . . . . . . . . . . . . . . . . 45
6 Persistent spin currents in antiferromagnet Heisenberg rings 47
6.1 Classical ground state . . . . . . . . . . . . . . . . . . . . . . . . 48
6.2 Spin-wave spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.3 Persistent magnetization current . . . . . . . . . . . . . . . . . . . 52
6.4 Modified spin-wave theory . . . . . . . . . . . . . . . . . . . . . . 53
7 Magnetization currents in electric fields 57
8 Conclusions 59
- vii -II Functional RG with collective fields 61
9 Introduction 63
9.1 Functional renormalization group . . . . . . . . . . . . . . . . . . 63
9.2 Electrons in one dimension . . . . . . . . . . . . . . . . . . . . . . 66
9.3 Outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
10 Interacting fermions as coupled Fermi-Bose systems 73
10.1 Path-integral formulation . . . . . . . . . . . . . . . . . . . . . . . 73
10.2 Hubbard-Stratonovich transformation . . . . . . . . . . . . . . . . 75
10.3Compact notation for Fermi and Bose fields . . . . . . . . . . . . . 76
11 Field-theoretical formalism in compact notation 79
11.1 Generating functionals . . . . . . . . . . . . . . . . . . . . . . . . 79
11.1.1 Connected Green’s functions . . . . . . . . . . . . . . . . . 79
11.1.2 Amputated connected Green’s functions . . . . . . . . . . 81
11.1.3 One-line irreducible vertices . . . . . . . . . . . . . . . . . 82
11.1.4 Tree expansion . . . . . . . . . . . . . . . . . . . . . . . . 84
11.2 Functional RG flow equations . . . . . . . . . . . . . . . . . . . . 87
12 Symmetries of the Fermi-Bose theory 93
12.1 Translational invariance . . . . . . . . . . . . . . . . . . . . . . . 93
12.2 Global gauge invariance . . . . . . . . . . . . . . . . . . . . . . . 95
12.3 Dyson-Schwinger equations . . . . . . . . . . . . . . . . . . . . . . 96
12.4 Local gauge transformations . . . . . . . . . . . . . . . . . . . . . 96
13 Functional RG for mixed field theory 101
13.1 Definition of physical vertices . . . . . . . . . . . . . . . . . . . . 101
13.2 Cutoff schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
13.3 Flow equations for physical vertices . . . . . . . . . . . . . . . . . 109
13.4 Rescaling and classification of vertices . . . . . . . . . . . . . . . . 109
13.5 Truncation to skeleton elements of two-point functions . . . . . . 116
13.5.1 Skeleton graphs . . . . . . . . . . . . . . . . . . . . . . . . 116
13.5.2 Truncation scheme . . . . . . . . . . . . . . . . . . . . . . 118
14 Interaction cutoff scheme 121
14.1 Exact flow equations for an interaction cutoff . . . . . . . . . . . 121
14.2 Initial condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
14.3 Ward identities as solutions of flow equations . . . . . . . . . . . . 127
14.4 Exact fRG solution of the Tomonaga-Luttinger model . . . . . . . 130
14.5 Truncation scheme based on relevance . . . . . . . . . . . . . . . 132
15 Summary and outlook 137
- viii -Bibliography 141
Deutsche Zusammenfassung 155
Ver¨offentlichungen 163
Lebenslauf 165
Danksagung 167
- ix -- x -