A Numerical Renormalization Group Approach to Dissipative ...

More documents

Recommendations

Info

Contents Contents i Introduction 1 I Experimental Motivation 5 1 Kondo Effect 7 1.1 Kondo Effect in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Kondo Effect in Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Coherent Control of Single Spins 13 2.1 Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Color Centers in Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Molecular Electronics 19 3.1 C60-Molecule Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Single Molecule Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 II Numerical Renormalization Group 25 4 Fermionic Baths 27 4.1 Reducing to One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Logarithmic Discretization of Conduction Band . . . . . . . . . . . . . . . . . 28 4.3 Mapping the Hamiltonian on a Semi-infinite Chain . . . . . . . . . . . . . . . 29 4.4 Iterative Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5 Bosonic Baths 37 5.1 Star Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.2 Chain Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6 Density Matrix Numerical Renormalization Group 43 6.1 Energy Scale Separation for Dynamical Quantities . . . . . . . . . . . . . . . 43 6.2 Reduced Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 i
ii CONTENTS 7 Complete Basis 49 7.1 Over-counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7.2 Complete Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 7.3 Resummation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.4 Application to Spectral Functions . . . . . . . . . . . . . . . . . . . . . . . . . 53 8 Time-dependence 57 8.1 Quantum Quenches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 8.2 Time Evolution Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 8.3 Transforming ˆρeq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 8.4 Implementing the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 9 Finite Temperatures 65 9.1 Single Shell Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 9.2 Full Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 9.3 Spectral Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 10 Improving Accuracy 69 10.1 Using Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 10.2 Oliveira’s z-Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 10.3 Damping and Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 10.4 Self-Energy Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 10.5 High-Accuracy NRG Spectral Functions as Benchmark . . . . . . . . . . . . . 74 III Applications 77 11 Ferromagnetic Kondo Model 79 11.1 Experimental Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 11.1.1 Schrieffer-Wolff Transformation . . . . . . . . . . . . . . . . . . . . . . 80 11.1.2 Underscreened Kondo Effect . . . . . . . . . . . . . . . . . . . . . . . 81 11.1.3 Single Molecule Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . 82 11.2 Equilibrium Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 11.3 Nonequilibrium Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . 85 11.3.1 Isotropic Kondo Exchange . . . . . . . . . . . . . . . . . . . . . . . . . 87 11.3.2 Anisotropic Kondo Exchange . . . . . . . . . . . . . . . . . . . . . . . 88 11.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 12 Fermionic vs. Bosonic Bath 93 12.1 Spin-Boson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 12.1.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 12.1.2 Experimental Realizations . . . . . . . . . . . . . . . . . . . . . . . . . 95 12.1.3 Physical Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 12.2 Bosonization Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 12.3 Static Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 12.4 Comparing Spin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 12.4.1 Weak Coupling 0 < α < 0.5 . . . . . . . . . . . . . . . . . . . . . . . . 102 12.4.2 Toulouse Limit α = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Page 1: A Numerical Renormalization Group A
Page 6 and 7: CONTENTS iii 12.4.3 Intermediate Co
Page 9 and 10: Introduction The miniaturization of
Page 11: Table of Contents 3 treatment of th
Page 15 and 16: Chapter 1 Kondo Effect 1.1 Kondo Ef
Page 17 and 18: 1.1 Kondo Effect in Metals 9 Figure
Page 19 and 20: 1.2 Kondo Effect in Quantum Dots 11
Page 21 and 22: Chapter 2 Coherent Control of Singl
Page 23 and 24: 2.1 Quantum Dots 15 Figure 2.2: Sca
Page 25: 2.2 Color Centers in Diamond 17 An
Page 28 and 29: 20 CHAPTER 3. Molecular Electronics
Page 35 and 36: Chapter 4 Fermionic Baths In the fi
Page 37 and 38: 4.3 Mapping the Hamiltonian on a Se
Page 39 and 40: 4.3 Mapping the Hamiltonian on a Se
Page 41 and 42: 4.4 Iterative Diagonalization 33 4.
Page 43 and 44: 4.4 Iterative Diagonalization 35 Fi
Page 45 and 46: Chapter 5 Bosonic Baths One disting
Page 47 and 48: 5.1 Star Hamiltonian 39 Figure 5.2:
Page 49 and 50: 5.2 Chain Hamiltonian 41 For an ill
Page 51 and 52: Chapter 6 Density Matrix Numerical
Page 53 and 54: 6.2 Reduced Density Matrix 45 magne
Page 55:
6.2 Reduced Density Matrix 47 where
Page 58 and 59:
50 CHAPTER 7. Complete Basis A comm
Page 60 and 61:
52 CHAPTER 7. Complete Basis Figure
Page 62 and 63:
54 CHAPTER 7. Complete Basis G(ω)
Page 65 and 66:
Chapter 8 Time-dependence In this c
Page 67 and 68:
8.2 Time Evolution Formula 59 we en
Page 69 and 70:
8.3 Transforming ˆρeq Now we empl
Page 71 and 72:
8.4 Implementing the Algorithm 63 I
Page 73 and 74:
Chapter 9 Finite Temperatures 9.1 S
Page 75 and 76:
9.3 Spectral Functions 67 density m
Page 77 and 78:
Chapter 10 Improving Accuracy In th
Page 79 and 80:
10.3 Damping and Broadening 71 A↑
Page 81 and 82:
10.4 Self-Energy Trick 73 A↑(ω)
Page 83:
10.5 High-Accuracy NRG Spectral Fun
Page 87 and 88:
Chapter 11 Ferromagnetic Kondo Mode
Page 89 and 90:
11.1.2 Underscreened Kondo Effect 8
Page 91 and 92:
11.1.3 Single Molecule Magnets 83 c
Page 93 and 94:
11.3 Nonequilibrium Magnetization 8
Page 95 and 96:
11.3.1 Isotropic Kondo Exchange 87
Page 97 and 98:
11.3.2 Anisotropic Kondo Exchange 8
Page 99:
11.4 Summary 91 spin at time t < 0
Page 102 and 103:
94 CHAPTER 12. Fermionic vs. Bosoni
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
100 CHAPTER 12. Fermionic vs. Boson
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 125 and 126:
Chapter 13 Two Spins in a Bosonic B
Page 127 and 128:
13.1 Generalized Two-Spin Boson Mod
Page 129 and 130:
13.2.2 Qualitative Understanding of
Page 131 and 132:
13.2.2 Qualitative Understanding of
Page 133 and 134:
13.3.1 Static Entanglement Entropy
Page 135 and 136:
13.3.2 Subohmic System: Scaling of
Page 137 and 138:
13.4.1 Decoherence Without Transver
Page 139 and 140:
13.4.2 Beating and Decoherence: Bre
Page 141 and 142:
13.4.3 Synchronization of Spin Dyna
Page 143 and 144:
13.4.4 Vanishing Ising Interaction
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
13.4.6 Generation of Highly Entangl
Page 153 and 154:
Chapter 14 Decoherence in an Aharan
Page 155 and 156:
14.1.1 Mapping to Symmetric/Antisym
Page 157 and 158:
14.2.1 Implementing the Kubo Formal
Page 159 and 160:
14.4 “Diagonal” AC Conductance
Page 161 and 162:
14.4.2 Renormalization of GLL(ωac)
Page 163 and 164:
14.5.2 “Polaron Shift” of Peak
Page 165 and 166:
14.5.3 Multiple Phonon Excitations
Page 167:
14.6 Summary 159 ReGLR/(g 2 ph e2
Page 170 and 171:
162 CHAPTER 14. Decoherence in an A
Page 172 and 173:
164 BIBLIOGRAPHY [12] P. Nozières
Page 174 and 175:
166 BIBLIOGRAPHY [36] D. D. Awschal
Page 176 and 177:
168 BIBLIOGRAPHY [58] A. Hewson: Th
Page 178 and 179:
170 BIBLIOGRAPHY [81] A. Isidori, D
Page 180 and 181:
172 BIBLIOGRAPHY [104] M. Heyl and
Page 182 and 183:
174 BIBLIOGRAPHY [128] M. Garst, S.
Page 184 and 185:
176 BIBLIOGRAPHY [152] D. Braun:
Page 186 and 187:
178 BIBLIOGRAPHY [174] T. L. Schmid
Page 188 and 189:
180 BIBLIOGRAPHY [198] M. E. Fisher
Page 191 and 192:
Appendix A Details of the Mapping t
Page 193 and 194:
Using the definition of ˆ f † m+
Page 195 and 196:
Appendix B Detailed Derivation of S
Page 197 and 198:
B.1 Full Density Matrix 189 Fig. B.
Page 199 and 200:
B.1 Full Density Matrix 191 m frequ
Page 201 and 202:
B.2 Single Shell Approximation 193
Page 203 and 204:
Appendix C Details of Bosonization
Page 205 and 206:
C.3 Bosonic Reorganization of Fock
Page 207 and 208:
C.5 Derivation of the Bosonization
Page 209:
C.5 Derivation of the Bosonization
Page 212 and 213:
204 APPENDIX D. Derivation of Scali
Page 215 and 216:
Appendix E Mapping the Two-Spin Bos
Page 217:
nian (E.1) thus reads in terms of f
Page 221:
Publications Part of the results pr
Page 225:
Acknowledgments I would like to tak
Page 228 and 229:
Der dritte Teil der Arbeit beinhalt
Page 230 and 231:
Darüber hinaus haben wir dargelegt
Page 233:
Erklärung Hiermit erkläre ich, da
show all

A Numerical Renormalization Group Approach to Dissipative ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?