A Numerical Renormalization Group Approach to Dissipative ...

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158 CHAPTER 14. Decoherence in an Aharanov-Bohm Interferometer resonances at sidebands of the level energy ωac = ε + nωph, where n = 1, 2, . . .. We have verified that these additional features are not caused by band edge effects, scaling all parameters (Γν,σ, ωph, gph, ε) by a factor 2 (cf. Fig. 14.12). The additional features can be observed independent of the relative size of the electronic conduction band W , which is a strong indication that they are connected to processes involving several phonons. ReGLR/((gph/ωph) 2 e 2 ¯h −1 ) 0.15 0.1 0.05 0 -0.05 0 0.5 1 1.5 2 2.5 ωac/(ε + ωph) xgph/W = analytic 0.06 0.08 0.1 ImGLR/((gph/ωph) 2 e 2 ¯h −1 ) 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 0 0.5 1 1.5 2 2.5 ωac/(ε + ωph) xgph/W = analytic 0.06 0.08 0.1 Figure 14.12: In the real part of the transconductance GLR(ωac) (left panel) the small negative peak at ωac = ε+ωph is clearly visible. System parameters are chosen xΓν,σ = ±0.02W, xωph = 0.1W, xε = 0.05W , where x = 1 (lines) and x = 2 (points). Note that the described features do not change scaling the system parameters. For large xgph = 0.1W we find an additional sideband at ωac = ε + 2ωph in the NRG results for the real part of GLR(ωac). The corresponding features can be extracted from the imaginary part of GLR(ωac) as well, though they are not as clearly distinguishable as in the real part (cf. left panel). 14.5.4 Vanishing Transconductance for ωac → 0. We have discussed the appearance of sub-threshold transconductance, i.e. a finite conductance for ωac < ωph in the AC case, in Sec. 14.5.1. While the arguments given there explain the vanishing of conductance in the DC limit in lowest order in the electron-phonon coupling gph, it is not obvious that higher-order processes do not change this picture. One may naturally ask about the effects of the broadening of the phonon propagator due to coupling to the electrons. Some (though not all) higher-order effects can thus be captured by calculating the electronic self-energies with a phonon propagator dressed with electronic polarization bubbles. In the DC case, one finds that electrons can indeed absorb an energy up to eV from the phononic bath broadened by coupling to the electrons, instead of needing to emit ωph as in lowest order. Nonetheless, the weight of such contributions is small leading to a current dependence I ∝ (eV ) 3 for small bias voltage for the considered higher-order contributions in the electron-phonon coupling and consequently a vanishing linear conductance at zero frequency. This agrees with the DC limit of our NRG results for the AC conductance presented in Fig. 14.13, which indicates as discussed that all higher-order contributions to the linear zero-temperature conductance vanish. We thus confirm the naive perturbative argument, that there are no incoherent transport processes at vanishing frequency, temperature and transport voltage.
14.6 Summary 159 ReGLR/(g 2 ph e2 ¯h −1 ) 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 1e-06 1e-05 0.0001 0.001 0.01 0.1 ωac/W gph/W = 0.05 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 Figure 14.13: Real part of the transconductance GLR(ωac) on a logarithmic scale. Notice that even though for gph > ωph the peak in the transconductance shifts to smaller AC frequencies ωac, in the limit of very small frequencies we always obtain limωac→0 GLR(ωac) = 0. (Γν,σ = ±0.01, εν = 0.05, ωph = 0.1) 14.6 Summary In this last chapter of this thesis, we have investigated the influence of an Einstein phonon mode on the transport through an Aharonov-Bohm interferometer consisting of two quantum dots coupled in parallel to two lead electrodes. We applied the numerical renormalization group to calculate the conductance through the interferometer applying an AC bias to the lead electrodes. The main numeric results of our investigation confirm and back up the findings of a perturbative nonequilibrium Green’s function method (see Ref. [193]). In contrast to the case of DC conductance, we find transport below the threshold ωac < ωph, i.e. transport through the interferometer without real emission of phonons is possible (analogous to the case of a plate capacitance, where the electrostatic interaction leads to a finite AC conductance, but a vanishing DC conductance). We confirm the naive perturbative argument, that there are no incoherent transport processes at vanishing frequency, temperature and transport voltage. Thus limωac→0 GLR(ωac) = 0, which is equivalent to stating that all higher-order contributions to the linear zero-temperature conductance vanish. The peak in the real part of 2 gph transconductance was investigated in some detail. We found that the height scales like ωph and is shifted to smaller AC frequencies ωac < ε at larger electron-phonon coupling reminiscent of polaron shift. Additional resonances at sidebands of the level energy ε = ε + nωph, where n = 1, 2, . . ., stem from processes involving multiple phonons and could be verified using our numerical renormalization group method calculations.
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A Numerical Renormalization Group A
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Contents Contents i Introduction 1
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CONTENTS iii 12.4.3 Intermediate Co
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Introduction The miniaturization of
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Table of Contents 3 treatment of th
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Chapter 1 Kondo Effect 1.1 Kondo Ef
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1.1 Kondo Effect in Metals 9 Figure
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1.2 Kondo Effect in Quantum Dots 11
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Chapter 2 Coherent Control of Singl
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2.1 Quantum Dots 15 Figure 2.2: Sca
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2.2 Color Centers in Diamond 17 An
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20 CHAPTER 3. Molecular Electronics
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Chapter 4 Fermionic Baths In the fi
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4.3 Mapping the Hamiltonian on a Se
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4.3 Mapping the Hamiltonian on a Se
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4.4 Iterative Diagonalization 33 4.
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4.4 Iterative Diagonalization 35 Fi
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Chapter 5 Bosonic Baths One disting
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5.1 Star Hamiltonian 39 Figure 5.2:
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5.2 Chain Hamiltonian 41 For an ill
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Chapter 6 Density Matrix Numerical
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6.2 Reduced Density Matrix 45 magne
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6.2 Reduced Density Matrix 47 where
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50 CHAPTER 7. Complete Basis A comm
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52 CHAPTER 7. Complete Basis Figure
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54 CHAPTER 7. Complete Basis G(ω)
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Chapter 8 Time-dependence In this c
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8.2 Time Evolution Formula 59 we en
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8.3 Transforming ˆρeq Now we empl
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8.4 Implementing the Algorithm 63 I
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Chapter 9 Finite Temperatures 9.1 S
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9.3 Spectral Functions 67 density m
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Chapter 10 Improving Accuracy In th
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10.3 Damping and Broadening 71 A↑
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10.4 Self-Energy Trick 73 A↑(ω)
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10.5 High-Accuracy NRG Spectral Fun
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Chapter 11 Ferromagnetic Kondo Mode
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11.1.2 Underscreened Kondo Effect 8
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11.1.3 Single Molecule Magnets 83 c
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11.3 Nonequilibrium Magnetization 8
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11.3.1 Isotropic Kondo Exchange 87
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11.3.2 Anisotropic Kondo Exchange 8
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11.4 Summary 91 spin at time t < 0
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94 CHAPTER 12. Fermionic vs. Bosoni
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100 CHAPTER 12. Fermionic vs. Boson
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Page 116 and 117: 108 CHAPTER 12. Fermionic vs. Boson
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 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: 14.5.3 Multiple Phonon Excitations
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
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nian (E.1) thus reads in terms of f
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Publications Part of the results pr
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Acknowledgments I would like to tak
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Der dritte Teil der Arbeit beinhalt
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Darüber hinaus haben wir dargelegt
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Erklärung Hiermit erkläre ich, da
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