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nils-uni@bluemer.name
Prof. Dr. Nils Blümer |
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Attention: these pages are mostly outdated! For current information on Nils Blümer, see web pages at the KU.
Last changed:
12-Sep-13
Mott-Hubbard Metal-Insulator Transition and Optical Conductivity in High Dimensions
by Nils Blümer
(Shaker Verlag, ISBN 3-8322-2320-7)
Table of Contents
Introduction 1 1 Models and Methods 5 1.1 Hubbard Model 5 1.1.1 Solid State Theory for Crystals 6 1.1.2 Electronic Lattice Models 7 1.1.3 Wannier Representation 8 1.1.4 One-band Hubbard Model 9 1.2 Dynamical Mean-Field Theory 11 1.2.1 Limit Z→∞ for Spin Models 12 1.2.2 Limit Z→∞ for Fermions 13 1.2.3 Simplifications for the Hubbard Model in Z→∞ 15 1.3 Quantum Monte Carlo Algorithm 19 1.3.1 Wick's Theorem for the Discretized Impurity Problem 19 1.3.2 Monte Carlo Importance Sampling 21 1.4 Maximum Entropy Method 23 2 Lattice and Density of States 27 2.1 Hypercubic Lattice and Extensions 29 2.1.1 Definitions and Analytical Considerations 29 2.1.2 Numerical Results 32 2.1.3 Magnetic Frustration and Asymmetry of the DOS 38 2.2 Bethe Lattice, RPE, and Disorder 39 2.2.1 Bethe Tree, Cayley Tree, and Husimi Cactus 40 2.2.2 Renormalized Perturbation Expansion 43 2.3 General Density of States in d=∞ 47 2.4 Redefinition of the Bethe Lattice 53 2.4.1 Model in d=∞ 54 2.4.2 Truncating the Hopping Range 57 2.4.3 Finite Dimensionality 59 2.4.4 Application to Asymmetric Model DOS 62 2.5 Conclusion 64 3 Mott Metal-Insulator Transition in the d→∞ Hubbard Model 67 3.1 Motivation 68 3.1.1 Experiment 68 3.1.2 Theory 71 3.2 Characterization of Phase Transitions within the DMFT 73 3.2.1 Transitions of First or Higher Order 73 3.2.2 Convergence of Fixed Point Methods 75 3.2.3 Observables 76 3.3 Phase Diagram: Development until 1999 79 3.4 Discussion of QMC Algorithms 82 3.4.1 Fourier Transformation and Smoothing 84 3.4.2 Overrelaxation and Sweeping Strategies 89 3.4.3 Estimation of Errors 90 3.4.4 Parallelization 92 3.5 Results: Coexistence Region 93 3.5.1 Choice of Observables and Extrapolation 93 3.5.2 Properties of the Insulating Phase 96 3.5.3 Internal Energy 100 3.5.4 Coexistence Phase Diagram 108 3.5.5 Double Occupancy 113 3.6 Results: Thermodynamic Phase Transition Line 121 3.6.1 Differential Equation for dUc/dT and Linearization 121 3.6.2 Low-temperature Asymptotics of U(T) 125 3.6.3 Full Phase Diagram 131 3.6.4 Implications of Partial Frustration 138 3.7 Landau Theory and Criticality 142 3.7.1 Free Energy Functional for the Bethe Lattice 143 3.7.2 Direct Evaluation of Free Energy Differences 144 3.7.3 Critical Behavior Near the MIT 146 3.8 Spectra 155 3.8.1 Maximum Entropy Method for Spectral Functions 155 3.8.2 Algorithmic Choices and Numerical Tests 160 3.8.3 Numerical Results for the Bethe DOS 166 3.9 Conclusion 173 4 Optical Conductivity 175 4.1 Definition and General Properties of the Optical Conductivity 176 4.1.1 Connection between Conductivity and Reflectivity 177 4.1.2 Optical f-sum Rules 179 4.1.3 Experiments 181 4.1.4 Impact of Electronic Model Abstractions 183 4.2 Kubo Formalism 186 4.2.1 Kubo Formalism in the Continuum 186 4.2.2 Kubo Formalism on a Lattice 188 4.2.3 General Confirmation of the f-sum Rule 190 4.3 Optical Conductivity in the Limit d→∞ 192 4.3.1 Optical Conductivity for the Hypercubic Lattice 194 4.3.2 f-sum Rule within the DMFT 195 4.3.3 f-sum Rule and General Dispersion Formalism 197 4.4 Optical Conductivity for the Bethe Lattice 198 4.4.1 Treelike Layout of the Bethe Lattice 200 4.4.2 Single-Chain Stacked Bethe Lattice 204 4.4.3 Periodically Stacked Lattices 209 4.4.4 Offdiagonal Disorder 211 4.4.5 General Dispersion Method 213 4.5 Generalizations 216 4.5.1 Coherent versus Incoherent Transport in High Dimensions 216 4.5.2 Optical Conductivity in Finite Dimensions 219 4.5.3 Impact of Frustration by t-t' Hopping 222 4.6 QMC Results for the Bethe Lattice 225 4.6.1 Numerical Procedure for QMC Data 226 4.6.2 Results: Self-Energy on the Real Axis 230 4.6.3 Results: Optical Conductivity 234 4.7 Conclusion 242 5 Realistic Modeling of Strongly Correlated Materials 245 5.1 La1-xSrxTiO3 246 5.2 DFT and LSDA 249 5.3 LDA+DMFT 251 5.4 Results for La1-xSrxTiO3 255 5.4.1 Density of States and Photoemission Spectra 255 5.4.2 Influence of Discretization Errors 261 5.4.3 Optical Conductivity 268 5.5 Conclusion 276 Summary 279 A Additions to ``Models and Methods'' 281 A.1 Extensions of the Hubbard Model 281 A.2 Characterization of Generic Momenta 284 A.3 DCA, CDMFT, and RDA 286 B Hyperdiamond Lattice 291 C Fourier-Transforming Imaginary-Time Green Functions 297 D Linear Response to Electromagnetic Fields 305 D.1 Electromagnetic Interaction Hamiltonian and Choice of Gauge 305 D.2 Linear Response Theory 306 Bibliography 309 Index 321 List of Publications 331 Curriculum Vitae 333 Acknowledgements 335
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