The Mott Metal-Insulator Transition

Transcription

1 Florian Gebhard The Mott Metal-Insulator Transition Models and Methods With 38 Figures Springer

2 1. Metal Insulator Transitions Classification of Metals and Insulators Definition of Metal and Insulator Classes of Insulators Gap Criterion for the Insulator Electrical Conductivity at Low Temperature Gap at Finite Temperature Types of Metal-Insulator Transitions Quantum Phase Transition Thermodynamic Phase Transition Band, Peierls, and Anderson Insulators Band Insulators Peierls Insulators Anderson Insulators Mott Insulators: Basic Theoretical Concepts Electron-Electron Interaction in Metals Exchange and Correlations Magnetic Moments Slater Insulator Mott Insulator Mott-Hubbard Insulator Mott-Heisenberg Insulator Mott Insulators: Some Experimental Observations Wigner Insulator Verwey Insulator Multi-Band Mott Insulator Charge-Transfer Insulator Whither Theory? Hubbard Model Electronic Many-Particle Problem Hamilton Operator of Solid-State Physics Hamilton Operator of the Electronic Problem 51

3 X Contents 2.2 Band Structure Calculations Independent Electrons Hartree Theory Local Density Approximation Improvements to the Local Density Approximation Derivation of the Hubbard Model Second Quantization Hubbard Model Hubbard Model and Real Materials Polymers Transition Metals and Their Oxides Fullerenes High-Temperature Superconductors Heavy Fermion Materials Liquid Helium Relevance of the Hubbard Model Model Properties Model Parameters Model Variants Symmetries Basic Properties of the Hubbard Model Magnetism Approximate Methods Hartree-Fock Theory Method and Exact Statements Slater's Theory of the Antiferromagnet Stoner's Theory of the Ferromagnet Simplified Hartree-Fock Phase Diagram Advantages and Problems Improvements Local Density Approximation Site Occupation Functional Theory Site Occupation Local Density Approximation XQ Local Density Approximation LSDA with Self-Interaction Corrections Green-Function Decoupling Method Atomic Limit and Hubbard Bands Hubbard-I Approximation Advantages and Problems Improvements Alloy-Analogy Approximation Hubbard-III Approximation Advantages and Problems Improvements 108

4 XI 3.4 Variational Wave Functions Gutzwiller-Correlated Wave Functions Variational Characterization of an Insulator Exact Statements Gutzwiller Wave Function in One Dimension Variational Wave Functions in Infinite Dimensions Brinkman-Rice Transition Expansion Around the Limit of Infinite Dimensions Variational Phase Diagram in Infinite Dimensions Extensions and Improvements Advantages and Problems Slave Boson Approach Slave Bosons and Functional Integrals Kotliar-Ruckenstein Slave Bosons Kotliar-Ruckenstein Approximation Advantages and Problems Barnes-Coleman Slave Bosons and Approximation Advantages and Problems One-Dimensional Hubbard Models Hubbard Model with Nearest-Neighbor Hopping Two-Electron Problem for Two Lattice Sites General Two-Electron Problem in One Dimension Bethe Ansatz for the IV-Electron Problem Ground-State Energy at Half Band-Filling MetaHnsulator Transition at Half Band-Filling Drude Weight Below Half Band-Filling MetaHnsulator Transition Approaching Half Filling Hubbard Model with Long-Range Hopping Basic Properties General Two-Electron Problem in One Dimension Effective Hamiltonian for the JV-Electron Problem Ground-State Energy and MetaHnsulator Transition Thermodynamic Properties Hubbard Model with Variable-Range Hopping Basic Properties General Two-Electron Problem in One Dimension S Matrix and Generalized Lieb-Wu Equations Ground-State Energy at Half Band-Filling MetaHnsulator Transition at Half Band-Filling Mott-Hubbard Transition in One Dimension gr-ology Hamiltonian Perfect Nesting in One Dimension Long-Range Hopping Comparison with Basic Theoretical Concepts 182

5 Hubbard Model in Infinite Dimensions Limit of Infinite Dimensions Spin Models in Infinite Dimensions Non-Interacting Itinerant Electrons Generalized Mean-Field Approaches Mean-Field Spin Models Random Dispersion Approximation Simplifications in Infinite Dimensions Position-Space Collapse of Diagrams Irrelevance of Momentum Conservation Effective Models with Single-Site Interaction Analytical Results in Infinite Dimensions Perturbation Theory for Small Interactions Perturbation Theory for Large Interactions Variational Wave Functions Falicov-Kimball Model Spinless Fermions Approximate Solutions of the Hubbard Model Model Specifications Constraints on Approximate Treatments Quantum Monte-Carlo Calculations Exact Diagonalization Studies Iterated Perturbation Theory Non-Crossing Approximation Local Moments in Hubbard's Approximations Local-Moment Approach Metal-Insulator Transitions Mott-Heisenberg Insulator Correlated Metal Mott-Hubbard Insulator Mott-Hubbard Transition Comparison with Basic Theoretical Concepts 240 Further Models with Hubbard Interaction Degenerate and Extended Hubbard Models Band-Degenerate Hubbard Model in One Dimension Extended Hubbard Model Generalized Hubbard Models Models with Conserved Double Occupancies Bond-Charge Interactions and Harris-Lange Model Harris-Lange Model in One Dimension Montorsi-Rasetti Model Supersymmetric Hubbard Model Models with Commuting Operators Hubbard Model with Infinite-Range Hopping 256

6 XIII Stoner Model with Long-Range Hopping Landau-Hubbard Model Conclusions 263 Appendix 267 A.l Calculations for Hartree-Fock Theory 267 A.1.1 Diagonalization of the Hartree-Fock Hamiltonian 267 A.1.2 Ground-State Properties for Half Band-Filling 268 A.2 Green Functions and Hubbard-I Approximation 270 A.2.1 Green Functions 270 A.2.2 Hubbard-I Approximation 272 A.3 Generalized Lieb-Wu Integral Equations 274 A.3.1 Derivation of the Integral Equations 274 A.3.2 Solution for Half-Filled Bands 276 A.3.3 Quasi-Particle Dispersions 277 A.3.4 Limit of Long-Range Hopping 279 References 281 Index 309