Correlated Electrons in Quantum Matter > 물리학

1 Introduction 1

2 Independent Electrons 9

2.1 Many-Electron Hamiltonian 10

2.2 Basis Sets 11

2.3 Self-consistent Field Equations 12

2.4 Unrestricted SCF Approximation 18

2.5 Missing Features of the Independent-Electron Approximation 20

3 Homogeneous Electron Gas 25

3.1 Uncorrelated Electrons 26

3.2 Random-Phase Approximation 31

3.3 Wigner Crystal 34

4 Density Functional Theory 39

4.1 Theory of Hohenberg, Kohn and Sham 40

4.2 Local-Density Approximation and Extensions 43

4.3 Strong Electron Correlations: LDA+U 47

4.4 The Energy Gap Problem 50

4.5 Time-Dependent DFT 53

5 Wavefunction-Based Methods 57

5.1 Method of Configuration Interactions 59

5.2 Cumulants and their Properties 63

5.3 Ground-State Wavefunction and Energy 64

5.3.1 Method of Increments 68

5.4 Different Approximation Schemes 70

5.4.1 Partitioning and Projection Methods 71

5.4.2 Coupled Cluster Method 72

5.4.3 Selection of Excitation Operators 75

5.4.4 Trial Wavefunctions 78

6 Correlated Ground-State Wavefunctions 83

6.1 Semiconductors 84

6.1.1 Model for Interatomic Correlations 84

6.1.2 Estimates of Intra-Atomic Correlations 89

6.1.3 Ab Initio Results 90

6.2 Ionic and van der Waals Solids 93

6.2.1 Three Oxides: MgO, CaO and NiO 93

6.2.2 Rare-Gas Solids 96

6.3 Simple Metals 97

6.4 Ground States with Strong Correlations: CASSCF 99

7 Quasiparticle Excitations 101

7.1 Single-particle Green\'s Function 102

7.1.1 Perturbation Expansions 106

7.1.2 Temperature Green\'s Function 111

7.2 Quasiparticles in Metals 117

7.3 Quasiparticles in Semiconductors and Insulators 123

7.3.1 Quasiparticle Approximation 124

7.3.2 A Simple Model: Bond-Orbital Approximation 126

7.3.3 Wavefunction-Based Ab Inito Calculations 132

8 Incoherent Excitations 137

8.1 Projection Method 138

8.2 An Example: Hubbard Model 141

9 Coherent-Potential Approximations 145

9.1 Static Disorder 146

9.2 Dynamical Disorder: DMFT and Beyond 148

10 Strongly Correlated Electrons 157

10.1 Measure of Correlation Strengths 159

10.2 Indicators of Strong Correlations 162

10.2.1 Low-Energy Scales: a Simple Model 163

10.2.2 Effective Hamiltonians 167

10.3 Kondo Effect 168

10.4 The Hubbard Model Revisited 178

10.4.1 Spin-Density Wave Ground State 178

10.4.2 Gutzwiller\'s Ground-State Wavefunction 183

10.4.3 Hubbard\'s Approximations and their Extensions 186

10.4.4 Kanamori Limit 189

10.5 The t-J Model 191

10.6 Mean-Field Approximations 207

10.6.1 Test of Different Approximation Schemes 212

10.7 Metal-Insulator Transitions 220

10.8 Numerical Studies 224

10.9 Break-down of Fermi Liquid Description 231

10.9.1 Marginal Fermi Liquid Behavior 232

10.9.2 Charged and Neutral Quasiparticles 234

10.9.3 Hubbard Chains 235

10.9.4 Quantum Critical Point 239

11 Transition Metals 241

11.1 Ground-State Wavefunction 242

11.2 Satellite Structures 248

11.3 Temperature-Dependent Magnetism 250

11.3.1 Local Spin Fluctuations 251

11.3.2 Long-Wavelength Spin Fluctuations 264

12 Transition-Metal Oxides 281

12.1 Doped Charge-Transfer Systems: the Cuprates 282

12.1.1 Quasiparticle-like Excitations 284

12.2 Orbital Ordering 299

12.2.1 Manganites: LaMnO3 and related Compounds 307

12.2.2 Vanadates: LaVO3 314

12.2.3 Ladder Systems: α\'-NaV2O5 314

12.2.4 Other Oxides 319

13 Heavy Quasiparticles 321

13.1 Kondo Lattice Systems 324

13.1.1 Renormalized Band Theory 325

13.1.2 Large Versus Small Fermi Surface 333

13.1.3 Mean-Field Treatment 336

13.2 Charge Ordering in Yb4As3: an Instructive Example 340

13.3 Partial Localization: Dual Role of 5f Electrons 348

13.4 Heavy d Electrons: LiV2O4 355

14 Excitations with Fractional Charges 363

14.1 Trans-Polyacetylene 364

14.2 Fractional Quantum Hall Effect 368

14.3 Correlated Electrons on Frustrated Lattices 375

14.3.1 Loop Models 378

14.3.2 Dimer Models 386

14.3.3 Mapping to a U(1) Gauge Theory 390

14.3.4 Magnetic Monopoles 393

15 Superconductivity 399

15.1 The Superconducting State 402

15.1.1 Pair States 405

15.1.2 BCS Ground State 410

15.2 Cooper Pair Breaking 417

15.2.1 Ergodic vs. Nonergodic Perturbations 418

15.2.2 Pairing Electrons with Population Imbalance 423

15.3 Cooper Pairing without Phonons 435

15.3.1 Filled Skutterudite PrOs4Sb12 437

15.3.2 UPd2Al3: Pairing and Time-Reversal Symmetry Breaking 440

15.4 Magnetic Resonances 441

15.5 High-Tc Superconductors 448

15.5.1 Suppression of Antiferromagnetic Order by Holes 450

15.5.2 Pseudogap Regime 451

15.5.3 Strange-Metal 454

15.5.4 Optical Properties: Drude Peak 455

15.5.5 Pairing Interactions 460

15.5.6 Stripe Formation 467

A Some Relations for Cumulants 473

B Scattering Matrix in Single-Centre and Two-Centre Approximation 475

C Intra-atomic Correlations in a C Atom 479

D Landau Parameter: Quasiparticle Mass 481

E Kondo Lattices: Quasiparticle Interactions 483

F Lanczos Method 485

G Density Matrix Renormalization Group 489

H Monte Carlo Methods 497

H 1 Sampling Techniques 498

H 2 Ground-State Energy 500

I Computing the Memory Function by Increments 505

J Kagome Lattice at 1/3 Filling 507

References 509

Index 525