1 Introduction 1
1.1 Magnetism and Other Effects of Electron-Electron Inter-action 1
1.2 Sources of Magnetic Fields 5
1.3 Getting Acquainted:Magnetite 7
1.3.1 Charge States 8
1.3.2 Spin States 9
1.3.3 Charge Ordering 11
1.4 Variety of Correlated Systems:An Outline of the Course 14
2 Atoms,Ions,and Molecules 17
2.1 Hydrogen Atom in a Magnetic Field 18
2.1.1 Non-Relativistic Treatment 18
Motion in a Magnetic Field 19
Zeeman Effect（Ⅰ） 21
2.1.2 Relativistic Effects 22
Spin-Orbit Coupling 24
Zeeman Effect （Ⅱ） 25
Problem 2.1 28
2.2 Direct Exchange 28
Problem 2.2 36
2.3 Many-Electron Ions 36
Problem 2.3 40
2.3.1 Coupling to the Magnetic Field 40
Digression:The Bohr-Van Leeuwen Theorem 42
2.3.2 Hund's Rules 43
Problem 2.4 45
2.4 Paramagnetism and Diamagnetism 45
2.4.1 Paramagnetic Susceptibility 46
Magnetization Curve 51
Problems 2.5-2.8 52
2.4.2 Diamagnetism 53
Digression:Superstrong Fields 54
2.5 Hydrogen Molecule 56
2.5.1 Direct Exchange in Non-Orthogonal Orbitals 57
2.5.2 Kinetic Exchange 60
2.5.3 Molecular Orbitals versus Heitler-London 64
Solutions to the Problems 66
3 Crystal Field Theory 75
3.1 Incomplete Shells in an Anisotropic Environment:Crys-tal Fields 75
3.2 The Role of Symmetry Arguments in Quantum Mechanics 80
3.2.1 Irreducible Representations 82
3.3 The Octahedral Group 86
Problems 3.1-3.2 89
3.4 Symmetry Properties of Atomic States 89
3.5 Splitting of a d-Level in Cubic Field 91
3.5.1 Quenching the Orbital Angular Momentum 94
3.5.2 Partial Restoration of Orbital Momentum by Spin Orbit Coupling 96
Problems 3.3-3.4 98
3.5.3 High-Spin versus Low-Spin States 98
3.6 Jahn-Teller Effect 100
3.7 Time Reversal Invariance 104
3.8 The f2 Configuration 110
3.8.1 Cubic Crystal Field 111
3.8.2 Tetragonal Crystal Field 115
3.8.3 Metamagnetic Transition 118
3.8.4 Exchange Induced Magnetism 121
Problems 3.5-3.6 122
3.9 Double Groups 122
Problems 3.7-3.8 128
3.10 Crystal Field Potentials 128
3.10.1 Quadrupole Moments 132
Solutions to the Problems 134
4 Mott Transition and Hubbard Model 147
4.1 Metals and Insulators:Breakdown of the Independent-Electron Description 147
4.2 Mott Transition 150
4.3 The Hubbard Model 157
4.3.1 Local Basis 161
4.3.2 Which Electrons Do We Mean? 162
4.4 Limiting Cases 163
4.4.1 The Band Limit 164
Problems 4.1-4.4 166
4.4.2 The Atomic Limit 167
What Causes the Ordering? 169
4.5 Symmetries 170
4.5.1 Spin-Rotational Invariance 170
4.5.2 Electron-Hole Symmetry 171
Problem 4.5 173
4.6 Infinite-Dimensional Hubbard Model 173
Problems 4.6-4.8 177
4.7 Hubbard Subbands 178
4.7.1 The Mott-Hubbard Transition 180
4.8 Ground State Phase Diagram 182
Solutions to the Problems 187
5 Mott Insulators 199
5.1 The Large-U Limit 199
5.1.1 Classification of Hopping Events 200
5.1.2 The Canonical Transformation 203
5.1.3 Hubbard Operators 205
5.1.4 The t-J Model 207
5.1.5 Half-Filled Band:The Heisenberg Model 211
Problems 5.1-5.2 213
5.1.6 Higher-Order Exchange 214
5.2 Superexchange 217
5.3 The Extended Hubbard Model 221
Problems 5.3-5.5 226
5.4 Orbital Degeneracy,Orbital Ordering 226
Problems 5.6-5.8 234
5.5 Correlated Insulators 234
5.5.1 Mott Transitions in Transition Metal Oxides 235
5.5.2 Mott Insulators versus Charge Transfer Insulators 243
Hole Doping 248
Solutions to the Problems 250
6 Heisenberg Magnets 263
6.1 Ferromagnetic Heisenberg Model 264
6.1.1 Ground State:Symmetry Breaking 265
6.1.2 Excitations:Spin Waves 268
Effects of Anisotropy 275
Magnon-Magnon Interactions 276
Finite Temperatures 277
Problem 6.1 279
6.2 Antiferromagnetic Heisenberg Model 280
6.2.1 Introduction 280
6.2.2 Spin Waves 283
Sublattice Magnetization 290
Anisotropy 297
A Glimpse at the Strange World of D=1 299
6.3 Néel Order versus Valence Bond States 306
6.3.1 Resonating Valence Bonds 307
Problems 6.2-6.4 311
Digression:Symmetry Breaking 312
6.3.2 Valence Bond Solids 315
6.3.3 The CaV4O9 Story 326
6.3.4 Frustration 330
Problem 6.5 335
Solutions to the Problems 336
7 Itinerant Electron Magnetism 341
7.1 Introduction 341
7.2 Magnetic Order 344
7.2.1 Digression:Symmetry Breaking 347
7.3 Mean Field Theories 349
7.4 Stoner Model 352
Problems 7.1-7.3 357
7.5 Generalized Susceptibility 357
7.5.1 Criteria for q≠0 Instabilities 360
Problems 7.4-7.5 362
7.6 Spin Density Waves 363
7.6.1 Gap Equation 367
7.6.2 Finite Temperatures 370
7.6.3 Strong Coupling Limit 376
7.6.4 Away from Half-Filling 381
Problems 7.6-7.9 385
7.7 Transition Metals and Alloys 386
7.7.1 Introduction 386
7.7.2 LSDA versus Lattice Fermion Models 389
7.7.3 Nearly Ferromagnetic Metals 394
7.7.4 Ferromagnetic Metals 396
Weak Itinerant Ferromagnets 396
Iron Group Elements 399
Paramagnetic Susceptibility 401
7.7.5 Chromium 402
Solutions to the Problems 406
8 Ferromagnetism in Hubbard Models 419
8.1 Preliminary Overview 419
8.1.1 The Low-Density Limit 423
8.2 Exactly Proven Cases of High-Spin Ground States 425
8.2.1 Lieb's Ferrimagnetism 426
8.2.2 Flat-Band Ferromagnetism 428
8.2.3 Nagaoka Ferromagnetism 430
Probem 8.1 433
8.2.4 Ferromagnetism in a Nearly Flat Band 433
8.3 The Ring Exchange Mechanism 435
Probem 8.2 440
8.4 Instability of the Nagaoka State 441
8.4.1 The Single-Spin-Flip State 442
8.4.2 Improved Variational Methods:Square Lattice 448
8.4.3 Lattice Structure Dependence of Ferromagnetism in a Single-Band Model 451
The fcc Lattice 455
Discussion 458
8.5 Effects of Degeneracy 463
8.5.1 Double Exchange 463
Problem 8.3 470
8.5.2 Two-Band Model 470
8.5.3 The LaMnO3 System 473
Magnetostructural Transition 478
Charge Ordering 480
Solutions to the Problems 485
9 The Gutzwiller Variational Method 497
9.1 Minimum Polarity Principle 499
9.1.1 Digression:The Early History 502
9.2 The Variational Ground State 505
9.2.1 The Gutzwiller Trial State 506
Problems 9.1-9.2 513
9.2.2 The Cluster Method 513
9.3 Brinkman-Rice Transition 520
Solutions to the Problems 525
10 The Correlated Metallic State 527
10.1 The Reduced Fermi Step 527
10.2 Heavy Fermions:Half-Filled Band 530
The Effect of a Strong Magnetic Field 535
Probems 10.1-2 537
10.3 Arbitrary Band Filling 537
Problems 10.3-10.4 542
10.4 The Fermi Volume 542
10.5 The La1-xSrxTiO3 System 543
10.6 Discussion and Outlook 548
10.6.1 Gutzwiller Method:Exact Treatment 549
10.6.2 Metallic and Insulating States 553
Problem 10.5 556
10.6.3 Digression:The l/r Hubbard Chain 556
Problem 10.6 560
10.6.4 Gutzwiller States with Magnetic Order 561
Ferromagnetism 561
Antiferromagnetism 563
Phase Segregation versus Stripe Phases 568
10.6.5 The Infinite-Dimensional Hubbard Model 574
The Correlated Metal 577
Phase Diagram 582
Solutions to the Problems 590
11 Mixed Valence and Heavy Fermions 597
11.1 Lanthanides and Actinides 598
11.2 The Kinds of Valence:Integral,Mixed,Nearly Integral 600
11.2.1 Digression:Valence Skipping 604
11.2.2 Valence Mixing in Rare Earth Compounds 605
11.2.3 Renormalized Hybridized Bands 611
11.2.4 Nearly Integral Valence:Heavy Fermions 619
11.3 Kondo Lattice 628
11.3.1 Kondo Impurity 632
11.3.2 Indirect Exchange 641
11.3.3 Kondo Singlet versus RKKY Magnetism 644
Doniach Phase Diagram 646
Variational Approach 651
Additional Remarks 660
11.4 Rare Earth Magnetism 662
12 Quantum Hall Effect 669
12.1 Introduction 670
12.2 Motion in a Magnetic Field 677
12.2.1 Landau Levels 678
12.2.2 Algebraic Approach 683
12.2.3 Symmetrical Gauge 685
12.3 Integer Quantum Hall Effect 688
12.3.1 High-Field Argument 692
12.3.2 The Gauge Argument 695
12.3.3 Periodic Potential 700
12.4 Fractional Quantum Hall Effect 703
12.4.1 Correlated Many-Electron States 704
Fractionally Charged Quasiparticles 707
Jain States 712
12.5 Discussion and Outlook 715
Phase Diagram 717
Spin Effects 721
A Hydrogen Atom 725
A.1 Hydrogenic Wave Functions 725
A.2 J-Eigenstates 727
B Single-Spin-Flip Ansatz 729
C Gutzwiller Approximation 733
D Schrieffer-Wolff Transformation 741
Bibliography 747
Index 773