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1 Introduction1

1.1 Magnetism and Other Effects of Electron-Electron Inter-action1

1.2 Sources of Magnetic Fields5

1.3 Getting Acquainted:Magnetite7

1.3.1 Charge States8

1.3.2 Spin States9

1.3.3 Charge Ordering11

1.4 Variety of Correlated Systems:An Outline of the Course14

2 Atoms,Ions,and Molecules17

2.1 Hydrogen Atom in a Magnetic Field18

2.1.1 Non-Relativistic Treatment18

Motion in a Magnetic Field19

Zeeman Effect（Ⅰ）21

2.1.2 Relativistic Effects22

Spin-Orbit Coupling24

Zeeman Effect （Ⅱ）25

Problem 2.128

2.2 Direct Exchange28

Problem 2.236

2.3 Many-Electron Ions36

Problem 2.340

2.3.1 Coupling to the Magnetic Field40

Digression:The Bohr-Van Leeuwen Theorem42

2.3.2 Hund's Rules43

Problem 2.445

2.4 Paramagnetism and Diamagnetism45

2.4.1 Paramagnetic Susceptibility46

Magnetization Curve51

Problems 2.5-2.852

2.4.2 Diamagnetism53

Digression:Superstrong Fields54

2.5 Hydrogen Molecule56

2.5.1 Direct Exchange in Non-Orthogonal Orbitals57

2.5.2 Kinetic Exchange60

2.5.3 Molecular Orbitals versus Heitler-London64

Solutions to the Problems66

3 Crystal Field Theory75

3.1 Incomplete Shells in an Anisotropic Environment:Crys-tal Fields75

3.2 The Role of Symmetry Arguments in Quantum Mechanics80

3.2.1 Irreducible Representations82

3.3 The Octahedral Group86

Problems 3.1-3.289

3.4 Symmetry Properties of Atomic States89

3.5 Splitting of a d-Level in Cubic Field91

3.5.1 Quenching the Orbital Angular Momentum94

3.5.2 Partial Restoration of Orbital Momentum by Spin Orbit Coupling96

Problems 3.3-3.498

3.5.3 High-Spin versus Low-Spin States98

3.6 Jahn-Teller Effect100

3.7 Time Reversal Invariance104

3.8 The f2 Configuration110

3.8.1 Cubic Crystal Field111

3.8.2 Tetragonal Crystal Field115

3.8.3 Metamagnetic Transition118

3.8.4 Exchange Induced Magnetism121

Problems 3.5-3.6122

3.9 Double Groups122

Problems 3.7-3.8128

3.10 Crystal Field Potentials128

3.10.1 Quadrupole Moments132

Solutions to the Problems134

4 Mott Transition and Hubbard Model147

4.1 Metals and Insulators:Breakdown of the Independent-Electron Description147

4.2 Mott Transition150

4.3 The Hubbard Model157

4.3.1 Local Basis161

4.3.2 Which Electrons Do We Mean?162

4.4 Limiting Cases163

4.4.1 The Band Limit164

Problems 4.1-4.4166

4.4.2 The Atomic Limit167

What Causes the Ordering?169

4.5 Symmetries170

4.5.1 Spin-Rotational Invariance170

4.5.2 Electron-Hole Symmetry171

Problem 4.5173

4.6 Infinite-Dimensional Hubbard Model173

Problems 4.6-4.8177

4.7 Hubbard Subbands178

4.7.1 The Mott-Hubbard Transition180

4.8 Ground State Phase Diagram182

Solutions to the Problems187

5 Mott Insulators199

5.1 The Large-U Limit199

5.1.1 Classification of Hopping Events200

5.1.2 The Canonical Transformation203

5.1.3 Hubbard Operators205

5.1.4 The t-J Model207

5.1.5 Half-Filled Band:The Heisenberg Model211

Problems 5.1-5.2213

5.1.6 Higher-Order Exchange214

5.2 Superexchange217

5.3 The Extended Hubbard Model221

Problems 5.3-5.5226

5.4 Orbital Degeneracy,Orbital Ordering226

Problems 5.6-5.8234

5.5 Correlated Insulators234

5.5.1 Mott Transitions in Transition Metal Oxides235

5.5.2 Mott Insulators versus Charge Transfer Insulators243

Hole Doping248

Solutions to the Problems250

6 Heisenberg Magnets263

6.1 Ferromagnetic Heisenberg Model264

6.1.1 Ground State:Symmetry Breaking265

6.1.2 Excitations:Spin Waves268

Effects of Anisotropy275

Magnon-Magnon Interactions276

Finite Temperatures277

Problem 6.1279

6.2 Antiferromagnetic Heisenberg Model280

6.2.1 Introduction280

6.2.2 Spin Waves283

Sublattice Magnetization290

Anisotropy297

A Glimpse at the Strange World of D=1299

6.3 Néel Order versus Valence Bond States306

6.3.1 Resonating Valence Bonds307

Problems 6.2-6.4311

Digression:Symmetry Breaking312

6.3.2 Valence Bond Solids315

6.3.3 The CaV4O9 Story326

6.3.4 Frustration330

Problem 6.5335

Solutions to the Problems336

7 Itinerant Electron Magnetism341

7.1 Introduction341

7.2 Magnetic Order344

7.2.1 Digression:Symmetry Breaking347

7.3 Mean Field Theories349

7.4 Stoner Model352

Problems 7.1-7.3357

7.5 Generalized Susceptibility357

7.5.1 Criteria for q≠0 Instabilities360

Problems 7.4-7.5362

7.6 Spin Density Waves363

7.6.1 Gap Equation367

7.6.2 Finite Temperatures370

7.6.3 Strong Coupling Limit376

7.6.4 Away from Half-Filling381

Problems 7.6-7.9385

7.7 Transition Metals and Alloys386

7.7.1 Introduction386

7.7.2 LSDA versus Lattice Fermion Models389

7.7.3 Nearly Ferromagnetic Metals394

7.7.4 Ferromagnetic Metals396

Weak Itinerant Ferromagnets396

Iron Group Elements399

Paramagnetic Susceptibility401

7.7.5 Chromium402

Solutions to the Problems406

8 Ferromagnetism in Hubbard Models419

8.1 Preliminary Overview419

8.1.1 The Low-Density Limit423

8.2 Exactly Proven Cases of High-Spin Ground States425

8.2.1 Lieb's Ferrimagnetism426

8.2.2 Flat-Band Ferromagnetism428

8.2.3 Nagaoka Ferromagnetism430

Probem 8.1433

8.2.4 Ferromagnetism in a Nearly Flat Band433

8.3 The Ring Exchange Mechanism435

Probem 8.2440

8.4 Instability of the Nagaoka State441

8.4.1 The Single-Spin-Flip State442

8.4.2 Improved Variational Methods:Square Lattice448

8.4.3 Lattice Structure Dependence of Ferromagnetism in a Single-Band Model451

The fcc Lattice455

Discussion458

8.5 Effects of Degeneracy463

8.5.1 Double Exchange463

Problem 8.3470

8.5.2 Two-Band Model470

8.5.3 The LaMnO3 System473

Magnetostructural Transition478

Charge Ordering480

Solutions to the Problems485

9 The Gutzwiller Variational Method497

9.1 Minimum Polarity Principle499

9.1.1 Digression:The Early History502

9.2 The Variational Ground State505

9.2.1 The Gutzwiller Trial State506

Problems 9.1-9.2513

9.2.2 The Cluster Method513

9.3 Brinkman-Rice Transition520

Solutions to the Problems525

10 The Correlated Metallic State527

10.1 The Reduced Fermi Step527

10.2 Heavy Fermions:Half-Filled Band530

The Effect of a Strong Magnetic Field535

Probems 10.1-2537

10.3 Arbitrary Band Filling537

Problems 10.3-10.4542

10.4 The Fermi Volume542

10.5 The La1-xSrxTiO3 System543

10.6 Discussion and Outlook548

10.6.1 Gutzwiller Method:Exact Treatment549

10.6.2 Metallic and Insulating States553

Problem 10.5556

10.6.3 Digression:The l/r Hubbard Chain556

Problem 10.6560

10.6.4 Gutzwiller States with Magnetic Order561

Ferromagnetism561

Antiferromagnetism563

Phase Segregation versus Stripe Phases568

10.6.5 The Infinite-Dimensional Hubbard Model574

The Correlated Metal577

Phase Diagram582

Solutions to the Problems590

11 Mixed Valence and Heavy Fermions597

11.1 Lanthanides and Actinides598

11.2 The Kinds of Valence:Integral,Mixed,Nearly Integral600

11.2.1 Digression:Valence Skipping604

11.2.2 Valence Mixing in Rare Earth Compounds605

11.2.3 Renormalized Hybridized Bands611

11.2.4 Nearly Integral Valence:Heavy Fermions619

11.3 Kondo Lattice628

11.3.1 Kondo Impurity632

11.3.2 Indirect Exchange641

11.3.3 Kondo Singlet versus RKKY Magnetism644

Doniach Phase Diagram646

Variational Approach651

Additional Remarks660

11.4 Rare Earth Magnetism662

12 Quantum Hall Effect669

12.1 Introduction670

12.2 Motion in a Magnetic Field677

12.2.1 Landau Levels678

12.2.2 Algebraic Approach683

12.2.3 Symmetrical Gauge685

12.3 Integer Quantum Hall Effect688

12.3.1 High-Field Argument692

12.3.2 The Gauge Argument695

12.3.3 Periodic Potential700

12.4 Fractional Quantum Hall Effect703

12.4.1 Correlated Many-Electron States704

Fractionally Charged Quasiparticles707

Jain States712

12.5 Discussion and Outlook715

Phase Diagram717

Spin Effects721

A Hydrogen Atom725

A.1 Hydrogenic Wave Functions725

A.2 J-Eigenstates727

B Single-Spin-Flip Ansatz729

C Gutzwiller Approximation733

D Schrieffer-Wolff Transformation741

Bibliography747

Index773