Electron Correlation

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1 Series in Modern Condensed Matter Physics Vol. 5 Lecture Notes an Electron Correlation and Magnetism Patrik Fazekas Research Institute for Solid State Physics & Optics, Budapest lb World Scientific h Singapore New Jersey. London Hong Kong

2 Preface xiii 1 Introduction Magnetism and Other Effects of Electron-Electron Interaction Sources of Magnetic Fields Getting Acquainted: Magnetite Charge States Spin States Charge Ordering Variety of Correlated Systems: An Outline of the Course 14 2 Atoms, Ions, and Molecules Hydrogen Atom in a Magnetic Field Non-Relativistic Treatment 18 Motion in a Magnetic Field 19 Zeeman Effect (I) Relativistic Effects 22 Spin-Orbit Coupling 24 Zeeman Effect (II) 25 Problem Direct Exchange 28 Problem Many-Electron Ions 36 Problem Coupling to the Magnetic Field 40 Digression: The Bohr-Van Leeuwen Theorem. 42 v

3 vi Contents Hund's Rules 43 Problem Paramagnetism and Diamagnetism Paramagnetic Susceptibility 46 Magnetization Curve 51 Problems Diamagnetism 53 Digression: Superstrong Fields Hydrogen Molecule Direct Exchange in Non-Orthogonal Orbitals Kinetic Exchange Molecular Orbitals versus Heitler-London Solutions to the Problems 66 3 Crystal Field Theory Incomplete Shells in an Anisotropic Environment: Crystal Fields The Role of Symmetry Arguments in Quantum Mechanics Irreducible Representations The Octahedral Group 86 Problems Symmetry Properties of Atomic States Splitting of a d-level in Cubic Field Quenching the Orbital Angular Momentum Partial Restoration of Orbital Momentum by Spin- Orbit Coupling 96 Problems High-Spin versus Low-Spin States Jahn-Teller Effect Time Reversal Invariance The f 2 Configuration Cubic Crystal Field Tetragonal Crystal Field Met amagnetic Transition Exchange Induced Magnetism 121 Problems Double Groups 122

4 vii Problems Crystal Field Potentials Quadrupole Moments 132 Solutions to the Problems Mott Transition and Hubbard Model Metals and Insulators: Breakdown of the Independent- Electron Description Mott Transition The Hubbard Model Local Basis Which Electrons Do We Mean? Limiting Cases The Band Limit 164 Problems The Atomic Limit 167 What Causes the Ordering? Symmetries Spin-Rotational Invariance Electron-Hole Symmetry 171 Problem Infinite-Dimensional Hubbard Model 173 Problems Hubbard Subbands The Mott-Hubbard Transition Ground State Phase Diagram 182 Solutions to the Problems Mott Insulators The Large-U Limit Classification of Hopping Events The Canonical Transformation Hubbard Operators The t - J Model Half-Filled Band: The Heisenberg Model 211 Problems Higher-Order Exchange 214

5 viii Contents 5.2 Superexchange The Extended Hubbard Model 221 Problems Orbital Degeneracy, Orbital Ordering 226 Problems Correlated Insulators Mott Transitions in Transition Metal Oxides Mott Insulators versus Charge Transfer Insulators 243 Hole Doping Solutions to the Problems Heisenberg Magnets Ferromagnetic Heisenberg Model Ground State: Symmetry Breaking Excitations: Spin Waves 268 Effects of Anisotropy 275 Magnon-Magnon Interactions 276 Finite Temperatures 277 Problem Antiferromagnetic Heisenberg Model Introduction Spin Waves 283 Sublattice Magnetization 290 Anisotropy 297 A Glimpse at the Strange World of D = Neef Order versus Valence Bond States Resonating Valence Bonds 307 Problems Digression: Symmetry Breaking Valence Bond Solids The CaV4 09 Story Frustration 330 Problem Solutions to the Problems 336

6 Itinerant Electron Magnetism 7.1 Introduction 7.2 Magnetic Order Digression: Symmetry Breaking 7.3 Mean Field Theories 7.4 Stoner Model Problems Generalized Susceptibility Criteria for q 0 Instabilities Problems Spin Density Waves Gap Equation Finite Temperatures Strong Coupling Limit Away from Half-Filling Problems Transition Metals and Alloys Introduction LSDA versus Lattice Fermion Models Nearly Ferromagnetic Metals Ferromagnetic Metals Weak Itinerant Ferromagnets Iron Group Elements Paramagnetic Susceptibility Chromium Solutions to the Problems 8 Ferromagnetism in Hubbard Models 8.1 Preliminary Overview The Low-Density Limit 8.2 Exactly Proven Cases of High-Spin Ground States Lieb's Ferrimagnetism Fiat-Band Ferromagnetism Nagaoka Ferromagnetism Problem Ferromagnetism in a Nearly Flat Band 8.3 The Ring Exchange Mechanism ix

7 Problem Instability of the Nagaoka State The Single-Spin-Flip State Improved Variational Methods: Square Lattice Lattice Structure Dependence of Ferromagnetism in a Single-Band Model 451 The fcc Lattice 455 Discussion Effects of Degeneracy Double Exchange 463 Problem Two-Band Model The LaMnO 3 System 473 Magnetostructural Transition 478 Charge Ordering 480 Solutions to the Problems The Gutzwiller Variational Method Minimum Polarity Principle Digression: The Early History The Variational Ground State The Gutzwiller Trial State 506 Problems The Cluster Method Brinkman-Rice Transition 520 Solutions to the Problems The Correlated Metallic State The Reduced Fermi Step Heavy Fermions: Half-Filled Band 530 The Effect of a Strong Magnetic Field 535 Problems Arbitrary Band Filling 537 Problems The Fermi Volume The Lai_r,SrxTiO3 System Discussion and Outlook 548

8 xi Gutzwiller Method: Exact Treatment Metallic and Insulating States 553 Problem Digression: The 1/r Hubbard Chain 556 Problem Gutzwiller States with Magnetic Order 561 Ferromagnetism 561 Antiferromagnetism 563 Phase Segregation versus Stripe Phases The Infinite-Dimensional Hubbard Model 574 The Correlated Metal 577 Phase Diagram 582 Solutions to the Problems Mixed Valence and Heavy Fermions Lanthanides and Actinides The Kinds of Valence: Integral, Mixed, Nearly Integral Digression: Valence Skipping Valence Mixing in Rare Earth Compounds Renormalized Hybridized Bands Nearly Integral Valence: Heavy Fermions Kondo Lattice Kondo Impurity Indirect Exchange Kondo Singlet versus RKKY Magnetism 644 Doniach Phase Diagram 646 Variational Approach 651 Additional Remarks Rare Earth Magnetism Quantum Hall Effect Introduction Motion in a Magnetic Field Landau Levels Algebraic Approach Symmetrical Gauge Integer Quantum Hall Effect 688

9 xii Contents High-Field Argument The Gauge Argument Periodic Potential Fractional Quantum Hall Effect Correlated Many-Electron States 704 Fractionally Charged Quasiparticies 707 Jain States 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

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