Lecture Notes on Electron Correlation and Magnetism

The following sections are included:

• Contents

• Preface

The following sections are included:

• 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

The following sections are included:

• Hydrogen Atom in a Magnetic Field
  • Non-Relativistic Treatment
  • Relativistic Effects

• Direct Exchange

• Many-Electron Ions
  • Coupling to the Magnetic Field
  • Hund's Rules

• Paramagnetism and Diamagnetism
  • Paramagnetic Susceptibility
  • Diamagnetism

• Hydrogen Molecule
  • Direct Exchange in Non-Orthogonal Orbitals
  • Kinetic Exchange
  • Molecular Orbitals versus Heitler–London

• Solutions to the Problems

The following sections are included:

• Incomplete Shells in an Anisotropic Environment: Crystal Fields

• The Role of Symmetry Arguments in Quantum Mechanics
  • Irreducible Representations

• The Octahedral Group

• 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
  • High-Spin versus Low-Spin States

• Jahn–Teller Effect

• Time Reversal Invariance

• The f² Configuration
  • Cubic Crystal Field
  • Tetragonal Crystal Field
  • Metamagnetic Transition
  • Exchange Induced Magnetism

• Double Groups

• Crystal Field Potentials
  • Quadrupole Moments

• Solutions to the Problems

The following sections are included:

• 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
  • The Atomic Limit

• Symmetries
  • Spin-Rotational Invariance
  • Electron–Hole Symmetry

• Infinite-Dimensional Hubbard Model

• Hubbard Subbands
  • The Mott–Hubbard Transition

• Ground State Phase Diagram

• Solutions to the Problems

The following sections are included:

• The Large-U Limit
  • Classification of Hopping Events
  • The Canonical Transformation
  • Hubbard Operators
  • The t - J Model
  • Half-Filled Band: The Heisenberg Model
  • Higher-Order Exchange

• Superexchange

• The Extended Hubbard Model

• Orbital Degeneracy, Orbital Ordering

• Correlated Insulators
  • Mott Transitions in Transition Metal Oxides
  • Mott Insulators versus Charge Transfer Insulators

• Solutions to the Problems

The following sections are included:

• Ferromagnetic Heisenberg Model
  • Ground State: Symmetry Breaking
  • Excitations: Spin Waves

• Antiferromagnetic Heisenberg Model
  • Introduction
  • Spin Waves

• Néel Order versus Valence Bond States
  • Resonating Valence Bonds
  • Valence Bond Solids
  • The CaV₄O₉ Story
  • Frustration

• Solutions to the Problems

The following sections are included:

• Introduction

• Magnetic Order
  • Digression: Symmetry Breaking

• Mean Field Theories

• Stoner Model

• Generalized Susceptibility
  • Criteria for q ≠ 0 Instabilities

• Spin Density Waves
  • Gap Equation
  • Finite Temperatures
  • Strong Coupling Limit
  • Away from Half-Filling

• Transition Metals and Alloys
  • Introduction
  • LSDA versus Lattice Fermion Models
  • Nearly Ferromagnetic Metals
  • Ferromagnetic Metals
  • Chromium

• Solutions to the Problems

The following sections are included:

• Preliminary Overview
  • The Low-Density Limit

• Exactly Proven Cases of High-Spin Ground States
  • Lieb's Ferrimagnetism
  • Flat-Band Ferromagnetism
  • Nagaoka Ferromagnetism
  • Ferromagnetism in a Nearly Flat Band

• The Ring Exchange Mechanism

• Instability of the Nagaoka State
  • The Single-Spin-Flip State
  • Improved Variational Methods: The Case of the Square Lattice
  • Lattice Structure Dependence of Ferromagnetism in a Single-Band Model

• Effects of Degeneracy
  • Double Exchange
  • Two-Band Model
  • The LaMnO₃ System

• Solutions to the Problems

The following sections are included:

• Minimum Polarity Principle
  • Digression: The Early History

• The Variational Ground State
  • The Gutzwiller Trial State
  • The Cluster Method

• Brinkman–Rice Transition

• Solutions to the Problems

The following sections are included:

• The Reduced Fermi Step

• Heavy Fermions: Half-Filled Band

• Arbitrary Band Filling

• The Fermi Volume

• The La_1-xSr_xTiO₃ System

• Discussion and Outlook
  • Gutzwiller Method: Exact Treatment
  • Metallic and Insulating States
  • Digression: The 1/r Hubbard Chain
  • Gutzwiller States with Magnetic Order
  • The Infinite-Dimensional Hubbard Model

• Solutions to the Problems

The following sections are included:

• Lanthanides and Actinides

• The Kinds of Valence: Integral, Mixed, and 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

• Rare Earth Magnetism

The following sections are included:

• Introduction

• Motion in a Magnetic Field
  • Landau Levels
  • Algebraic Approach
  • Symmetrical Gauge

• Integer Quantum Hall Effect
  • High-Field Argument
  • The Gauge Argument
  • Periodic Potential

• Fractional Quantum Hall Effect
  • Correlated Many-Electron States

• Discussion and Outlook

The following sections are included:

• Appendix A Hydrogen Atom
  • Hydrogenic Wave Functions
  • J-Eigenstates

• Appendix B Single-Spin-Flip Ansatz

• Appendix C Gutzwiller Approximation

• Appendix D Schrieffer–Wolff Transformation

• Bibliography

• Index