MAGNETISM MADE SIMPLE. An Introduction to Physical Concepts and to Some Useful Mathematical Methods. Daniel C. Mattis


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1 THE THEORY OF MAGNETISM MADE SIMPLE An Introduction to Physical Concepts and to Some Useful Mathematical Methods Daniel C. Mattis Department of Physics, University of Utah lb World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI
2 Contents Prologue Chapter 1. History of Magnetism Physics and Metaphysics Gilbert and Descartes Rise of Modern Science Electrodynamics The Electron The Demise of Classical Physics Quantum Theory Modern Foundations Magnetic Bubbles Ultimate Thin Films Dilute Magnetic Alloys New Directions 49 Chapter 2. Currents or Spins? Charge Currents or Spins? The Magnetic Dipole The Magnetic DipoleDipole Interactions The Exchange Interactions Metals and Their Alloys Superconductivity The Need to Study Spin Angular Momentum 69 ix
3 x The Theory of Magnetism Made Simple Chapter 3. Quantum Theory of Angular Momentum Kinetic Angulo'. Momentum Spherical Harmonics Reason for Integer / and m Matrices of Angular Momentum Pauli Spin Matrices Compounding Angular Momentum Equations of Motion of Interacting Angular Momenta Coupled Boson Representation Rotations More an Compound Angular Momentum Other Representations Spins OneHalf: Special Tricks Spins One Quadratic Forms Pro j ection Operators 101 Chapter 4. Magnetism and the ManyBody Problem Hamiltonian Physics and Degeneracy The Pauli Principle The TwoFermion Problem Electrons in One Dimension: A Theorem The Wronskian States of Three Electrons Eigenfunctions of Total S 2 and Sz Hund's Rules p3 Configuration p2 and p4 Configurations Second Quantization Double Exchange Superexchange Jellium Hubbard Model: An Introduction Nagaoka's Ferromagnetism One Dimension: Exact Solutions Ferrimagnetism SpinDependent Tunneling 172
4 Contents xi Chapter 5. From Magnons to Solitons: Spin Dynamics Spin Waves as Harmonie Oscillators OneMagnon Eigenstates in Ferromagnets TwoMagnon States and Eigenstates in Ferromagnets Bound States in One Dimension Bound States in Two and Three Dimensions Antiferromagnetic Magnons: The OneDimensional XY Model Bethe's Solution of OneDimensional Heisenberg Antiferromagnet Linearized Antiferromagnetic Magnons Ferrimagnetism Some Rigorous Notions in Antiferro and FerriMagnetism Vortices Solitons and Bloch Domain Walls: Introductory Material Solitary Wave Solution More an Magnetic Domains TimeDependent Phenomena MajumdarGhosh Model and "Quantum Frustration" Integer Spins The AKLT Spin One Chain Defective Antiferromagnets 255 Chapter 6. Magnetism in Metals Bloch and Wannier States TightBinding Weak Magnetic Properties Exchange in Solids: Construction of a Model Hamiltonian PerturbationTheoretic Derivation of Heisenberg Hamiltonian Heisenberg Hamiltonian in Metals Ordered Magnetic Metals: Deriving the Ground State Kondo Effect Spin Glasses Magnetism without Localized Spins: Preliminaries Degenerate Bands and IntraAtomic Exchange Forces Magnons in Metals Marginal Magnetism of Impurities Correlations and Equivalence to s c/ Model Periodic Anderson Model 342
5 xii The Theory of Magnetism Made Simple Chapter 7. Statistical Thermodynamics Spins in a Magnetic Field The Partition Function The Concept of the Molecular Field Discontinuity in Specific Heat Magnetic Susceptibility and Spontaneous Magnetization Adiabatic Demagnetization Antiferromagnetism ShortRanged versus LongRanged Interactions Fermions, Bosons, and all that 375 Fermions 375 Bosons 375 Gaussian 376 Legendre Transformations Gaussian and Spherical Models 378 Gaussian Model 379 Spherical Model 382 Watson's Integrals Generalized Magnetic Susceptibility in Gaussian and Spherical Models Spherical Antiferromagnet Spherical Model Spin Glass 395 Magnetic Properties of Spin Glass Thermodynamics of Magnons Magnetism in Two Dimensions The XY Model: 1D The XY Model: 2D Transfer Matrix of Plane Rotator Model 433 Chapter 8. The Ising Model High Temperature Expansions Graph Theory Low Temperature Expansions and the Duality Relations Peierls' Proof of Long Range Order D Ising Model in Longitudinal Fields D Ising Model in Transverse Fields Concerning Quadratic Forms of Fermion Operators TwoDimensional Ising Model: The Transfer Matrix Solution of TwoDimensional Ising Model in Zero Field 485
6 Contents xiii Spontaneous Magnetization and Magnetic Susceptibility Zeros of the Partition Function Miscellania, Including 2D Antiferromagnets The ThreeDimensional Ising Model Ising Gauge Glass Frustration 528 Chapter 9. Miscellaneous Advanced Topics Critical Phenomena Green Functions Formalism Nonlinear Responses and Chaos Kondo Phenomenon: The sd Model Redux Scaling, Renormalization and Information Theory 547 Bibliography 549 Index 561
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Chapter 1 HartreeFock Theory 1.1 Formalism For N electrons in an external potential V ext (r), the manyelectron Hamiltonian can be written as follows: N H = [ p i i=1 m +V ext(r i )]+ 1 N N v(r i r j
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