The Theory of Magnetism Made Simple: An Introduction to Physical Concepts and to Some Useful mathematical methods

by Mattis, Daniel C.

• ISBN13: 9789812386717
• ISBN10: 9812386718
• Format: Paperback
• Copyright: 2006-05-30
• Publisher: World Scientific Pub Co Inc

Summary

The original edition of The Theory of Magnetism was the first book to develop the various relevant topics using modern methods adapted for the many-body problem and thus it became popular (reportedly the "most-stolen" book from the exhibition stalls at the March meeting of the American Physical Society!). It presented and taught the fermionic field theory central to Onsager's analysis of the statistical mechanics of the two-dimensional Ising model of magnetism. In its pages the Lieb-Mattis theorems on magnetic ordering of electronic energy levels and on the absence of ferromagnetism in one dimension were restated and proved in a form accessible to students. The exchange mechanism in insulators and the Ruderman-Kittel interaction in metals were some of the innovative topics presented to the reader. Spin waves and their interactions were analyzed in some detail. The first chapter, on the history of physics as seen through the prism of research in magnetism, co-authored with Dr Noemi Mattis, proved especially popular. In this new edition, while retaining much of the material in earlier editions, especially the first chapter, the author has eliminated some of the bulk (the most recent edition was in two volumes) and added a number of new subjects. Among these are the effects of lowering the dimensionality (exact solutions of some important models in zero and one dimension are exhibited and contrasted with the three-dimensional versions) and the importance of the two-body Coulomb interactions. The reader is introduced to the topic of critical exponents, which has been so marvellously worked out in recent decades. Quoting a novel theorem by Lieb and exotic band structures, the authorre-examines the origins of ferromagnetism. In the presentation, physical principles come first, the mathematics second. Developing the reader's intuition and mastery of the subject takes precedence. Because of this the book was not

Author Biography

Daniel C. Mattis, B.S., M.S., Ph.D., Fellow of the American Physical Society (APS), is a frequent lecturer at research institutions

Table of Contents

Prologue
Chapter 1. History of Magnetism	1	(52)
1.1. Physics and Metaphysics	3	(3)
1.2. Gilbert and Descartes	6	(6)
1.3. Rise of Modern Science	12	(4)
1.4. Electrodynamics	16	(7)
1.5. The Electron	23	(3)
1.6. The Demise of Classical Physics	26	(5)
1.7. Quantum Theory	31	(4)
1.8. Modern Foundations	35	(4)
1.9. Magnetic Bubbles	39	(5)
1.10. Ultimate Thin Films	44	(3)
1.11. Dilute Magnetic Alloys	47	(2)
1.12. New Directions	49	(4)
Chapter 2. Currents or Spins?	53	(17)
2.1. Charge Currents or Spins?	53	(3)
2.2. The Magnetic Dipole	56	(1)
2.3. The Magnetic Dipole-Dipole Interactions	57	(6)
2.4. The Exchange Interactions	63	(2)
2.5. Metals and Their Alloys	65	(3)
2.6. Superconductivity	68	(1)
2.7. The Need to Study Spin Angular Momentum	69	(1)
Chapter 3. Quantum Theory of Angular Momentum	70	(34)
3.1. Kinetic Angular Momentum	70	(4)
3.2. Spherical Harmonics	74	(4)
3.3. Reason for Integer / and m	78	(2)
3.4. Matrices of Angular Momentum	80	(2)
3.5. Pauli Spin Matrices	82	(1)
3.6. Compounding Angular Momentum	83	(4)
3.7. Equations of Motion of Interacting Angular Momenta	87	(1)
3.8. Coupled Boson Representation	87	(3)
3.9. Rotations	90	(2)
3.10. More on Compound Angular Momentum	92	(3)
3.11. Other Representations	95	(1)
3.12. Spins One-Half: Special Tricks	96	(2)
3.13. Spins One	98	(2)
3.14. Quadratic Forms	100	(1)
3.15. Projection Operators	101	(3)
Chapter 4. Magnetism and the Many-Body Problem	104	(71)
4.1. Hamiltonian Physics and Degeneracy	104	(3)
4.2. The Pauli Principle	107	(4)
4.3. The Two-Fermion Problem	111	(2)
4.4. Electrons in One Dimension: A Theorem	113	(3)
4.5. The Wronskian	116	(2)
4.6. States of Three Electrons	118	(2)
4.7. Eigenfunctions of Total S² and Sz	120	(5)
4.8. Hund's Rules	125	(2)
4.9. p³ Configuration	127	(6)
4.10. p² and p4 Configurations	133	(6)
4.11. Second Quantization	139	(4)
4.12. Double Exchange	143	(3)
4.13. Superexchange	146	(3)
4.14. Jellium	149	(5)
4.15. Hubbard Model: An Introduction	154	(9)
4.16. Nagaoka's Ferromagnetism	163	(4)
4.17. One Dimension: Exact Solutions	167	(1)
4.18. Ferrimagnetism	168	(4)
4.19. Spin-Dependent Tunneling	172	(3)
Chapter 5. From Magnons to Solitons: Spin Dynamics	175	(82)
5.1. Spin Waves as Harmonic Oscillators	176	(9)
5.2. One-Magnon Eigenstates in Ferromagnets	185	(1)
5.3. Two-Magnon States and Eigenstates in Ferromagnets	186	(9)
5.4. Bound States in One Dimension	195	(1)
5.5. Bound States in Two and Three Dimensions	196	(4)
5.6. Antiferromagnetic Magnons: The One-Dimensional XY Model	200	(6)
5.7. Bethe's Solution of One-Dimensional Heisenberg Antiferromagnet	206	(9)
5.8. Linearized Antiferromagnetic Magnons	215	(9)
5.9. Ferrimagnetism	224	(2)
5.10. Some Rigorous Notions in Antiferro- and Ferri-Magnetism	226	(2)
5.11. Vortices	228	(3)
5.12. Solitons and Bloch Domain Walls: Introductory Material	231	(4)
5.13. Solitary Wave Solution	235	(6)
5.14. More on Magnetic Domains	241	(1)
5.15. Time-Dependent Phenomena	242	(5)
5.16. Majumdar-Ghosh Model and "Quantum Frustration"	247	(3)
5.17. Integer Spins	250	(4)
5.18. The AKLT Spin One Chain	254	(1)
5.19. Defective Antiferromagnets	255	(2)
Chapter 6. Magnetism in Metals	257	(87)
6.1. Bloch and Wannier States	259	(1)
6.2. Tight-Binding	260	(7)
6.3. Weak Magnetic Properties	267	(6)
6.4. Exchange in Solids: Construction of a Model Hamiltonian	273	(8)
6.5. Perturbation-Theoretic Derivation of Heisenberg Hamiltonian	281	(3)
6.6. Heisenberg Hamiltonian in Metals	284	(3)
6.7. Ordered Magnetic Metals: Deriving the Ground State	287	(9)
6.8. Kondo Effect	296	(8)
6.9. Spin Glasses	304	(6)
6.10. Magnetism without Localized Spins: Preliminaries	310	(4)
6.11. Degenerate Bands and Intra-Atomic Exchange Forces	314	(6)
6.12. Magnons in Metals	320	(7)
6.13. Marginal Magnetism of Impurities	327	(10)
6.14. Correlations and Equivalence to s-d Model	337	(5)
6.15. Periodic Anderson Model	342	(2)
Chapter 7. Statistical Thermodynamics	344	(94)
7.1. Spins in a Magnetic Field	344	(5)
7.2. The Partition Function	349	(5)
7.3. The Concept of the Molecular Field	354	(4)
7.4. Discontinuity in Specific Heat	358	(3)
7.5. Magnetic Susceptibility and Spontaneous Magnetization	361	(4)
7.6. Adiabatic Demagnetization	365	(1)
7.7. Antiferromagnetism	366	(3)
7.8. Short-Ranged versus Long-Ranged Interactions	369	(6)
7.9. Fermions, Bosons, and all that	375	(3)
Fermions	375	(1)
Bosons	375	(1)
Gaussian	376	(1)
Legendre Transformations	377	(1)
7.10. Gaussian and Spherical Models	378	(11)
Gaussian Model	379	(3)
Spherical Model	382	(5)
Watson's Integrals Generalized	387	(2)
7.11. Magnetic Susceptibility in Gaussian and Spherical Models	389	(3)
7.12. Spherical Antiferromagnet	392	(3)
7.13. Spherical Model Spin Glass	395	(12)
Magnetic Properties of Spin Glass	399	(8)
7.14. Thermodynamics of Magnons	407	(7)
7.15. Magnetism in Two Dimensions	414	(4)
7.16. The XY Model: 1D	418	(8)
7.17. The XY Model: 2D	426	(7)
7.18. Transfer Matrix of Plane Rotator Model	433	(5)
Chapter 8. The Ising Model	438	(95)
8.1. High Temperature Expansions	441	(4)
8.2. Graph Theory	445	(3)
8.3. Low Temperature Expansions and the Duality Relations	448	(4)
8.4. Peierls' Proof of Long Range Order	452	(2)
8.5. 1D Ising Model in Longitudinal Fields	454	(8)
8.6. 1D Ising Model in Transverse Fields	462	(12)
8.7. Concerning Quadratic Forms of Fermion Operators	474	(4)
8.8. Two-Dimensional Ising Model: The Transfer Matrix	478	(7)
8.9. Solution of Two-Dimensional Ising Model in Zero Field	485	(9)
8.10. Spontaneous Magnetization and Magnetic Susceptibility	494	(8)
8.11. Zeros of the Partition Function	502	(5)
8.12. Miscellania, Including 2D Antiferromagnets	507	(10)
8.13. The Three-Dimensional Ising Model	517	(9)
8.14. Ising Gauge Glass	526	(2)
8.15. Frustration	528	(5)
Chapter 9. Miscellaneous Advanced Topics	533	(16)
9.1. Critical Phenomena	533	(3)
9.2. Green Functions Formalism	536	(3)
9.3. Nonlinear Responses and Chaos	539	(1)
9.4. Kondo Phenomenon: The s-d Model Redux	539	(8)
9.5. Scaling, Renormalization and Information Theory	547	(2)
Bibliography	549	(12)
Index	561