Quantum Theory of Magnetism Wolfgang Nolting · Anupuru Ramakanth

Home , Antiferromagnetism, Diamagnetism, Identical particles

Quantum Theory of Magnetism

123 Prof. Dr. Wolfgang Nolting Prof. Dr. Anupuru Ramakanth Humboldt-Universitat¨ zu Berlin Kakatiya University Institut fur¨ Physik Department of Physics Newtonstr. 15 Warangal-506009 12489 Berlin India Germany [email protected]

ISBN 978-3-540-85415-9 e-ISBN 978-3-540-85416-6 DOI 10.1007/978-3-540-85416-6 Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2009927326

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Springer is part of Springer Science+Business Media (www.springer.com) Preface

In the literature, theory of magnetism is available in books at two levels. At the beginners level it is dealt as a chapter in books on solid state physics. There certain basic facts about magnetism are presented and some of the theoretical ideas are mentioned almost in passing. If one wants to persue the subject in depth then one has to refer to the books at the higher level. At this level there already do exist excel- lent books which have either a large canvas or limit the area but go to considerable depth. These books, however, are useful to those who already are practitioners of theoretical research in magnetism and are not exactly suitable for those who want to prepare themselves for research. In other words, we find that to our knowledge, there does not exist a book on the quantum theory of magnetism which serves as a text book and also one which helps and guides one in self-study. That is, a book where every step is worked out in detail and also contains a number of problems which help in self-assessment and also which supplement the material dealt with in the text of the book. It is precisely to fill this gap, we have attempted to write this book. This need was felt by one of the authors (WN) while giving special courses to advanced students almost two decades ago. As a result he published a book with exactly the same title but in German. The book was well received and many a reader has been asking since then if there is a revised edition. As a result, the present book is an english, thoroughly revised and updated version of the original German version. The book presumes the reader to have certain basic understanding of the concepts and techniques of quantum mechanics and statistical mechanics. Except for that, the book is, we hope, self-contained in the sense that every single step has been worked out both in the main text and also in the solutions to the problems. Some of the problems have rather long solutions. Then, these represent the results which most often are assumed to be well known but in fact they need to be worked out somewhere. Some of the problems are such that the solutions impart considerable training to one who wants to start out on research and helps him in learning certain “standard” tricks in order to understand some seminal papers and also in implementing his own research programme. The first chapter introduces, starting from Maxwell’s equations, certain basic facts about magnetism such as magnetic moment, magnetization and susceptibility and also contains a section on thermodynamics as applicable to magnetic systems. If one wants to understand magnetism of materials, one has to be first clear about

v vi the magnetism of individual atoms. This purpose is served by Chap. 2 (Atomic magnetism), in which all the important magnetic properties of atoms are discussed. We derive the electron spin and spinÐorbit interaction from the relativistic Dirac equa- tion and investigate the behaviour of an atomic electron in the field of the nucleus and also in the presence of an external magnetic field. The third chapter is devoted to “diamagnetism” which is a property possessed by all materials, which, however, is observable only when it is not overwhelmed by some other forms of magnetism (para-, ferro-, ferri- or antiferromagnetism). Diamagnetism in some measure can be explained as an induction effect, particularly because of the negative susceptibility. Chapter 4 deals with paramagnetism, which in contrast to diamagnetism, presumes the existence of permanent magnetic moments. These moments can either be localized stemming from partially filled electron shells of the ions in solid or they can be the moments of the quasi-freely moving band electrons. An external magnetic field tries to order them, whereas the thermal motion opposes the ordering tendency. The result is a positive susceptibility which in general is temperature dependent. Param- agnets are characterized by the fact that a direct interaction between the permanent moments, to a good approximation, can be neglected. In contrast to this, collective magetism (ferro, ferri and antiferromagnetism) is characterized by a spontaneous ordering of the magnetic moments below a critical temperature, and therefore a necessary precondition for this is the existence of a microscopic interaction between the moments. The so-called exchange interaction even though has its origin in pure electrostatic interaction cannot be understood from classical point of view. The general experience is that for a beginner this poses certain conceptual difficulties but, at the same time, it is the basis of understanding collective magnetism. There- fore exchange interaction has been discussed in considerable detail in Chap. 5. The so-called direct exchange interaction is determined via the overlap integrals of the wavefunctions of the participating magnetic ions. As a result it is of very short range and therefore is seldom realized in nature as compared to certain indirect exchange interactions which use the electrons in the conduction band (RKKY interaction) or the diamagnetic ions (superexchange, double exchange) as “catalysts” for an interaction between the localized moments. The coupling mechanisms are explained using simple cluster models and it is shown that ultimately all the interactions have the same operator form (Heisenberg model). Having established the required conceptual basis, the last three chapters are devoted to the three important models of magnetism, namely, the Ising, the Heisen- berg and the Hubbard model, respectively. In these chapters, an attempt has been made to present material such that the approach is pedagogic and at the same time gives the latest results available in literature. In doing this care is taken to derive all the results systematically and in every detail. Some of the important derivations are treated as problems whose complete solutions are given. While discussing quantum theory of magnetism it is imperative that one uses the techniques of many-body theory. In order to famialiarize the reader with these techniques, two appendices are added. The first one deals with the formalism of second quantization where all the results are worked out and to provide sufficient training to the self-learner, a set of problems is added. The second apendix is concerned with the many-body theory. Preface vii

The topics in this appendix are so chosen that they are directly relevant to the theory of magnetism. Again in this appendix, too, problems are provided, some of which elucidate certain further results which are left out in the actual text. The preperation of this book took about 3 years during which the authors were able to get together at either Berlin or Warangal for short periods, the ﬁnancial support for which was provided by the Volkswagen Foundation and Kakatiya Uni- versity. It is a pleasure to acknowledge the help of Dr. G. Gangadhar Reddy in various forms during the entire period of writing the book.

Berlin, Germany W. Nolting Warangal, India A. Ramakanth July 2009 Contents

1 Basic Facts ...... 1 1.1 Macroscopic Maxwell Equations ...... 1 1.2 Magnetic Moment and Magnetization ...... 7 1.3 Susceptibility ...... 13 1.4 Classiﬁcation of Magnetic Materials ...... 15 1.4.1 Diamagnetism ...... 15 1.4.2 Paramagnetism ...... 15 1.4.3 Collective Magnetism ...... 17 1.5 Elements of Thermodynamics ...... 19 1.6 Problems...... 22

2 Atomic Magnetism ...... 25 2.1 Hund’s Rules ...... 25 2.1.1 RussellÐSaunders (LS-)Coupling ...... 25 2.1.2 Hund’s Rules for LS Coupling...... 27 2.2 DiracEquation...... 28 2.3 ElectronSpin...... 34 2.4 SpinÐOrbitCoupling...... 40 2.5 WignerÐEckart Theorem ...... 45 2.5.1 Rotation ...... 45 2.5.2 RotationOperator ...... 47 2.5.3 Angular Momentum ...... 48 2.5.4 RotationMatrices...... 50 2.5.5 TensorOperators...... 52 2.5.6 WignerÐEckart Theorem ...... 53 2.5.7 ExamplesofApplication...... 55 2.6 Electron in an External Magnetic Field ...... 56 2.7 Nuclear Quadrupole Field ...... 62 2.8 Hyperﬁne Field ...... 67 2.9 Magnetic Hamiltonian of the Atomic Electron ...... 72 2.10 Many-ElectronSystems ...... 74 2.10.1 CoulombInteraction...... 74

ix x Contents

2.10.2 SpinÐOrbitCoupling...... 75 2.10.3 FurtherCouplings...... 76 2.11 Problems...... 81

3 Diamagnetism ...... 85 3.1 BohrÐvan Leeuwen Theorem ...... 85 3.2 Larmor Diamagnetism (Insulators) ...... 87 3.3 The Sommerfeld Model of a Metal ...... 90 3.3.1 Properties of the Model ...... 91 3.3.2 Sommerfeld Expansion ...... 99 3.4 Landau Diamagnetism (Metals) ...... 104 3.4.1 Free Electrons in Magnetic Field (Landau Levels) ...... 104 3.4.2 Grand Canonical Potential of the Conduction Electrons . . 109 3.4.3 Susceptibility of the Conduction Electrons ...... 117 3.5 The de HaasÐVan Alphen Effect ...... 121 3.5.1 Oscillations in the Magnetic Susceptibility ...... 121 3.5.2 Electron Orbits in Magnetic Field ...... 124 3.5.3 Physical Origin of the Oscillations ...... 128 3.5.4 Onsager Argument ...... 131 3.6 Problems...... 134 References ...... 136

4 Paramagnetism ...... 137 4.1 Pauli Spin Paramagnetism ...... 138 4.1.1 “Primitive” Theory of the Pauli Spin Paramagnetism . . . . 138 4.1.2 TemperatureCorrections...... 141 4.1.3 Exchange Corrections ...... 142 4.2 Paramagnetism of the Localized Electrons ...... 155 4.2.1 WeakSpinÐOrbitInteraction...... 158 4.2.2 StrongSpinÐOrbitCoupling...... 164 4.2.3 Van Vleck Paramagnetism ...... 166 4.3 Problems...... 171 References ...... 174

5 Exchange Interaction ...... 175 5.1 Phenomenological Theories ...... 178 5.1.1 The Exchange Field ...... 178 5.1.2 Weiss Ferromagnet ...... 180 5.2 Direct Exchange Interaction ...... 184 5.2.1 Pauli’sPrinciple...... 184 5.2.2 The HeitlerÐLondon Method ...... 188 5.2.3 Dirac’s Vector Model...... 195 5.3 Indirect Exchange Interaction ...... 200 5.3.1 RudermannÐKittelÐKasuyaÐYosida (RKKY) Interaction . 200 5.3.2 Superexchange ...... 209 Contents xi

5.3.3 Double Exchange ...... 217 5.4 Problems...... 226 References ...... 231

6 Ising Model ...... 233 6.1 The Model ...... 234 6.2 The One Dimensional Ising Model ...... 236 6.2.1 Spontaneous Magnetization ...... 236 6.2.2 One Dimensional Ising Model in External Field ...... 240 6.3 The Phase Transition of Two-Dimensional Ising Model ...... 245 6.3.1 TheMethodofProof...... 245 6.3.2 Finite Ising Lattice with Special Boundary Conditions . . . 247 6.3.3 Probabilities ...... 247 6.3.4 Realization Possibilities for the Polygons ...... 249 6.3.5 Magnetization of the Finite Lattice ...... 252 6.3.6 Thermodynamic Limit...... 252 6.4 The Free Energy of the Two-Dimensional Ising Model ...... 254 6.4.1 High-Temperature Expansion ...... 255 6.4.2 Spin Products as Graphs ...... 256 6.4.3 Loops ...... 257 6.4.4 DirectedPaths ...... 261 6.4.5 Matrix M1 ...... 264 6.4.6 FreeEnergyperSpin...... 266 6.4.7 Curie Temperature Tc ...... 267 6.4.8 Speciﬁc Heat ...... 268 6.4.9 Spontaneous Magnetization ...... 269 6.5 Problems...... 270 References ...... 271

7 Heisenberg Model ...... 273 7.1 Model Hamiltonian ...... 273 7.1.1 SpinOperators...... 273 7.1.2 Model Extensions ...... 279 7.2 Exact Statements ...... 283 7.2.1 MerminÐWagner Theorem ...... 283 7.2.2 One-Magnon States of a Ferromagnet ...... 291 7.3 Molecular Field Approximations ...... 296 7.3.1 Ferromagnet ...... 297 7.3.2 Antiferromagnet ...... 303 7.3.3 Ferrimagnet ...... 317 7.4 SpinWaves...... 322 7.4.1 Linear Spin Wave Theory for the Isotropic Ferromagnet . 322 7.4.2 “Renormalized” Spin Waves...... 329 7.4.3 Harmonic Approximation for Antiferromagnets ...... 336 7.4.4 Harmonic Approximation for a Ferromagnet with Dipolar Interaction ...... 345 xii Contents

7.5 Thermodynamics of S = 1/2 Ferromagnet ...... 351 7.5.1 Tyablikov Decoupling ...... 351 7.5.2 Spontaneous Magnetization ...... 354 7.5.3 Thermodynamic Potentials ...... 359 7.6 Thermodynamics of S ≥ 1/2 Ferromagnets ...... 361 7.6.1 Green’s Functions ...... 361 7.6.2 Spontaneous Magnetization ...... 363 7.6.3 TheCallenMethod...... 371 7.7 Problems...... 381 References ...... 386

8 Hubbard Model ...... 387 8.1 Introduction ...... 387 8.2 Model for Band Magnets ...... 388 8.2.1 SolidasaMany-BodySystem...... 388 8.2.2 Electrons in Narrow Energy Bands ...... 389 8.2.3 Hubbard Model ...... 393 8.3 Stoner Model ...... 395 8.3.1 Stoner Ansatz (Ferromagnet) ...... 395 8.3.2 Stoner Excitations ...... 397 8.3.3 Magnetic Phase Transition ...... 398 8.3.4 Static Susceptibility ...... 405 8.4 Exact Statements and General Properties ...... 409 8.4.1 MerminÐWagner Theorem ...... 410 8.4.2 TheInfinitelyNarrowBand...... 415 8.4.3 The Two-Site Model ...... 420 8.4.4 The Exactly Half-Filled Band...... 427 8.4.5 Strong-Coupling Regime ...... 431 8.4.6 Spectral Moments ...... 437 8.4.7 High-Energy Expansions ...... 439 8.4.8 Weak-Coupling Regime ...... 441 8.4.9 InfiniteDimensions...... 444 8.4.10 Effective“impurity”-Problem...... 448 8.5 Magnetism and Electronic Correlations ...... 451 8.5.1 Hubbard-I Approximation ...... 451 8.5.2 InterpolationMethod...... 454 8.5.3 Correlation Effects and Ferromagnetism ...... 455 8.5.4 Criterion for Ferromagnetism ...... 457 8.5.5 Static Susceptibility and Ferromagnetism ...... 461 8.5.6 Spin-Dependent Band Shift ...... 464 8.5.7 QuasiparticleDamping...... 470 8.5.8 Dynamical Mean Field Theory ...... 475 8.5.9 Modified Perturbation Theory ...... 479 Contents xiii

8.5.10 Curie Temperature, Magnetization and Static Susceptibility ...... 482 8.6 Problems...... 485 References ...... 490

A Second Quantization ...... 491 A.1 IdenticalParticles...... 492 A.2 Continuous Fock Representation ...... 494 A.2.1 SymmetrizedMany-ParticleStates...... 494 A.2.2 ConstructionOperators...... 495 A.2.3 Many-BodyOperators...... 497 A.3 Discrete Fock Representation (Occupation Number Representation)...... 501 A.3.1 SymmetrizedMany-ParticleStates...... 501 A.3.2 ConstructionOperators...... 503 A.4 Examples ...... 506 A.4.1 BlochElectrons...... 506 A.4.2 Wannier Electrons ...... 508 A.4.3 DensityOperator...... 509 A.4.4 CoulombInteraction...... 510 A.5 Problems...... 511

B The Method of Green’s Functions ...... 515 B.1 Linear Response Theory ...... 515 B.1.1 KuboFormula ...... 515 B.1.2 Magnetic Susceptibility ...... 518 B.1.3 Dielectric Function ...... 520 B.2 Spectroscopies and Spectral Densities ...... 523 B.3 Double-Time Green’s Functions ...... 528 B.3.1 DeﬁnitionsandEquationsofMotion...... 528 B.3.2 Spectral Representations ...... 531 B.3.3 Spectral Theorem ...... 534 B.3.4 Spectral Moments ...... 536 B.3.5 Kramer’sÐKronigRelations...... 537 B.3.6 SimpleApplications ...... 539 B.4 The Quasiparticle Concept ...... 545 B.4.1 InteractingElectrons...... 545 B.4.2 Electronic Self-energy ...... 548 B.4.3 Quasiparticles...... 551 B.4.4 QuasiparticleDensityofStates ...... 555 B.4.5 Thermodynamics ...... 558 B.5 Problems...... 559

C Solutions to Problems ...... 563

Index ...... 743