Group Theory in Solid State Physics and Photonics

Das Prinzip der Gruppentheorie ist gut einführt. Dennoch werden in diesem Lehrbuch zwei komplett neue Aspekte vorgestellt: besseres Verständnis durch die Konzentration auf Problemlösungsstrategien und der weitreichende Einsatz von Mathematica sowie die Einführung eines neuen Tools für die Photonik durch die Anwendung von Konzepten auf photonische Fasern und Kristalle.

Autorentext
Wolfram Hergert, extraordinary professor in Computational Physics, is member of the Theoretical Physics group at University Halle-Wittenberg, Germany. Main subjects of his work are solid state theory, electronic and magnetic structure of nanostructures and photonics. Prof. Hergert has experience in teaching group theory and in applying Mathematica to physical problems. He has published in renowned journals, like Nature and Physical Review Letters, and edited a books on Computational Materials Science and Mie Theory. He is also coauthor of a book on Quantum Theory.

Matthias Geilhufe studied physics at the Martin Luther University Halle-Wittenberg (Germany) with specialization in theoretical and computational physics. From 2012-2015 he was employed as a PhD student at the Max Planck Institute of Microstructure Physics in Halle. In 2015 he obtained his PhD at the Martin Luther University Halle-Wittenberg. Currently, he is working at the Nordita Institute in Stockholm, Sweden. His work is based on the investigation of electronic and magnetic properties of complex materials. For his research, methods based on group theory or density functional theory are applied.

Zusammenfassung
While group theory and its application to solid state physics is well established, this textbook raises two completely new aspects. First, it provides a better understanding by focusing on problem solving and making extensive use of Mathematica tools to visualize the concepts. Second, it offers a new tool for the photonics community by transferring the concepts of group theory and its application to photonic crystals.
Clearly divided into three parts, the first provides the basics of group theory. Even at this stage, the authors go beyond the widely used standard examples to show the broad field of applications. Part II is devoted to applications in condensed matter physics, i.e. the electronic structure of materials. Combining the application of the computer algebra system Mathematica with pen and paper derivations leads to a better and faster understanding. The exhaustive discussion shows that the basics of group theory can also be applied to a totally different field, as seen in Part III. Here, photonic applications are discussed in parallel to the electronic case, with the focus on photonic crystals in two and three dimensions, as well as being partially expanded to other problems in the field of photonics.
The authors have developed Mathematica package GTPack which is available for download from the book's homepage. Analytic considerations, numerical calculations and visualization are carried out using the same software. While the use of the Mathematica tools are demonstrated on elementary examples, they can equally be applied to more complicated tasks resulting from the reader's own research.

Inhalt

Preface VII

1 Introduction 1

1.1 Symmetries in Solid-State Physics and Photonics 4

1.2 A Basic Example: Symmetries of a Square 6

Part One Basics of Group Theory 9

2 Symmetry Operations and Transformations of Fields 11

2.1 Rotations and Translations 11

2.1.1 Rotation Matrices 13

2.1.2 Euler Angles 16

2.1.3 EulerRodrigues Parameters and Quaternions 18

2.1.4 Translations and General Transformations 23

2.2 Transformation of Fields 25

2.2.1 Transformation of Scalar Fields and Angular Momentum 26

2.2.2 Transformation of Vector Fields and Total Angular Momentum 27

2.2.3 Spinors 28

3 Basics Abstract Group Theory 33

3.1 Basic Definitions 33

3.1.1 Isomorphism and Homomorphism 38

3.2 Structure of Groups 39

3.2.1 Classes 40

3.2.2 Cosets and Normal Divisors 42

3.3 Quotient Groups 46

3.4 Product Groups 48

4 Discrete Symmetry Groups in Solid-State Physics and Photonics 51

4.1 Point Groups 52

4.1.1 Notation of Symmetry Elements 52

4.1.2 Classification of Point Groups 56

4.2 Space Groups 59

4.2.1 Lattices, Translation Group 59

4.2.2 Symmorphic and Nonsymmorphic Space Groups 62

4.2.3 Site Symmetry, Wyckoff Positions, and WignerSeitz Cell 65

4.3 Color Groups and Magnetic Groups 69

4.3.1 Magnetic Point Groups 69

4.3.2 Magnetic Lattices 72

4.3.3 Magnetic Space Groups 73

4.4 Noncrystallographic Groups, Buckyballs, and Nanotubes 75

4.4.1 Structure and Group Theory of Nanotubes 75

4.4.2 Buckminsterfullerene C60 79

5 Representation Theory 83

5.1 Definition of Matrix Representations 84

5.2 Reducible and Irreducible Representations 88

5.2.1 The Orthogonality Theorem for Irreducible Representations 90

5.3 Characters and Character Tables 94

5.3.1 The Orthogonality Theorem for Characters 96

5.3.2 Character Tables 98

5.3.3 Notations of Irreducible Representations 98

5.3.4 Decomposition of Reducible Representations 102

5.4 Projection Operators and Basis Functions of Representations 105

5.5 Direct Product Representations 112

5.6 WignerEckart Theorem 120

5.7 Induced Representations 123

6 Symmetry and Representation Theory in k-Space 133

6.1 The Cyclic Bornvon Kármán Boundary Condition and the Bloch Wave 133

6.2 The Reciprocal Lattice 136

6.3 The Brillouin Zone and the Group of the Wave Vector k 137

6.4 Irreducible Representations of Symmorphic Space Groups 142

6.5 Irreducible Representations of Nonsymmorphic Space Groups 143

Part Two Applications in Electronic Structure Theory 149

7 Solution of the SCHRÖDINGER Equation 151

7.1 The Schrödinger Equation 151

7.2 The Group of the Schrödinger Equation 153

7.3 Degeneracy of Energy States 154

7.4 Time-Independent Perturbation Theory 157

7.4.1 General Formalism 159

7.4.2 Crystal Field Expansion 160

7.4.3 Crystal Field Operators 164

7.5 Transition Probabilities and Selection Rules 169

8 Generalization to Include the Spin 177

8.1 The Pauli Equation 177

8.2 Homomorphism between SU(2) and SO(3) 178

8.3 Transformation of the SpinOrbit Coupling Operator 180

8.4 The Group of the Pauli Equation and Double Groups 183

8.5 Irreducible Representations of Double Groups 186

8.6 Splitting of Degeneracies by SpinOrbit Coupling 189

8.7 Time-Reversal Symmetry 193

8.7.1 The Reality of Representations 193

8.7.2 Spin-Independent Theory 194

8.7.3 Spin-Dependent Theory 196

9 Electronic Structure Calculations 197 9.1 Solution o...