Group Theory in Solid State Physics and Photonics

Problem Solving with Mathematica

Wolfram Hergert, R. Matthias Geilhufe

• Publisher: Wiley-VCH
• Year: 2018
• ISBN:
  9783527413010
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Condensed Matter

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.

1
Introduction

When the original German version was first published in 1931, there was a great reluctance among physicists toward accepting group theoretical arguments and the group theoretical point of view. It pleases the author, that this reluctance has virtually vanished in the meantime and that, in fact, the younger generation does not understand the causes and the bases of this reluctance.

E.P. Wigner (Group Theory, 1959)

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Symmetry is a far-reaching concept present in mathematics, natural sciences and beyond. Throughout the chapter the concept of symmetry and symmetry groups is motivated by specific examples. Starting with symmetries present in nature, architecture, fine arts and music a transition will be made to solid state physics and photonics and the symmetries which are of relevance throughout this book. Finally the square is taken as a first explicit example to explore all transformations leaving this object invariant.

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Symmetry and symmetry breaking are important concepts in nature and almost every field of our daily life. In a first and general approach symmetry might be defined as: Symmetry is present when one cannot determine any change in a system after performing a structural or any other kind of transformation.

Nature, Architecture, Fine Arts, and Music

One of the most fascinating examples for symmetry in nature is the manifold and beauty of the mineral skeletons of Radiolaria, which are tiny unicellular species. Figure 1.1a shows a table from HAECKEL’s “Art forms in Nature” [4] presenting a special group of Radiolaria called Spumellaria.

The concept of symmetry can also be found in architecture. Our urban environment is characterized by a mixture of buildings of various centuries. However, every epoch reflects at least some symmetry principles. For example, the Art déco style buildings, like the Chrysler Building in New York City (cf. Figure 1.1b), use symmetry as a design element in a particularly striking manner.

Figure 1.1 Symmetry in nature and architecture. (a) Table 91 from HAECKEL’s ‘Art forms in Nature’ [4]; (b) Chrysler Building in New York City [5] (© JORGE ROYAN, www.royan.com.ar, CC BY-SA 3.0).

Within the fine arts, the works of M.C. ESCHER (1898–1972) gain their special attraction from an intellectually deliberate confusion of symmetry and symmetry breaking.

In ESCHER’s woodcut Snakes [6], a threefold rotational symmetry can be easily detected in the snake pattern. A rotation by 120^◦ transforms the painting into itself. A considerable amount of his work is devoted to mathematical principles and symmetry. The series “Circle Limits” deals with hyperbolic regular tessellations, but they are also interesting from the symmetry point of view. The woodcut, entitled Circle Limit III [6], the most interesting under the four circle limit woodcuts, shows a twofold rotational axis. If the figure is transformed into a black and white version a fourfold rotational axis appears. Obviously, the color leads to a reduction of symmetry [7]. The change of symmetry by inclusion of additional degrees of freedom like color in the present example or the spin, if we consider a quantum mechanical system, leads to the concept of color or SHUBNIKOV groups. A comprehensive overview on symmetry in art and sciences is given by SHUBNIKOV [8]. WEYL [9] and ALTMANN [10] start their discussion of symmetry principles from a similar point of view.

Also in music symmetry principles can be found. Tonal and temporal reflections, translations, and rotations play an important role. J.S. BACH’s crab canon from The Musical Offering (BWV1079) is an example for reflection. The brilliant effects in M. RAVEL’s Boléro achieved by a translational invariant theme represent an impressive example as well.

Physics

The conservation laws in classical mechanics are closely related to symmetry. Table 1.1 gives an overview of the interplay between symmetry properties and the resulting conservation laws.

A general formulation of this connection is given by the NOETHER theorem. That symmetry principles are the primary features that constrain dynamical laws was one of the great advances of EINSTEIN in his annus mirabilis 1905 [11]. The relevance of symmetry in all fields of theoretical physics can be seen as a major achievement of twentieth century physics.

In parallel to the development of quantum theory, the direct connection between quantum theory and group theory was understood. Especially E. WIGNER revealed the role of symmetry in quantum mechanics and discussed the application of group theory in a series of papers between 1926 and 1928 [11] (see also H. WEYL 1928 [12]). Symmetry accounts for the degeneracy of energy levels of a quantum system. In a central field, for example, an energy level should have a degeneracy of 2l +1 (l – angular momentum quantum number) because the angular momentum is conserved due to the rotational symmetry of the potential. However, considering the hydrogen atom a higher ‘accidental’ symmetry can be found, where levels have a degeneracy of n², the square of the principle quantum number. The reason was revealed by PAULI [13, 14] in 1926 using the conservation of the quantum mechanical analogue of the LENZ–RUNGE vector and by FOCK in 1935 by the comparison of the SCHRÖDINGER equation in momentum space with the integral equation of four-dimensional spherical harmonics [15]. Fock showed that the electron effectively moves in an environment with the symmetry of a hypersphere in four-dimensional space. The symmetry of the hydrogen atom is mediated by transformations of the entire Hamiltonian and not of its parts, the kinetic and the potential energy alone. Such dynamical symmetries cannot be found by the analysis of forces and potentials alone. The basic equations of quantum theory and electromagnetism are time dependent, i.e., dynamic equations. Therefore, the symmetry properties of the physical systems as well as the symmetry properties of the fundamental equations have to be taken into account.

Table 1.1 Conservation laws and symmetry in classical mechanics.

Symmetry property		Conserved quantity
Homogeneity of time (translations in time)	⇒	Energy
Homogeneity of space (translations in space)	⇒	Momentum
Isotropy of space (rotations in space)	⇒	Angular momentum
Invariance under Galilei transformations	⇒	Center of gravity

1.1 Symmetries in Solid-State Physics and Photonics

In Figure 1.2, two representative examples of solid-state systems are shown. The scanning t...

Table of contents

• Cover
• Table of Contents
• Preface
• 1 Introduction
• Part One: Basics of Group Theory
• Part Two: Applications in Electronic Structure Theory
• Part Three: Applications in Photonics
• Part Four: Other Applications
• Appendix A: Spherical Harmonics
• Appendix B: Remarks on Databases
• Appendix C: Use of MPB together with GTPack
• Appendix D: Technical Remarks on GTPack
• References
• Index
• End User License Agreement