利用Mathematica解决固态物理与光子学中的群论问题

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"Group Theory in Solid State Physics and Photonics: Problem Solving with mma" 是一本由 Wolfram Hergert 和 R. Matthias Geilhufe 合著的专业书籍，旨在通过 Mathematica 工具来深入理解和解决固体物理学和光子学中的问题。这本书的独特之处在于它不仅介绍了群论的基础知识，还将其应用于固态物理（特别是材料电子结构）和光子学领域，特别是针对光子晶体。在第一部分，书中详细阐述了群论的基础概念，不仅覆盖了标准示例，还拓宽了应用范围，以展示群论在各种情况下的适用性。这部分旨在为读者建立坚实的理论基础。第二部分将这些理论应用于凝聚态物理学，重点是材料的电子结构。通过结合使用 Mathematica 计算机代数系统与传统的手算方法，作者提供了一种更直观、更快速理解复杂概念的方法。这种结合使用软件和传统方法的方式有助于读者深入理解群论的基本原理，并能将其应用于更广泛的领域。第三部分，作者将注意力转向光子学，尤其是二维和三维光子晶体的应用。这部分内容扩展了群论在光子学中的应用，同时也探讨了光子学领域的其他相关问题。书中提到的 Mathematica 包 GTPack 可在书的官方网站上下载，该包支持分析计算、数值计算和可视化，为读者提供了强大的工具，以解决他们自己研究中的复杂任务。书中所有的例子，无论简单还是复杂，都展示了如何使用 Mathematica 进行操作，这为读者提供了一种直观的学习方式，使他们能够将所学知识应用于实际问题。同时，作者提醒读者，虽然书中示例侧重于基础知识，但这些工具同样适用于读者在个人研究中遇到的更复杂的挑战。 "Group Theory in Solid State Physics and Photonics: Problem Solving with mma" 是一本创新性的教材，它将数学工具与物理理论紧密结合，为学习者提供了一种新的理解和解决问题的途径，特别是在固态物理和光子学这两个领域。借助 Mathematica 的强大功能，读者可以更有效地掌握群论并将其应用于实际问题，无论是固态材料的性质探索，还是新型光子器件的设计和分析。

Introduction

When the original German version was ﬁrst published in 1931, there was a

great reluctance among physicists toward accepting group theoretical ar-

guments 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 re-

luctance.

E.P. Wigner (Group eory, 1959)

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.

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, andMusic

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 H’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 envi-

ronment 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.

Group eory in Solid State Physics and Photonics, First Edition. Wolfram Hergert and R. Matthias Geilhufe.

KGaA.

31Introduction

Physics

The conservation laws in classical mechanics are closely related to symmetry. Ta-

ble 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 N theorem.

That symmetry principles are the primary features that constrain dynamical laws

was one of the great advances of E 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 be-

tween quantum theory and group theory was understood. Especially E. W

revealed the role of symmetry in quantum mechanics and discussed the applica-

tion of group theory in a series of papers between 1926 and 1928 [11] (see also

H. W 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 de-

generacy of 2l + 1(l – angular momentum quantum number) because the angular

momentum is conserved due to the rotational symmetry of the potential. Howev-

er, considering the hydrogen atom a higher ‘accidental’ symmetry can be found,

wherelevelshaveadegeneracyofn

, the square of the principle quantum num-

ber. The reason was revealed by P [13, 14] in 1926 using the conservation

of the quantum mechanical analogue of the L–R vector and by F

in 1935 by the comparison of the S equation in momentum space

with the integral equation of four-dimensional spherical harmonics [15]. F

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 quan-

tum theory and electromagnetism are time dependent, i.e., dynamic equations.

Therefore, the symmetry properties of the physical systems as well as the symme-

try properties of the fundamental equations have to be taken into account.

Table 1.1 Conservation laws and symmetry in classicalmechanics.

Symmetry propertyConserved 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 G transformations ⇒ Center of gravity

51.1 Symmetries in Solid-State Physics andPhotonics

Next to the optimization of numerical calculations, group theory can be ap-

plied to classify promising systems for further investigations, like in the case of the

search for multiferroic materials [19, 20]. In general, four primary ferroic prop-

erties are known: ferroelectricity, ferromagnetism, ferrotoroidicity, and ferroelas-

ticity. The magnetoelectric coupling, of special interest in applications, is a sec-

ondary ferroic eﬀect. The occurrence of multiple ferroic properties in one phase

is connected to speciﬁc symmetry conditions a material has to accomplish.

Defects in solids and at solid surfaces play a continuously increasing role in ba-

sic research and applications (diluted magnetic semiconductors, p-magnetism in

oxides). For example, group theory allows to get useful information in a general

and eﬃcient way (cf. [21, 22]) treating defect states in the framework of perturba-

tion theory.

More recently, a close connection between high-energy physics and condensed

matter physics has been established, where eﬀective elementary excitations with-

in a crystal behave as particles that were formally described in elementary particle

physics. A promising class of materials are Dirac materials like graphene, where

the elementary electronic excitations behave as relativistic massless Dirac fermi-

ons [23, 24]. Degeneracies and crossings of energy bands within the electronic

band structure together with the dispersion relation in the neighborhood of the

crossing point are closely related to the crystalline symmetry [25, 26].

In Figure 1.2b, a scanning electron microscope (SEM) image of macroporous

silicon is shown. The special etching technique provides a periodically structured

dielectric material that is referred to as a photonic crystal. The propagation of

electromagnetic waves in such structures can be calculated starting from M-

’s equations [27, 28]. The resulting eigenmodes of the electromagnetic ﬁeld

are closely connected to the symmetry of the structured dielectric. Group theory

can be applied in various cases within the ﬁeld of photonics. Subsequently, a few

examples are mentioned. The photonic bands of two-dimensional photonic crys-

tals can be classiﬁed with respect to the symmetry of the lattice. The symmetry

properties of the eigenmodes, found by means of group theory, decide whether

this mode can be excited by an external plane wave [29]. Metamaterials are com-

posite materials that have peculiar electromagnetic properties that are diﬀerent

from the properties of their constituents. Group theory can be used for design

and optimization of such materials [30]. Group theoretical arguments also help

to discuss the dispersion in photonic crystal waveguides in advance. Clearly, this

approach represents a more sophisticated strategy in comparison to relying on a

trial and error approach [31, 32]. If a magneto-optical material is used for a pho-

tonic crystal, time-reversal symmetry is broken due to the intrinsic magnetic ﬁeld.

In this case, the theory of magnetic groups can be used to study the properties of

such systems [33].

The goal of this book is to discuss the variety of possible applications of com-

putational group theory as a powerful tool for actual research in photonics and

electronic structure theory. Speciﬁc examples usingthe Mathematica package

GT-

Pack

will be provided.

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