Fundamentals of the Physics of Solids / 16-back-matter

638 D Fundamentals of Group Theory

Since the character is the same for elements in the same class, the sum over the group elements can be replaced by a sum over the conjugacy classes. Denoting the number of elements in the ith class by gi, and the character in the μth irreducible representation of the elements in the ith class by χ(iμ),

r	χi(ν)	= gδμν .	(D.1.23)
giχi(μ)

i=1

The characters of the irreducible representations satisfy another orthogonality relation,

r	χj(μ)	= g δij .	(D.1.24)
χi(μ)

μ=1 gi

For the point groups most often encountered in solid-state physics, the characters of the irreducible representations can be looked up in a various references. The present appendix presents one such example: the character table of the group Oh is given in Table D.1 on page 648.

D.1.5 The Reduction of Reducible Representations

According to (D.1.22), the characters of irreducible representations must satisfy the relation

1
	|χ(μ)(R)|2 = 1 .	(D.1.25)
g
	RG

This necessary condition is at the same time the su cient condition for the irreducibility of the representation. If the characters χ(R) of a representation do not satisfy the relation

1
	|χ(R)|2 = 1	(D.1.26)
g
	RG

then the representation is reducible.

The matrix D of any reducible representation D can be block diagonalized with a suitably chosen unitary transformation U in such a way that the matrices of the irreducible representations appear in the individual blocks. The reducible representation is then said to have been decomposed into the direct sum of irreducible representations:

D(R) = D(1)(R) D(1)(R) · · · D(2)(R) D(2)(R) . . .

r	(D.1.27)
= nμD(μ)(R) .
μ=1

It follows from this decomposition that the characters of the reducible representation can be expressed with the characters of the irreducible representations:

D.1 Basic Notions of Group Theory

639

r
χ(R) = nμχ(μ)(R) .	(D.1.28)
μ=1

Since their character is the same, for any element in the ith conjugacy class

	r
χi =	nμχi(μ) .	(D.1.29)

μ=1

The number nμ, which specifies how many times the irreducible representation D(μ) appears in the reduction formula, can be determined from the relation

	nμ =	1		χ(μ) (R)χ(R) =	1	r	giχi(μ) χi .	(D.1.30)
		g RG			g i=1

Based on this formula, the reduction of an arbitrary representation is straightforward when its character and the characters of the irreducible representations are known.

D.1.6 Compatibility Condition

In applications, where certain symmetries are broken by the perturbing potential the relationship between the irreducible representations of a group and of one of its subgroups may become particularly important. Consider a group G and a subgroup G1 thereof. An irreducible representation of the group G may be reducible on the elements of the subgroup G1. Usually only a small number of the irreducible representations of the subgroup G1 appear in the reduction formula. These irreducible representations of the subgroup are said to be compatible with the corresponding irreducible representations of the group

G.

D.1.7 Basis Functions of the Representations

The representation matrices were defined to act on a linear space. However, no significance has been attached to the basis vectors of the space up to now. Physical applications are especially interesting when group elements act on a function space rather than an abstract linear space. Then the basis functions of irreducible representations exhibit special properties.

To define the action of the group elements on a function space, consider the coordinate transformations that are of particular importance in solid-state physics. The coordinate transformation R, which moves point r into r = Rr, is associated with a linear operator O(R) that acts on the space of functions

and takes the function ψ(r) into ψ (r)
O(R)ψ(r) = ψ (r) .	(D.1.31)

640 D Fundamentals of Group Theory

The transformation is defined by the requirement that the transformed function take the same value at r as the original function at r, that is

ψ (r ) = O(R)ψ(r ) = ψ(r) ,	(D.1.32)
which implies
O(R)ψ(r) = ψ(R−1r) .	(D.1.33)

The operators O(R) form a group with the same multiplication rule as the group of coordinate transformations. Since the mapping of the group of coordinate transformations on the group of these operators is homomorphic, any representation of the group of operators is also a representation of the group of coordinate transformations.

Now select an arbitrary function ψ(r), and act on it with the operators O(R) that belong to a group G of coordinate transformations. When the group is of order g, the number of functions obtained this way is also g. However, they are not necessarily linearly independent. In the space of transformed functions a linearly independent set of basis functions (φ1, φ2, . . . , φr ) can be chosen. Expanding ψ(r) in terms of these,

r
ψ(r) = aiφi(r) .	(D.1.34)
i=1
Applying the transformation O(R) on the basis functions,
r
O(R)φi (r) = φi(R−1r) = Dji(R)φj (r) .	(D.1.35)

j=1

The matrix D(R) in the previous formula provides a representation of the group on the space spanned by the functions φi(r). When the transformed functions ψ(R−1r) are also expanded using this basis, the coe cients in the formula

	r
ψ(R−1r) =
	bi(R)φi(r)	(D.1.36)
	i=1
and in the expansion of ψ(r) are found to be related by
	r
bi(R) =	Dij (R)aj .	(D.1.37)
j=1

The representation generated above is not necessarily irreducible, therefore an additional reduction procedure may be required. It may then be established that any function ψ(r) can be expanded into suitably chosen basis functions of the irreducible representations,

dμ
ψ(r) = ai(μ)φi(μ)(r) .	(D.1.38)
μ i=1

D.1 Basic Notions of Group Theory

641

There is a certain arbitrariness in the choice of the basis functions of the irreducible representations since they are not unambiguously determined. However, regardless of the particular choice of the basis functions, some im-

portant relationships among them are always valid.
	coordinate transformations is represented irreducibly on the
If a group G of(μ)		(μ)	(ν)	(ν)	(r), . . .	by the unitary
function spaces φ1		(r), φ2	(r), . . . and φ1	(r), φ2
irreducible representations D(μ)(R) and D(ν)(R), then
φi(μ)	, φj(ν)	= dr φi(μ) (r)φj(ν)(r) = A(μ)δμν δij ,				(D.1.39)

that is, two basis functions with di erent labels are orthogonal. The product of two basis functions with the same label depends only on the choice of the representation but not on the choice of the basis function.

It is not too di cult to construct an operator

P (μ) = dμ			D(μ)	(R)O(R)
ij			ij
		g	RG
such that
P (μ)	φ(ν)(r) = δμν δjk φ(μ)(r) .
ij	k			i
The diagonal elements
P (μ) = dμ			D(μ)	(R)O(R)
i			ii
		g	RG

(D.1.40)

(D.1.41)

(D.1.42)

of this operator behave as projection operators. When acting on an arbitrary function ψ(r), they project out the part proportional to the ith basis function of the μth irreducible representation:

P	(μ)	ψ(r) = a(μ)	φ(μ)	(r) .	(D.1.43)
	i	i	i

D.1.8 The Double Group

In addition to the position coordinate r, the wavefunction of electrons may also contain the spin variable σ. Writing the wave function as a two-component spinor,

ψ(r, σ) =	ψ−(r) .	(D.1.44)
	ψ+(r)

The action of an operator O(R) associated with a coordinate transformation R consists not only in the inclusion of R−1r in the argument of the wavefunction but also in the mixing of the two components:

O(R)ψσ (r) = Dσ1/σ2(R)ψσ (R−1r) .

(D.1.45)

σ

642 D Fundamentals of Group Theory

The form of the matrix Dσ1/σ2 (R) valid for arbitrary rotations is given in Appendix F. Here we shall content ourselves with the remark that a rotation through 2π around any axis takes the spinor not into itself but into its negative:

O Cn (2π)	ψ−(r)	= −	ψ−(r) .	(D.1.46)
	ψ+(r)		ψ+(r)

The initial state is recovered only after a rotation through 4π. Likewise, since mirror reflection can be considered as the product of an inversion and a rotation through π, two subsequent mirror reflections take the spinor into its negative. Therefore we introduce a new symmetry operation, denoted by E, which corresponds to a rotation through 2π around any axis. In the spinor space this is represented by the matrix

					(D.1.47)
			0	−1
O E =			−1	0 .

If for each transformation R the operation R = RE = ER is also allowed, then the corresponding operator O(R) acts on the spinor according to the formula

O	R ψσ (r) = −		Dσ1/σ2	(R)ψσ (R−1r) .	(D.1.48)

σ

The transformations R and R – as well as the corresponding operators – defined in this way form a group. Since this contains twice as many elements as the group G, this is called the double group1 G of the group G.

The irreducible representations and the character table of this group can be determined using the methods discussed above. Rotations that di er by 2π are associated with the same character in certain representations and with opposite characters in others. They are called singleand double-valued representations, respectively. Double-valued representations become important only when spin–orbit interactions have to be taken into account, too. Otherwise even for electrons it is su cient to know the single-valued representations.

D.1.9 Continuous Groups

Up to now we have always considered groups that contain a finite number of elements, since crystalline solids are invariant only under discrete translations and discrete rotations. However, since the angular momentum discussed in Appendix F is related to the continuous rotation group, we shall briefly present some concepts specific to continuous groups.

O(n), the orthogonal group of degree n is a group of real n × n matrices such that

AAT = I ,

(D.1.49)

1 H. Bethe, 1929.

D.1 Basic Notions of Group Theory

643

where the transpose of a matrix is defined as					AT	ij	= Aji . This is equivalent
to defining O	(n)	as the group of continuous		transformations that conserve the
length of vectors in n-dimensional space, since if
		xi =	Aij xj				(D.1.50)
		j
then, on account of the above property of A,
		x 2 =		x2 .			(D.1.51)
		i	i		i
		i

Therefore the group O(n) contains rotations through an arbitrary angle around axes of arbitrary orientation that pass through a fixed point in n- dimensional space, reflections on the planes that contain the same point, and inversion in the same point.

Owing to the condition imposed on matrix A, det(A) = ±1. By taking only those matrices for which det(A) = 1, one obtains SO(n), the special orthogonal group of degree n. This corresponds to pure rotations in n-space.

U(n), the unitary group of degree n is the group of n ×n complex matrices

such that
AA† = I ,		(D.1.52)
where the adjoint matrix is defined as A†	ij	= Aji . This is equivalent to

defining U(n) as the group of continuous transformations that conserve the

quantity

2

z

in n-dimensional complex space, that is

i

i

z

2

=

zi

2

.

(D.1.53)

i

i

i

On account of the condition imposed on matrix A, det(A) = ±1 here, too. By taking only those matrices for which det(A) = 1, one obtains SU(n), the special unitary group of degree n. In the n = 2 case their most general form is

ab

−b	a	,	(D.1.54)

where a and b are complex numbers and |a|2 + |b|2 = 1, or in another form

e−iζ sin η	e−iξ cos η
eiξ cos η	−eiζ sin η	.	(D.1.55)

These are special cases of continuous Lie groups. The r-parameter Lie

group of coordinate transformations can be given as
xi = fi(x1, . . . , xn, a1, . . . , ar ) .	(D.1.56)
The number of parameters is n(n − 1)/2 for SO(n), n2	for U(n), and n2 −

1 for SU(n) groups. Without loss of generality the parameters aν can be

cαβγ

644 D Fundamentals of Group Theory

chosen in such a way that their zero value should correspond to the identity transformation. For small values δaν of the parameters it is required that up to linear order the transformation could be written as

xi =	&1 + i ν=1 Iν δaν ' xi .	(D.1.57)
	r

The quantities Iν (ν = 1, . . . , r) in the previous expression are the generators of the transformation, since it can be shown that for arbitrary values of the parameters the new coordinates can be expressed in terms of these:

	r	$ xi .
xi	= exp #i ν=1 Iν aν		(D.1.58)

As usual, an operator O(R) is associated with the coordinate transformation in such a way that

O(R)ψ(r) = ψ(R−1r) ,		(D.1.59)
that is for ψ(r) = r	&1 − i ν=1 Iν δaν ' r .
O(R)r =		(D.1.60)
	r

The generators therefore appear as n × n matrices. For finite values of the

parameters,	#−i ν=1 Iν aν $ r .	(D.1.61)
O(R)r = exp
	r

The commutator of two generators is also a generator, and the coe cients present in the commutator

		r
			(D.1.62)
Iα, Iβ	= i	cαβγ Iγ

γ=1

are the structure constants of the Lie group’s Lie algebra.

For rotations in the (x, y) plane, when the transformation is characterized by a single parameter, the angle φ of the rotation,

x	= x cos φ	−	y sin φ ,
y			(D.1.63)
	= x sin φ + y cos φ ,

a single generator appears:

I =	1	x	∂	− y	∂	=	1 ∂			,	(D.1.64)
	i		∂y		∂x		i		∂φ

D.1 Basic Notions of Group Theory

645

since

eiφI x = x cos φ − y sin φ ,

(D.1.65)

eiφI y = x sin φ + y cos φ .

Note that I is just the z component of the dimensionless angular momentum operator. Writing the generator as a 2 × 2 matrix,

I =		0 −i		,	(D.1.66)
		i	0
and obviously	cos φ
sin φ
cos φ	− sin φ			= e−iφI .	(D.1.67)

For the group SO(3) the three generators belong to the infinitesimal rotations around the three axes. Rotation through angle φ around the z-axis corresponds to the transformation

x	= x cos φ	−	y sin φ ,
y
	= x sin φ + y cos φ ,			(D.1.68)
z = z ;

its generator is

Iz =	1		∂	− y	∂		(D.1.69)
		x				.
	i		∂y		∂x

The generators of rotations around the x- and y-axes are, likewise,

Ix = i	y ∂z		− z ∂y		,	Iy = i	z ∂x		− x ∂z .		(D.1.70)
1		∂		∂		1		∂		∂

It is readily seen that the generators of the rotation group are the same as the components of the angular momentum operator. Their commutator is known,

		3
			(D.1.71)
Iα, Iβ	= i	αβγ Iγ ,

γ=1

where αβγ is the completely antisymmetric (Levi-Civita) tensor.

Based on (D.1.60), the generators Iα can be represented by 3 × 3 matrices

as			−i									i			.
Ix =		0 0		,	Iy =		0	0 0		,	Iz =		0	0		(D.1.72)
		0 0	0				0	0 i				0	−i 0
	0 i		0			−i 0 0						0	0	0

The structure constant of the SU(2) group is also the Levi-Civita symbol, and the three generators satisfy the commutation relation

646	D Fundamentals of Group Theory
		3
				(D.1.73)
	Iα, Iβ	= 2i	αβγ Iγ ,

γ=1

while for SU(3) the structure constant cαβγ that relates the eight generators

is totally antisymmetric in its three indices. Its nonvanishing components are

√

c123 = 2, c147 = c165 = c246 = c257 = c345 = c376 = 1, and c458 = c678 = 3.

The simplest, fundamental representation of the SU(2) generators are the three 2 × 2 Pauli matrices,

	1 0		i 0			0 −1		(D.1.74)
σx =	0 1	,	σy = 0 −i	,	σz =	1 0	,

while that of the SU(3) group is given by the eight Gell-Mann matrices:2

λ1	=	1 0 0			,	λ2	=	i 0 0					,	λ3	=	0 −1 0				,
			0 1 0						0	−i 0							1	0	0
	=	0 0 0					=	0 0 0							=	0		0	0	(D.1.75)
λ4			0 0 0		,	λ5			0	0	0		,	λ6			0	0 1	,
			0 0 1						0	0	−i						0	0 0

1 0 0						i 0 0		0 1	0 1 0
λ7	=	0 0 −i			,	λ8 = √			0 .
		0	0	0		1		1 0	0
		0 i		0				0 0 −2
							3

D.2 Applications of Group Theory

While in specific calculations it is important to consider the group that corresponds to the true symmetries of the crystal, for simplicity, we shall almost invariably assume cubic symmetry in our examples. Therefore below we shall give the irreducible representations for the cubic Oh group. Then we shall list a number of statements and theorems that arise from the connection between group theory and quantum mechanics, and that are particularly useful in solid-state physics applications.

D.2.1 Irreducible Representations of the Group Oh

The rotations and rotoinversions of the group Oh are listed in Tables 5.1 and 5.4. From the multiplication law for symmetry elements it may be shown that rotations through 90◦, 120◦, and 180◦ belong to separate classes – and even

2Murray Gell-Mann (1929–) was awarded the Nobel Prize in 1969 “for his contributions and discoveries concerning the classification of elementary particles and their interactions”.

D.2 Applications of Group Theory

647

among 180◦ rotations separate classes are formed by rotations around edges and face diagonals. Since the identity element always constitutes a separate class, the 24 rotations can be divided into 5 classes:

1.the identity element E;

2.the 6 fourfold rotations, C4m and C43m (m = x, y, z);

3.the 3 twofold rotations C2m around the same axes;

4.the 6 twofold rotations C2p (p = a, b, c, d, e, f ) around the face diagonals;

5.the 8 threefold rotations C3j and C32j (j = a, b, c, d) around the space diagonals.

The other 24 elements – which can be obtained from rotations via a multiplication by inversion – form 5 classes analogously: just the elements of the above classes need to be multiplied by the inversion element.

As for finite groups the number of irreducible representations is the same as the number of classes, the group Oh has 10 irreducible representations. Using (D.1.20), the relation for the dimensionality of the irreducible representations, the equation

10

d2	= 48	(D.2.1)
i

i=1

can be satisfied by integers if there are 4 one-, 2 two-, and 4 three-dimensional irreducible representations.

Two conventions are used in the literature for the notation of irreducible representations. The first one is due to L. P. Bouckaert, R. Smoluchowski, and E. P. Wigner, who determined in 1936 the irreducible representations of the groups that are of importance in solid-state physics. A letter denotes the point of the Brillouin zone for which the group in question is the symmetry group. An additional number shows the dimensionality of the representation, and a prime ( ) if the representation is odd under inversion. In the other notation (called the chemical notation) the letters A, E, and T show the dimensionality of the representation, while the indices g and u show whether the representation is even (in German: gerade) or odd (ungerade). Using the orthogonality relations (D.1.23) and (D.1.24) of the characters of the irreducible representations, the characters can be determined. The character table of the group Oh is given in Table D.1.

Now consider the irreducible representations of the double group. We have seen that the group O that contains the 24 rotational symmetries of the cube has 5 irreducible representations. However, the corresponding 48-element double group O has only 8 irreducible representations (i.e., its elements can be divided into 8 classes). Besides the identity element E, the element E that corresponds to a rotation through 2π constitutes a group on its own. New classes are formed also by the rotations through π/2 + 2π and 2π/3 + 2π, however rotations through π and π + 2π belong to the same class. Of the 8 representations 5 are single-valued – these are the same as the representations