Marder M.P. Condensed Matter Physics

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Rotational Symmetry—Group Representations 193

Figure 7.8. First Brillouin zone of the fee lattice, with conventional notation for points of

special symmetry. The reciprocal lattice of fee is bec, with lattice spacing of 4π/α, so this

polyhedron is the same as in Figure 2.3(B). In units of 2π/α, Γ = (0 0 0), X = (0 1 0),

L = (l/2 1/2 1/2), W = (1/2

0),

= (3/4 3/4 0), and t/= (1/4

1/4).

capable of inducing transitions from one electron state to another. Predicting the

strength of an allowed transition requires lengthy, detailed, and often question-

able calculation. By contrast, distinguishing between allowed and forbidden tran-

sitions depends upon knowledge of symmetry alone. For this reason, the notation

of group representations decorates experimental diagrams in numerous branches

of condensed matter physics, particularly those involving optical absorption.

Figures 7.8 to 7.10 show the Brillouin zones of the fee, bec, and hexagonal lat-

tices,

annotated with points of special symmetry. For many purposes, it is enough

to refer back to these figures as a reference for the conventional notation, without

knowing the theory of symmetry that accompanies them. The theory is developed

at some length, from various points of view, by Bradley and Cracknell (1972),

Koster (1957), Lyubarskii (1960), Murnaghan (1938), or Tinkham (1964).

Group Operations. To develop the theory, suppose one has any set of symmetry

operations (matrices) {G} that form a group. The definition of a group demands

that the collection {G} satisfy three requirements.

The group must contain the unit matrix, which in this subject is traditionally

denoted by E.

The product of any two matrices G\ and Gj in the group must be another

matrix G3 in the group.

194

Chapter 7. Non—Interacting Electrons in a Periodic Potential

Figure 7.9. First Brillouin zone of the bcc lattice, with conventional notation for points of

special symmetry. The reciprocal lattice of bcc is fee, with lattice spacing of Απ/α, so this

polyhedron is the same as in Figure 2.2(B). In units of 2π/α, Γ = (0 0 0), H = (0 1 0),

N = (1/2 1/2 0), and P= (1/2 1/2 1/2).

Figure 7.10. First Brillouin zone of the hexagonal lattice, with conventional notation for

points of special symmetry. Using coordinates b\ = Απ/α\/3

(\/3/2

—

1/2 0),

t>2

Απ/aVÏ

(λ/3/2 1/2 0) and

2TT/C

(0 0 1) ; Γ = (0 0 0), A = (0 0 1/2), M = (1/2 0 0),

K=(l/3 1/3 0),// = (1/3 1/3 1/2), andL= (1/2 0 1/2).

Rotational Symmetry—Group Representations 195

The inverse of every matrix in the group must also belong to the group.

To obtain a representation of the group {G}, pick some function ipj(r), which

in practice might be a wave function. Next, create all the functions ipj(Gr); that

is,

take W>

ana

" rotate and reflect it using all the symmetry operations in {G}.

Actually, ipj{Gr) rotates ipj through G~\ not G, but because G

is also in the

group, it does not matter. Even if the number of matrices in the group, its

order,

the number / of linearly independent functions ip(r) generated in this way will

generally be less than or equal to h. For each element G in the group, one must

have

{Gr) = Y

A{G)

^j{r). (7.90)

In this fashion, one obtains a matrix A(G) corresponding to each symmetry oper-

ation G; the set of matrices {A} is called a representation of the original group of

symmetry operations. The dimensions of the matrices A generally are quite

dif-

ferent from the dimensions of the original matrices G, because the dimension of

A is determined by the number of linearly independent

V>'s

one happens to obtain

by applying rotations and reflections G to ipj, while all G's are three-dimensional.

Lyubarskii (1960), p. 44, shows that the matrices A can all be taken to be unitary,

so one can assume

/\* =z A as Well as G* = G This is just the definition of a unitary matrix.

(7.91)

in everything that follows.

Two group representations {A

} and {A

(m)

} are equivalent if there exists a

single invertible matrix M such that

(n)

{G)M = A

\G)

This relation has to hold with a single M for

all the different matrices A(G) produced by

varying G.

(7.92)

which means that one obtains {A

(n)

} from

(m)

}

just by changing coordinate sys-

tems.

By making a good choice of M, one will often be able to simplify the matrices

[A] substantially, by bringing all of them into the form

/ an

ain ... aw

>11

bi"\

b\i'

bi"i" Ì

l' and /" need not be the same, and there

could be more than two independent blocks.

(7.93)

196 Chapter 7. Non-Interacting Electrons in a Periodic Potential

The reason that putting all matrices {A} into form (7.93) provides such an advan-

tage is that it means that the wave functions ipj break into two subsets, one contain-

ing

wave functions, the other containing /" wave functions, each of which trans-

forms exclusively into

itself.

The original

—

I' + /"-dimensional representation

has been simplified into an /'-dimensional representation and an /"-dimensional

representation, and the original representation is called reducible.

If,

on the other

hand, no M exists which can put all matrices into the form (7.93) simultaneously,

the representation is called irreducible.

Faced with 24 ten-dimensional matrices, the task of finding whether there ex-

ists some

to bring them into block diagonal form seems rather daunting.

The

whole question can, however, be settled by diagonalizing a single ten-dimensional

matrix. Begin with an absolutely arbitrary matrix X, let {A

(m)

} and {A

(n)

} be any

two representations of {G} both of dimension /, and form

(m)

(G)XA

(G*). (7.94)

Then

{m)

{G)M

= Y^A

(m)

{G)A

\G')XA

{n)

{G'*)

=Σ

A(m)

(

GG>

)

A(n)

(

'*)

SeeProblem6

(m)

(G')

(a)

(G'*G) Change order of summation by

' sending G'^CG'

A(m)

(

A(n)

(G'*)A

(n)

(G)

(G).

Comparing with Eq. (7.92),

first

appears that all representations have been

proved to be equivalent. However, A

(m)

andA

(n)

are only equivalent if M has an in-

verse. One can use M to best advantage by remarking that

also obeys Eq. (7.98),

and therefore so does M +

P, which is Hermitian. So now one has a Hermi-

tian matrix P with

(G)/

(m)

(G). (7.99)

Because P is Hermitian, it can be diagonalized and has an orthonormal basis of real

eigenvalues. Express all the matrices

{m)

(G), A

{n)

(G),

and P in this basis. One is

now

in a

position to decompose any given representation into irreducible pieces

and to show that different irreducible representations are orthogonal.

To accomplish the first task, let n = m be the same, and drop this superscript

for the moment. In the basis where

is diagonal, one has for the

element of

Eq. (7.99)

(G)ijPjj = Pii

(G)ij => (P

-Pjj)A(G)

= 0 (7.100)

So there are two possibilities. Either P,·,·

Pjj, or else A(G)ij

0. However,

even

single off-diagonal element such as A(G)n vanishes for all G, then A

reducible. The reason is this: Saying that A(G)i2 vanishes means that no group

(7.95)

(7.96)

(7.97)

(7.98)

Rotational Symmetry—Group Representations

197

operation transforms ψ\ into

V>2-

This statement means in turn that as one scans

over all the wave functions Οφ\, none of them can be transformed into ^2 either,

for

one could, then the product of the two group operations would take ψ\ into

ip2-

Because the sets of wave functions generated by Ωψ\ and

G1P2

are disjoint, the

representation is reducible.

If there are, say, ten nonzero eigenvalues, of which three have value

\, two

have value

P44,

and the rest are all different from one another, then one has reduced

the original ten-dimensional representation into one three-dimensional represen-

tation, one two-dimensional representations, and five one-dimensional representa-

tions.

If,

on the other hand, {A}

irreducible, then all the

are the same, and

therefore P and M are simply multiples of the unit matrix.

When the matrix

in Eq. (7.95) is chosen randomly, then the representations

uncovered by Eq. (7.100) are almost certainly irreducible. There are definitive

tests that allow one to decide this question with certainty, resulting from the fact

that either two representations are equivalent or else they are orthogonal

one

another, shown as follows:

Suppose the representations

{m)

}

and {A

}

are both irreducible in Eq. (7.99).

Problem 6 shows that from Eq. (7.99) one can also deduce

(m)

{G)PP* = PP*A

\G) => (F$-PJj)A

(m)

(G)ij = 0. (7.101)

Because by assumption {A

(m)

}

irreducible, one cannot allow even

single

off-

diagonal element of

{m)

(G)

to vanish for all G, and accordingly all of the diagonal

elements

P»

are equal. If they are nonzero, then P must be invertible, and accord-

ingly {A

(m>

} and {A

(n)

} are equivalent.

If,

on the other hand, {A

(m)

} and {A

(n)

} are

inequivalent, then all diagonal elements

must vanish, for all starting matrices X

in Eq. (7.94).

It is interesting to examine the consequences of choosing for X a matrix whose

only nonzero element is at position

βγ

Xij =

iß

(7.102)

αδ

= Σ

{m)

(G)

iß

(G*)js (7.103)

G,i,j

= ^AW(G)

(G*)

7ä

. (7.104)

Suppose first that {i4

(m)

} and {Λ

(η)

} are not equivalent. Then according

the

discussion following Eq. (7.101), P and hence M must vanish. Therefore

(m)

(G)

a/3

(

)

(G*)

0. If

{A«»>}

and {A«} are not (7.105)

~? equivalent representations.

If, on the other hand, {A

(m)

} and {A

(n)

} are the same and irreducible, then M must

be a multiple of the unit matrix, and

V A(G)

aß

A(G*)

= C

ßl

αδ

The precise multiple

ßl

of (7.106)

~f the unit matrix depends

upon the choice of βη in

Eq. (7.102).

198

Chapter 7. Non-Interacting Electrons in a Periodic Potential

Y V A(G)

A(G*)-y

ICß^ Evaluate fora

5 and then (7.107)

—'

sum on

See Problem 6, Eq. (7.133).

A{G*G)

ßl

hö

ßl

=►

ßl

= -δ

βΊ

r^n

n0f

(

108

)

L·

h is the number of elements

=»

A(G)

a/3

A(G*)

-fy

. in the group

{G}.

(7 109)

Finally, Eqs. (7.105) and (7.109) can be combined into the grand orthogonality

theorem

u I is the dimension of

the

J2A^(G)

aß

A^(G%

ηηι

βΊ

αδ

Γ^^ΓΞΓο"

(

110

)

G group elements.

7.3.1 Classes and Characters

In the discussion

Section 2.4, transformations divided naturally into various

classes, which were all denoted by the same symbol. For example, all rotations

by 60° about z form one class. The formal definition of a class of operations con-

taining some group element Gì is that

is the set of all transformation matrices

obtained by subjecting G\

G~[

G\Gi, where

ranges over all elements of the

group. The intuitive meaning of this definition is that two transformations belong

to the same class

by changing coordinate systems according to symmetries of

the group, one transformation turns into the other. In this sense, all the symmetry

operations belonging to a class are really the same.

Example:

D^·

Consider the point group

Dj,d

Its symmetry elements are given in

Table 7.2:

Table 7.2. Symmetry elements in

E Identity

i Inversion

{2C3} Rotation about z through ±120°; two elements in class.

{253} Rotation-inversion about z through ±60°; two elements in class.

{3C2} Three two-fold rotation axes.

{3σ} Three mirror planes.

Because the symmetry operations within

class are physically indistinguish-

able from one another, it is valuable to find mathematical features of group repre-

sentations that are constant within a class. The most easily calculated quantity of

this type is the

character

of a representation,

X(G)

(G) =Tr[A(G)]. The character

the trace of

the

matrix

(7.111)

Rotational Symmetry—Group Representations 199

The character of two different members of a class must be the same because for

any G\ and

in the same class, one obtains

v(G|)=Tr[Giì =Tr[GÌG

G3Ì G, and G

are in the same class, by definition, (7.1 12)

if there exists Gj so that G\

G^GiG^,.

= Tr[G

G;] (7.113)

= TriG?l

= Y(GI)

Using the cyclic property of the trace,

(7 114)

J AV

Tr[ASC]=Tr[BCA], easily derived by

writing everything out in component

form.

In terms of the characters

χ,

the orthogonality theorem, Eq. (7.110) can be

recast as

Σ A

{m)

{G)

"\G*)

y<WSo7 (7.115)

GQ7

αη

{m)

(G)x

{

"\G*)=h5

(7.116)

)

(

)

*{C

)

Ιΐδ

ηιη

. The sum on* is over the distinct classes Qof(7.1 17)

*—' G, each of which has M elements.

One use of Eq. (7.116) is to provide a sure test of whether a representation is

irreducible or not. If the representation is irreducible, then taking the traces of all

the matrices that compose it, taking their absolute square, and summing over all

the matrices, one must get just the order of the group h.

If,

on the other hand,

the representation is reducible, when one forms the sum described by Eq. (7.116)

with n

must come out to more than h. The reason for this claim is that

the character of each matrix equals the sum of the characters of the irreducible

representations that compose it. If irreducible representation

occurs

times,

then the character corresponding to G is

ip)

(G), and instead of getting

upon performing the sum in Eq. (7.116), one gets h J2

A final orthogonality relation can be obtained from Eq. (7.117), if one assumes

a theorem whose proof is slightly too lengthy to include here, given by Murnaghan

( 1938)

p. 84, that the number of classes equals the number of irreducible represen-

tations. Given this fact, the characters can be used to form square matrices

{m)

)

= Q

- x^*(C

(7.118)

kQ'

= hö

(7.119)

Q'nkQkm

hÔ

Using the fact that the matrices (7.120)

are square, so Q' = Q

=» Σ HnX

\C„)x

*(C

)=hS

(7.121)

The square matrix Q

is called a

character

table.

All the traces x

{m)

) are

integers; using this fact with Eqs. (7.117) and (7.121), it is often possible to work

out all the entries of Q without further information.

200

Chapter 7. Non-Interacting Electrons in a Periodic Potential

Table 7.3. Character table for the

group D

U 1

-1

-2

—

-1

2/C3 3C2 3/C2

1 1 1

-1 1 -1

1 -1 -1

-1 -1 1

-10 0

1 0 0

Example: D34.

Consider again the point group D^. It has six classes, with a total of 12 sym-

metry elements. One irreducible representation is very easy to find: Map every

element of the group to 1. All the characters χ are

in this representation, giving

the top row of the character table. A column of the character table is easily deter-

mined by examining Eq. (7.121) when n = m and choosing for C

the class of the

identity operation E. In an /-dimensional representation, the identity operation is

always represented by an /-dimensional unit matrix, and its character is just /; that

is,

{m)

{E) = l

. Therefore, according to Eq. (7.121), one obtains

Y,l

(7.122)

When h = 12, and there are six integer /'s over which to sum, there is only one way

to make things work out, which is to have four representations with / =

and two

representations with / = 2. Accordingly, the first column of Table 7.3 is as shown.

There is one other class with only one element, inversion /. The column below

this class can only contain the same integers that lie under E, because the sum of

their squares must still be 12, but negative signs are also allowed. The only way to

make this column orthogonal to the preceding one is to distribute minus signs as

shown. The minus signs could be ordered a bit differently; for example, the first

occurrence of

could be negative rather than the second, but this would in the end

correspond only to relabeling the rows. Proceed next to the two classes with two

elements. The sum of squares of entries in this column must add up to six, a goal

that can only be achieved if all entries are ±

There are only two ways to make

vectors of ± 1 that are orthogonal to the preceding two columns, and these appear

underneath

2C3

and

2/C3.

Finally, one must find two columns of integers where the

sum of squares of the entries adds up to four, and which must again be orthogonal

to all preceding columns.

Notation for Cubic Group O^. A companion of the theory of group represen-

tations is conventional notation for symmetrical functions. The irreducible repre-

sentations of

are referred to particularly frequently because they describe wave

functions and lattice vibrations in cubic crystals, including those with the diamond

structure, as in Figures 23.15 and 23.16. Notation for these irreducible representa-

tions appears in Table 7.4.

Rotational Symmetry—Group Representations 201

Table 7.4. Irreducible representations of 0/,

BSW

Γ,

Γ12

Γ|

Γ,

K d

Vf 1

Γ+ 1

Γ+ 2

Γ+ 3

ιγ

Γί 2

- 3

Basis functions

x\y

-z

)+y\z

-x

)+z\x

-y

)

)/2]ì

y2]

[yz(y

-z

)], [zx{z

-x

)], [xy(x

-z

)}

[xy], [yz], [ζχ]

xyz {x\y

-z

) +y

-x

)+z

-y

)}

xyz

[xyz{z

-(x

)/2}}, [xyz{x

-y

}}

M, [z]

[x(y

-z

)},

[y(z

-x

)], [z(x

-y

Irreducible representations of

Of,,

using the notation of Bouck-

aert, Smoluchowski, and Wigner (1936)—BSW—and of Koster

(1957)—K; also listed are the dimension d of the representation,

along with a set of

basis

functions that transform into each other in

accord with each representation.

7.3.2 Consequences of point group symmetries for Schrödinger's equation

Point group symmetries and the theory of group representations have two imme-

diate implications for solutions of Schrödingers's equation in periodic potentials.

First, they may be used to reduce the computational effort in calculating energy

bands,

and second they may be used to identify places in k space where energy

bands will be degenerate.

The first application is fairly obvious and requires no knowledge of the repre-

sentations. Suppose one has a point group symmetry

G—for

example a rotation

matrix. No physical measurement can vary if one subjects the crystal to the op-

eration of G. In particular, if one begins with a wave function at wave vector k,

φ^ = e'

u^(r), then another wave function at wave vector Gk results from con-

structing

l = e'

U^(G~

r). Note that

~k-G*r

G~k-r

because the dot prod- (7.123)

uct is invariant when both vectors in the prod-

uct are rotated by the same matrix.

The two wave functions ψ^ and ψ

i can be made orthogonal to one another if

need be, and have the same energy eigenvalue. So for all point group symmetries,

£ r = £

. (7.124)

n,k η,υκ

Because the first Brillouin zone is invariant under such point group operations, it

makes sense to divide the Brillouin zone into regions, called irreducible zones,

that can be repeated under point group operations to fill out the complete Brillouin

202

Chapter 7. Non-Interacting Electrons in a Periodic Potential

zone.

Each irreducible zone is a copy of all the others and has identical energy

surfaces, so one can restrict attention to one such zone.

These arguments, however, rely upon the assumption that the symmetry oper-

ation G acting upon k produced a wave vector physically distinguishable from it,

which is not the case if k and Gk differ only by a reciprocal lattice vector. While

this assumption is true in general, it fails at special points in the irreducible zone.

For these points, a symmetry operation that leaves the wave vector unchanged (up

to a reciprocal lattice vector) may or may not produce a new wave function when

it acts on the wave function one begins with. If the new wave function produced

by the symmetry operation is just a multiple of the old one, there is no particularly

interesting conclusion. But if the symmetry operation produces a linearly inde-

pendent wave function, then more than one wave function corresponds to a single

wave vector in the irreducible zone at a given energy; one has a degeneracy.

It is intuitively clear that some points in the Brillouin zone are more symmet-

rical than others. In Figure 7.11, the point Γ is obviously the most symmetrical of

all;

points along T are symmetrical, but not as symmetrical as K. The point q is

not symmetrical at all. The formal definition of the degree of symmetry for a point

in the Brillouin zone is very simple. For a given vector k in the Brillouin zone, the

group of that vector is the group of point group operations

such that

Gk = k-\- K: The conventional notation for symmetry point K (7.125)

and reciprocal lattice vector K are easily confused.

for some reciprocal lattice vector £,·. For example, the points along T are left

invariant when the hexagon is reflected around Γ; the point K is left invariant by

that reflection, and also can be operated upon by two more reflections and two

rotations, which move it to the corners indicated in Figure 7.11. The point Γ is

invariant under all point group operations.

Thus,

the application of group representations to Schrödinger's equation pro-

ceeds by choosing some k, such as K in Figure 7.11, and taking the group of oper-

ations {G} which leaves k invariant up to a reciprocal lattice vector. Once {G} has

been determined, certain consequences follow from its irreducible representations.

The only necessary fact is that to the group corresponds a collection of integers,

which are the dimensions of its representations. At least one of these integers is

If some of these integers are greater than 1, then there must exist some energy

levels at this wave vector with the corresponding degeneracy. Unfortunately, the

information provided by these dimensions is rather like a political spokesman who

raises various possible scenarios, but refuses to confirm or deny any particular one.

Any particular energy level could have any degree of degeneracy made possible

by any of the dimensions. It could even have greater degeneracy than at first ap-

pears possible, if parameters of the Hamiltonian are specially chosen so that two

generically different levels coincide, leading to an accidental degeneracy.

Example: Symmetry Points in Figure 7.11. The symmetry group of Γ is C^

. In-

spection of Table 2.9 shows that this group has six classes: the identity operation,