Download Physical applications of group theory

PHYSICAL APPLICATIONS OF GROUP THEORY
TODD KARIN
Abstract. This paper summarizes the main results of group theory and how they apply to studying the physics of condensed
matter. The aim is to overview the most important results and take home messages. This summary does not attempt to teach
group theory – it will be most useful as a companion to those learning the subject. I made this summary while taking the
course Physical Applications of Group Theory from Prof. Steve Sharpe at the University of Washington in Spring 2013. Much
of this summary comes from the lecture notes and a few texts [2, 3, 1].
Defintions are written as defined-word.
1. Basic Mathematical Background
2. Representation Theory
1.1. Basic Ideas. A group is a collection of elements G and
a composition rule that satisfy four rules: closure, associativity, existence of identity and existence of inverses. An abelian
group is one where all the elements commute.
The order of a group G is denoted by [G], and is the number of elements in the group.
A subgroup of a group is a collection of elements of a group
that themselves form a group.
If in a finite group, an element g is multiplied by itself
enough times, then the identity g n = e is eventually recovered. The order of an element g is the smallest such n so that
g n = e.
2.1. Isomorphism and Homomorphism. A map between
two groups that preserves group multiplication, i.e.
A → Â, B → B̂, AB → ÂB̂
is called a homomorphism. If the map is 1-1, then it is called
an isomorphism. Isomorphisms are called faithful and homomorphisms are called unfaithful.
2.2. Representation. A representation of an abstract group
is a map D from the group to a set of square complex (nonsingular) matrices such that group multiplication is preserved;
i.e. ab = c implies D(a)D(b) = D(c). The dimensionality of
a representation is each of its matrices.
2.3. Irreducible Representation. A similarity transformation is a change of basis. If S is a non-singular matrix,
then we can create a new representation D0 (g) from D(g)
using the rule
D0 (g) = SD(g)S −1
for all g in G.
If by one and the same similarity transformation, all the
matrices in the representation of a group can be made to acquire a block diagonal form, then the representation is said
to be reducible. Otherwise, the representation is irreducible
(also called an irrep).
In physical problems, each irrep describes the transformation properties of a set of eigenstates and corresponds to a
distinct energy eigenvalue. We will show this later.
There are two ways that a representation can be reducible.
The following representation corresponds to the direct sum of
two representations Γr = Γ1 ⊕ Γ2 ,


Γ1 0


0 Γ2
1.2. Rearrangement Theorem. Let G = {e, g1 , g2 , ...gh }
and gk ∈ G, then the assembly of elements
{gk e, gk g1 , ..., gk gh }
contains each element of the group once and only once.
1.3. Cosets. If H is a subgroup of G and g ∈ G, then the
assembly
{h1 g, h2 g, ..., hn g}
is called the right coset of H with respect to g.
Two right cosets of a given subgroup either contain exactly
the same elements or else have no elements in common.
The order of a subgroup is a divisor of the order of the
group. If the order of a group is prime, then the group is
isomorphic to Zn , the cyclic group of order n.
1.4. Conjugation and Class. An element b is conjugate to
a if b = gag −1 for some g ∈ G. A class is the totality of
elements which can be obtained from a by conjugation. All
elements of the same class have the same order.
A subgroup H is normal or invariant if H = gHg −1 for
all g in G. The left and right cosets of an invariant subgroup
are the same.
This block diagonal stacking is also a representation because
the blocks do not talk to each other in matrix multiplication.
We would say that Γ1 and Γ2 are the irreps contained in Γr .
In what follows, we will use nµ for the dimensionality of
the µth irrep.
Date: March 2012.
Contact: Todd Karin. University of Washington. tkarin (at) uw (dot) edu.
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TODD KARIN
2.4. Theorem on Untarity of Representations. Every 3.3. Second Orthogonality Relation for Characters.
representation with matrices having non vanishing determi- The sum over all irreducible representations
h
i∗
nants can be brought into unitary form by an equivalence
X
(3)
χ(Γj ) (Ck ) χ(γj ) (Ck0 ) Nk = [G]δkk0
(similarity) transformation.
Γj
2.5. Schur’s Lemma (Part 1). A matrix which commutes holds, where C labels the classes and N is the number of
k
k
with all matrices in an irrep is a constant matrix (a constant elements in a class. his orthogonality theorem relates to the
times the unit matrix). Therefore, if a non-constant commut- columns of the character table.
ing matrix exists, the rep is reducible; if non exists, the rep is
irreducible.
3.4. Reducible Representations. The reduction of any reducible rep into its irreps is unique. The number of times the
(1)
2.6. Schur’s Lemma (Part 2). Given two irreps D and
irrep Γµ appears in the rep R is given by
D(2) of dimensionality n1 and n2 respectively, if there is an
h
i∗
1 X
n1 × n2 matrix such that
Nk χ(Γµ ) (Ck ) χR (Ck )
(4)
aµ =
[G]
k
M D(1) (gj ) = D(2) (gj )M
for all gj ∈ G, then M = 0 if n1 6= n2 . If n1 = n2 then where k labels the classes, Nk is the number of elements in
the class k and Ck denotes any element of the class k. This
either M = 0 or the reps D(1) and D(2) differ by a similarity
is also called the decomposition theorem.
transformation.
2.7. Wonderful Orthogonality Theorem. The orthonormality relation
h
i∗
X (Γ )
[G]
(Γ )
δΓ Γ δjj 0 δkk0
(1)
Djk µ (g) Dj 0 kν0 (g) =
nµ µ ν
g∈G
is obeyed for all inequivalent, irreducible, unitary representations of ha group. If
i the representation is not unitary, then the
(Γ )
∗
replace Dj 0 kν0 (g)
(Γ )
→ Dj 0 kν0 (g −1 )
2.8. Dimenions of irreducible representations. For
small finite groups, often the dimensionalities of the irreps
nµ can be determined using the fact that
X
n2µ = [G]
µ
since there is often a unique way to add up a few integers
squared to get a number.
3.5. Steps to finding the character table. Using the following facts, the character table can be set up. There are
in fact so many relations on character that there are usually
many ways of going about finding the character table of small
finite groups.
(1) Find the number of conjugacy classes.
(2) The number of irreducible representations is equal to
the number of conjugacy classes.
(3) Find the dimensions of the irreps using Sec. 2.8
(4) There is always the trivial (identity) representation
where every group element is represented by 1 – so
the first row is always all 1’s.
(5) The first column is always the trace of the identity
matrix, and is therefore equal to the dimension of the
irrep.
(6) For all irreducible representations other than the
identity, the sum on classes
X
Nk χ(Γi ) (Ck ) = 0.
k
3. Character of a Representation
3.1. Character. The character of the group element g in a
representation Γµ is the trace of of the matrix DΓµ (g). The
character of a group element is invariant under a similarity
transformation. The character of every element in a class is
the same.
3.2. First Orthogonality Relation for characters. The
characters of an irrep obey
X
(2)
χΓµ (g)χΓν (g)∗ = [G]δΓµ Γν
For all classes other than the identity class, the sum
on irreps
X
χ(Γµ ) (Ck )nµ = 0.
Γµ
(7) Use row (Eq. 2) and column (Eq. 3) orthogonality.
(8) The characters for 1D irreps must respect group multiplication since the 1D characters are the same as a
1D irrep.
3.6. Number of occurrences of the trivial irrep. Let
Γ(µ) and Γ(ν) be irreducible representations. If Γ(µ) = Γ(ν)∗ ,
Because of this, a necessary and sufficient condition that two then Γ(µ) ⊗ Γ(ν)∗ contains the trivial rep once. Otherwise,
irreps be equivalent is that the characters be the same. This Γ(µ) = Γ(ν)∗ does not contain the trivial rep.
This can be proven easily from the definition of product
orthogonality theorem relates to the rows of the character
representations and row orthogonality.
table.
g∈G
PHYSICAL APPLICATIONS OF GROUP THEORY
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4. Application to group theory to Quantum
Mechanics
one because the physical system is the same. A straightforward argument leads to the a relation for the transformed
tensor σ 0 ,
4.1. Overview. Group theory is used in quantum mechanics
h
i−1
0
both to obtain exact results and in applications of perturbaσil
= D(V ) (g)ij σjk D(V ) (g)
,
kl
tion theory. For the exact results, group theory helps to:
(1) Irreducible representations of the symmetry group of where V is the vector representation. By physical principles
Schrodinger’s equation labels the states and specifies we know that
their degeneracies.
(2) Determining how degeneracies are lifted when the
symmetry is lowered.
(3) Helps find the correct linear combination of states to
diagonalize the Hamiltonian.
4.2. Symmetry operations of the Hamiltonian. The set
of operators PR which leave the Hamiltonian invariant are
called the group of the Schrodinger equation. Let the action
of PR on |ψi be
PR |ψi = |ψ 0 i.
(5)
0
σij
= σij .
By Schur’s lemma, we know that σ must be block diagonal
with blocks proportional to the identity. The dimensions of
the irreps of V in the symmetry group determine the size of
these blocks.
For tensors with an arbitrary dimension, there is another
way to find their symmetry properties. We notice that σ
transforms in V ⊗ V ∗ , but also that σ shouldn’t change under
rotations. This is possible only if V ⊗ V ∗ contains the trivial
representation. If V ⊗ V ∗ contains the trivial representation
n times, then there are n independent constants in σ.
If a symmetry operation “leaves the Hamiltonian invariant,”
this means that the matrix elements of the Hamiltonian are
the same regardless whether the original or the transformed
5. Lie Groups
ket is used:
5.1. Lie Groups and Lie Algebras. In a Lie group, a rephψ|H|ψi = hψ 0 |H|ψ 0 i
resentation of a group element is written
This means that the operators PR will commute with H.
D(α) = eiαa Xa
Given an eigenstate |ψi of the Hamiltonian
H|ψi = E|ψi
H (PR |ψi) = E (PR |ψi) .
=⇒
Thus Pr |ψi is also an eigenstate of H with the same eigenvalue. Note that Pr |ψi is not necessarily different from |ψi.
where Xa are the generators of the group.
Because of the Campbell Baker Haussdorf formula, to multiply two group elements we just need to know how the Xa
commute. A Lie algebra comprises of the generators of the
Lie group with the composition law of commutation. The Lie
algebra allows us to construct almost everything about it’s
corresponding Lie group.
4.3. Transformation properties of Quantum States.
Suppose that the quantum level En is k-fold degenerate. Then
any linear combination of these k eigenstates is also an eigenstate of H. When PR acts on one of these eigenstates, it must
5.2. Structure Constants. The commutator of any two
return some linear combination of the eigenstates degenerate
generators is another generator. The structure constants fabc
th
with |ψi. Let the degeneracy of the n level be k. We can
are defined by
write the linear combination as
[Xa , Xb ] = ifabc Xc
k
X
(n)
PR |ψn,α i =
D (R)jα |ψn,j i
The structure constants are completely antisymmetric.
j=1
The matrices D(n) form a representation of the group. Is
this representation reducible? If it is, then by a change of basis, we can put the D matrices into block diagonal form. Then
we have found several sets of eigenstates that only transform
amonst themselves when acted on by the symmetry operations. We would then say that this set of states transforms
as an irreducible representation of the group.[?]
5.3. States. The generators can be thought of as linear operators acting on states such that
Xa |ii = |ji[Xa ]ij
5.4. Adjoint Representation. The adjoint representation
is the set of matrices Ta
(Ta )bc = −ifabc
4.4. Symmetry of Tensors. Suppose we have some system
with symmetry group G (e.g. a crystal) and a tensor σ de- We also define a scalar product on the linear space of the
scribing its physical properties (e.g. the conductivity tensor). generators in the adjoint representation: tr(Ta Tb )
In the adjoint representation,
If this tensor has two indices, then we can use Schur to determine the form σ is allowed to take.
|[Xa , Xb ]i = Taadj |Xb i
If we apply one of the symmetry operations of the crystal,
then the transformed tensor must be the same as the initial .
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TODD KARIN
5.5. Cartan Generators. In a particular irreducible representation D, there will be a number of hermitian generators
Hi , i = 1, ..., m that all commute. We can choose a basis such
that
tr(Hi Hj ) = kD δij
We call m, the number of independent Cartan generators the
rank of the algebra.
Since the Cartan generators are Hermitian, they can be simultaneously diagonalized. The eigenvalues µi of the Cartan
generators satisfying
Hi |µ, x, Di = µi |µ, x, Di
are called weights.
The roots αi are the weights of the adjoint representation. States corresponding to the Cartan generators have zero
weight vectors because
Hi |Hj i = |[Hi , Hj ]i = 0
with the same eignevalue. Is the converse true? If we
apply all the symmetry operations to a given eigenstate, will we generate all the eigenstates with that
eigenvalue? Answer: Not necessarily, applying all
the symmetry operations to an eigenstate will yield
all the other eigenstates in the same irreducible representation. In the case of accidental degeneracy (which
is “unlikely”) the energy of states in a different irreducible representation could be the same.
• Question: How to decompose the representations for
half-integer spins into irreps of a certain crystal symmetry? Answer: Can write the representation of a
rotation using our knowledge of how to rotate quantum mechanical states. Use the fact that spin is a
pseudo-vector, so under inversions it doesn’t change.
Need to use a double group (or a double cover) to
describe symmetry of the crystal.
.
References
States corresponding to different weights must be orthog[1] D. B. Chesnut. Finite Groups and Quantum Theory. John Wiley &
onal because they have different eigenvalues of at least one of
Sons Inc, December 1974.
the Cartan generators.
[2] Mildred S. Dresselhaus, Gene Dresselhaus, and Ado Jorio. Group
6. Questions
• Question: It is clear that if you apply a symmetry
operation to an eigenstate, you will get an eigenstate
Theory: Application to the Physics of Condensed Matter. Springer,
softcover reprint of hardcover 1st ed. 2008 edition, November 2010.
[3] Howard Georgi. Lie Algebras In Particle Physics: from Isospin To
Unified Theories (Frontiers in Physics). Westview Press, 2 edition,
October 1999.