Group theory 101
Suggested reading:
Landau & Lifshits, Quantum Mechanics, Ch. 12
Tinkham, Group Theory and Quantum Mechanics
Dresselhaus, Dresselhaus, Jorio, Group Theory:
Applications to the Physics of Condensed Matter
Ramond, Group Theory: a Physicist’s Survey
Definition�
A (finite or infinite) sequence of elements A,B,C…form a group,
if the following four conditions are satisfied
Group of rotations of an equilateral triangle
180 rotations (flips)�
1
2
3
1
2
3
2’
1
2
3
3’
1’
3
2
1
🡺A🡺
🡺B🡺
🡺C🡺
120 rotations
1
2
3
1
2
3
1
2
3
🡺D🡺
🡺F🡺
1
2
3
AB
2
1
3
=A
=
3
1
2
=D
1
2
3
Closure property
Six elements: identity, three filps, two rotations
Group of order 6
Cayley (multiplication) table
| I | A | B | C | D | F |
I | I | A | B | C | D | F |
A | A | I | D | F | B | C |
B | B | F | I | D | C | A |
C | C | D | F | I | A | B |
D | D | C | A | B | F | I |
F | F | B | C | A | I | D |
Three classes: 1) identity (E), 2) three 180 rotations (A,B,C),
3) 120 rotation (D) and 240 rotation (F)
Right
Left
classes
Isomorphic groups
Two groups G and G’ are called isomorphic, if there is one-to-one correspondence between their elements
Basis
Representation of a group
Applying a symmetry operation to the basis function,
we get a linear superposition of basis functions
Representation of a group is as arbitrary as the choice
of the basis function.
If a matrix of particular representation cannot be reduced to
a block-diagonal form by any similarity transformations, such
a representaton is called irreducible.
Irreducible represenations of
rotations or flips
This function generates a trivial 1D representation
2) consider a function which is invariant with respect to 120
rotations but changes its sign upon flips
This function generates another 1D representation
Irreducible representations of ,continued…
3) 2D representations are formed by two basis functions which
transform as elements of a vector (x,y)
1
2
3
3’
1’
2’
x
y
identity
22’ flip: x🡪-x,y🡪y
33’ flip
11’ flip
240 rotation
120 rotation
Characters
Character=trace of an irreducible representation matrix
Traces are invariant🡺characters do not depend
on the choice of basis functions
Reading character tables
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| | | |
| | | |
| | | |
A,B: 1D representations (A is even upon rotation, B is odd)
E: 2D representation (not to be confused with identity!)
F: 3D representation…
Group
Basis
function
Class
irrep
Trace of irrep
Orthogonality of characters
| | | |
| | | |
| | | |
| | | |
🡻
same irrep
🡻
Also,
🡻
different irreps
Van Vleck orthogonality theorem for irreps
Decomposition theorem
Applications in Quantum Mechanics
A,B: 1D representations 🡺 non-degenerate levels
E: 2D representation 🡺 two-fold degeneracy
F: 3D representation 🡺 three-fold degeneracy
Wavefunctions must obey all symmetry properties
of the Hamiltonian.
A proper description of a degenerate state is a linear
superposition of wavefunctions.
Basis functions of a given irrep are transformed
into each other under group operations🡺
Degenerate states form a basis of a given irrep🡺
Dimensionality of a given irrep gives us immediately
degeneracy of the corresponding energy level
Lifting of degeneracy by perturbation
Example: lifting of cubic degeneracy
Rotational group of a cube (without inversion and reflection symmetries)
non-degen.
2-fold
3-fold
A strain is applied along the main diagonal
How does the strain split the degenerate levels?
Classes:
Lifting of 3-fold degeneracy
🡺
Lattice symmetries
Rotational symmetries of building blocks (polygons)
must be consistent with translational symmetry
crystallographic restriction theorem:
lattice can have only 2, 3, 4, and 6-fold rotational symmetries
Crystal Structure
Crystal structure can be obtained by attaching atoms or groups of atoms --basis-- to lattice sites.
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Crystal Structure
23
Crystal Structure = Crystal Lattice + Basis
Partially from Prof. C. W. Myles (Texas Tech) course presentation
Bravais lattices: monoatomic basis
Non-Bravais lattices: polyatomic basis
Graphene: Honeycomb
Five 2D Bravais lattices
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Rhombohedral
Hexagonal (Triangular)
Tetragonal (Square)
Orthorhombic (Rectangular)
Oblique
Oblique
180
Elements of symmetry: C2 rotations
Group: C2
Rhombohedral
Orthorhombic (Rectangular)
180
180
180
D2
D2
Equivalently, one can do reflections in vertical planes
D2=C2v (=means “isomorphic”)
180
Tetragonal (Square)
Symmetry operations:
3×90 rotations
180 rotations about 4
horizontal axes
🡪D4
90
Symmetry operations:
3×90 rotations
Reflections in 4
Vertical planes
🡪C4v
D4=C4v
Vibrational modes of the H2O molecule
System of N particles (not on the same line):
3N degrees of freedom
3 translational
3 rotational
# of vibrational modes: Nv= 3N-3-3=3N-6
For H2O: N=3🡪Nv=3
What are those 3 modes?
H2O
C2 axis+2 vertical planes (σv and σ’v)
🡪C2v group
σv
σ'v
σv
σ'v
σv
σ'v
A1
A1
B1