(TuTh 9:30am -10:50pm online) Course Syllabus pdf file

Lecture notes

Elements: The language of symmetry - group

Lecture 1 : Why is symmetry important?

Lecture 2 : What is group?

Lecture 3 : From math to physics - representations of group

Supplemental Material : A little bit of math - Orthogonality

Lecture 4 : Important- character tables, representations for finite groups

Point groups -- Symmetry of Molecules

Lecture 5 : Point groups - dihedral, tetrahedral, octahedral, icosohedral groups

Lecture 6 : Decomposition of direct product representations, projection operators, crystal harmonics

Lecture 7 : Crystal field splitting, cystalline tensors

Lecture 8 : Molecular vibration

Supplemental materials -- Space group symmetries of crystals

Crystal symmetry (I) : Crystal system, Bravis lattice

Crystal symmetry (II) : - non-symmorphic operations

Crystal symmetry (III) : Representations of space group

Crystal symmetry (IV) : more examples of space group

Supplemental Materials -- Galois theory, the starting of groups

Galois theory (I) : Number field and field extension

Galois theory (II) : Automorphism and Galois groups

Galois theory (III) : More on Galois groups

Galois theory (IV) : Normal and composite group series, solvability

Galois theory (V) : Sufficient and necessary conditions of algebraic solutions

Galois theory (VI) : Compass-straightedge construction problem

Supplemental Material: Old friends- SU(2) and SO(3)

Angular momentum (I) : SO(3) and SU(2)

Angular momentum (II) : D-matrix and wavefunctions of rotating top

Angular momentum (III) : From D-matrix to Spherical harmonics and CG coefficients

Angular momentum (IV) : Monopole harmonics

Angular momentum (V) : Algebraic solution to the spectra of an anisotropic top

Supplemental Materials

Elements of Lie groups (I) : Local and global properties of Lie group

Elements of Lie groups (II) : Integral over group manifold

Elements of Lie groups (III) : Lie's three theorems, Lie algebra

SU(3) and SU(N)

Lecture 9 : SU(3), weight and root

Lecture 10 : Bilinear-biquadratic spin-1 Heisenberg model: SU(3) baryon and meson states

Lecture 11 : SU(N), tensor, Young tableau (I)

Lecture 12 : More on SU(N) group

Supplemental Material : Useful information on SU(N) representations

Further reading : large N v.s. large S

Further reading : Quantum magnetism of SU(4) and SU(6) Hubbard model

Further reading : Mott insulating state of SU(2N) Dirac fermions

Lie algebra

Lecture 13 Classification of Lie algebra, root, Dynkin diagram

Lecture 14 Construction of roots

Lecture 15 Representations of Lie algebra

Lecture 16 SO(5) and Sp(4), Gamma matrices

Lecture 17 SO(6) and SU(4), exact SO(5) symmetry in spin-3/2 Hubbard model

Further reading : "Exact SO(5) Symmetry in the Spin-3/2 Fermionic System", PRL 91, 186402 (2003)

Lecture 18 SO(2N+1) and SO(2N) group

howework assignment

HW1 : Due on April 7, 2020.

HW2 : Due on May 7, 2020. Solution HW2 solution.

HW3 : Due on Jun 4, 2020.

Midterm Due on May 16, 2020 Middle night
Middle exam Solution Posted Jun 5.


You need to apply group theory to guide the study of electronic states of a C60 modelcule. Suggested topics include but are not limited to the following contents. If you can go beyond the list below, you can receive extra credit. You may do literature search on line, but are not allowed to discuss with others before you turn in your solutions to me.

Please note that below we use the full symmetry group of icosahedral, which includes inversion. This group is denoted as Yh, which has 120 symmetry elements. It is just Y \times Z2, and the Z2 includes identity and inversion.

1. Approximate C60 as a spherically symmetric shell. Solve the electron orbital problem on the sphere The radius of the shell is R, and the thickness of the shell is neglected.

Load 60 electrons in these orbitals. What is the filling configuration of the ground state of 60 electrons? (Hint: For fully filled orbitals, each state is filled by a pair of spin up and down electrons. The HOMO (highest ocupied molecular orbital) is partially filled, and you can use Hund's rule to determin its configuration. )

Certainly this picture is oversimplified: You would get that C60 is gapless and magnetic in contradiction to experimental facts. In fact, C60 is non-magnetic and there is an excitation gap between HOMO and LUMO (lowest unoccupied molecular orbital).

2. To improve, let us break the spherical symmetry to the icosehedral symmetry by adding potential distributions on the sphere.

Figure out each orbital with angular momentum $l$ in the original spherical symmetry splits into what irreducible representations of the icosahedral group. (You can do for states up to $l$=5. Assume that the potential is not strong enough to change the Sequence of energies of orbitals with different $l$, but only splits the degeneracy of states within the same $l$.)

Based on the fact of a gap between HOMO and LUMO, can you figure about what is the representation of the HOMO?

3. Now let us work out the same problem from the discrete side. Suppose each site contribute an atomic orbital. The C60 molecular orbitals form a 60 dimensional representation of icosahedral group. Please decompose it into direct sum of irreducible ones.

4. Write down a tight-binding model with the neareast neighboring hopping amplitude -t. Diagonalize the Hamiltonian matrix. You may use either diagonalize numerically, or, first use group theory to reduce the size of the matrix, and then diagonalize.

Sketch a level diagram with labeling the degeneracy and the representation of each orbital solved from the tight-binding model. Compare your result here with that obtained in part 2. What is the representation of LUMO and what is its degenreacy?

For extra credit, if you can do this part.
5. The vibrational modes of the C60 forms a 180 dimensional representation of the icosahedral group Yh. Please decompose it into representation of Yh.

Final exam
Please note that I added a new part of problem 5. - Due on Jun 13, midnight.

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Last modified: Jan 7, 2010.