1. Symmetry of Single-walled Carbon Nanotubes Part II

2. Outline Part II (December 6) Irreducible representations Symmetry-based quantum numbers Phonon symmetries M. Damjanović, I. Milošević, T. Vuković, and J. Maultzsch, Quantum numbers and band topology of nanotubes, J. Phys. A: Math. Gen. 36, 5707-17 (2003)

3. Application of group theory to physics Representation:  : G  P homomorphism to a group of linear operators on a vector space V (in physics V is usually the Hilbert space of quantum mechanical states). If there exists a V 1  V invariant real subspace   is reducible otherwise it is irreducible. V can be decomposed into the direct sum of invariant subspaces belonging to the irreps of G: V = V 1  V 2  …  V m If G = Sym[H]  for all eigenstates |  of H |  = |i   V i (eigenstates can be labeled with the irrep they belong to, "quantum number")  i | j    ij  selection rules

4. Illustration: Electronic states in crystals Lattice translation group: T Group "multiplication": t 1 + t 2 (sum of the translation vectors)

5. Finding the irreps of space groups 1.Choose a set of basis functions that span the Hilbert space of the problem 2.Find all invariant subspaces under the symmetry group (Subset of basis functions that transfor between each other) Basis functions for space groups: Bloch functions Bloch functions form invariant subspaces under T  only point symmetries need to be considered "Seitz star": Symmetry equivalent k vectors in the Brillouin zone of a square lattice  8-dimesional irrep In special points "small group” representations give crossing rules and band sticking rules.

6. Line groups and point groups of carbon nanotubes Chiral nanotubs: Lq p 22 (q is the number of carbon atoms in the unit cell) Achiral nanotubes: L2n n /mcm n = GCD(n 1, n 2 )  q/2 Point groups: Chiral nanotubs: q22 (D q in Schönfliess notation) Achiral nanotubes: 2n /mmm (D 2nh in Schönfliess notation)

7. Symmetry-based quantum numbers (k x,k y ) in graphene  (k,m) in nanotube k : translation along tube axis ("crystal momentum") m : rotation along cube axis ("crystal angular momentum”) CpCp

8. Linear quantum numbers Brillouin zone of the (10,5) tube. q=70 a = (21) 1/2 a 0  4.58 a 0 

9. Helical quantum numbers Brillouin zone of the (10,5) tube. q=70 a = (21) 1/2 a 0  4.58 a 0  n = 5 q/n = 14

10. Irreps of nanotube line groups Translations and z-axis rotations leave |km  states invariant. The remaining symmetry operations: U and  Seitz stars of chiral nanotubes: |km , |–k–m   1d (special points) and 2d irreps Achiral tubes: |km  |k–m  |–km  |–k–m   1, 2, and 4d irreps Damjanović notations:

11.  point (|00  ): G = point group The optical selection rules are calculated as usual in molecular physics: Infraded active: A 2u + 2E 1u (zig-zag) 3E 1u (armchair) A 2 + 5E 1 (chiral) Raman active 2A 1g + 3E 1g + 3E 2g (zig-zag) 2A 1g + 2E 1g + 4E 2g (armchair) 3A 1 + 5E 1 + 6E 2 (zig-zag) Optical phonons at the  point

12. Raman-active displacement patterns in an armchair nanotube Calcutated with the Wigner projector technique