1. Band structure of Graphene
Note the cones at K and K’ points

2. But the 2 components are for the 2 sublattices
Expansion of band structure around K and K’ points
But the 2 components are for the 2 sublattices 2 2

3. In the Dirac theory the 4 components are due to spin and charge degrees of freedom; here they are due to the two Fermi points and to the amplitude on sites a and b. The analogy requires a massless Dirac particle.

4. Magnetic length Landau levels Consider a Magnetic field B
Take B perpendicular to plane of Graphene.

6. Recall the textbook elementary harmonic oscillator y: the annihilation operator is

7. 7

8. Some Concepts from Topology
A convex set is a set of points containing all line segments between each pair of its points.
Euler’s Characteristic of a surface

9. From Wikipedia

10. Euler’s Theorem for convex polyhedra

11. Euler’s Theorem for general genus
Genus g of a surface is the largest number of non-intersecting closed curves that can be drawn on it withput separating it.
sphere g=0
torus g=1
double-hole doghnut g=2
See e.g.
Euler’s Theorem for general genus
A graphene lattice with pbc and without holes has g=1. One can also insert pentagons and eptagonswithout changing g.

12. One can insert two heptagons and two pentagons without leaving the plane.

13. Each graphene vertex has 3 links
Each graphene vertex has 3 links. Let us consider only pentagonal or heptagonal deformations.
Pentagons are balanced by equal number of eptagons.

14. Non-Abelian Vector potential
from Jiannis Pachos cond-mat
The insertion of a pentagon forces us to connect two sites that are of the same type, e.g. two white sites in the figure. Recall the structure of spinor:

15. In the magnetic case one introduces a vector potential A to allow the wave function to collect a phase factor. Here we want the wave function to collect a jump to the opposite components and this requires a non abelian vector potential such that
is off diagonal.
One can make a unitary transformation such that the insertion of a pentagon or an eptagon introduces independent magnetic fields at K and K’.
The zero modes of H are the eigenstates with zero eigenvalue in the limit of infinite systems. The Atiyah-Singer index theorem says that 2 p times the number of zero modes is equal to the flux of the effective magnetic field. This gives insight on the low-energy sector in terms of the number of pentagons and heptagons for systems of any size.

16. Open faces
G=1,N=0
G=0,N=2
G=2,N=0
G=0,N=4

17. Anyons
See also Sumatri Rao, arXiv:hep-th/ ,Jiannis Pachos,Introduction to Topological Quantum Computation
In 2d there is only the z axis, say, so no commutation relations, but the condition that the wave function be eigenfunction of Lz leads to integer angular momentum. However we shall see that this is violated of the particle has a flux f ; then one finds

18. Simple Model for an anyonq r
Indeed consider a spinless particle with charge q orbiting around a thin solenoid at distance r. If the current in the solenoid vanishes ( i=0 ) then Lz= integer. Now turn on the current i. The particle feels an electric field such that
However this is too rough. The charge is actually being switched on at the same time
that the flux in the solenoid is being switched on, with q(t)=constant X f(t).

19. Statistics

20. Chern-Simons anyons Usual electromagnetism in 3+1 d

23. About Excitations of Graphene
Short account of current Theoretical work
We saw that in 1d the Peierls distortion
leads to a double minimum potential, that is to the existence of two vacua and to the possibility of charge fractionalization.
In 2d the analogous to the Peierls
distortion ia s Kekule distortion

24. 1/3 of the hexagons is undistorted

26. Hence it could produce a zero energy mode..
Such a scalar field should not violate the symmetry between positive and negative energy states, that really arise from expansions around K and K’, since K and K’ are treated in the same way.
Hence it could produce a zero energy mode..
. This should correspond to a ½ charge excitation, in analogy to the charge fractionalization mechanism in 1d.