Note the cones at K and K’ points

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

2. Expansion of band structure around K and K’ points But the 2 components are for the 2 sublattices 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. Magneticlength Consider a Magnetic field B Landau levels Take B perpendicular to plane of Graphene.

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

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

9. From Wikipedia

10. Euler’s Theorem for convex polyhedra

11. 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. http://www.solitaryroad.com/c775.html 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. Let us consider only pentagonal or heptagonal deformations. Pentagons are balanced by equal number of eptagons.

14. Non-AbelianVectorpotential from Jiannis Pachos cond-mat0812116 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 SeealsoSumatriRao, arXiv:hep-th/9209066,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 anyon 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 r q 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-Simonsanyons Usualelectromagnetism in 3+1 d electromagnetism in 2+1 d

23. AboutExcitations 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. 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. Henceitcould produce a zero energy mode.. . Thisshouldcorrespond to a ½ chargeexcitation, in analogy to the chargefractionalizationmechanism in 1d.