A monolayer of carbon is called graphene. It exhibits unusual properties in the Hall effect and in the cyclotron resonance. It is found that it exhibits fractional charge in the Hall effect. The interactions amongst electrons almost become constant at low temperatures. Hence, the Kohn's theorem, which shows that the interactions do not play much role in determining the cyclotron resonance, becomes operative at low temperatures. The experiments on graphene do not depend on the wave vector dependence of the frequency. Hence whether the dispersion depends on k2 or on k does not matter. The Hubbard model has been very successful in explaining the ground state of several electron systems. We consider a triangle with three vortices. Each vortex can be occupied by two electrons. By using the spin in a particular way, we can obtain new features in the Hubbard model. There is a doubling in the Peierls-Luttinger phase factor and eigen values acquire higher multiplicities than are known for the usual treatment of spin. The flux is distributed on the area of the triangle. The graphene consists of hexagons of carbon atoms but the Hall effect shows that there are defects on which electrons form clusters so that there is spin wave type behaviour. A cluster of electrons shows spin-waves leading to "spin deviation" of several per cent.
Simple mechanism that breaks the Hall-effect linearity at low temperatures
