An effective Landau level mixing hamiltonian for graphene in the spherical geometry

The quantum Hall effect has been a subject of intense study since its discovery thirty-five years ago. The more recent isolation of monolayer graphene and the observations of the integer and fractional quantum Hall effect brought new excitement to the field. In this thesis, we construct an effective Hamiltonian for graphene that incorporates Landau level mixing for Dirac fermions in the first two Landau levels entirely within the spherical geometry. In the process of constructing the effective Hamiltonian, we solve the long-standing problem of the Landau level quantization in the spherical geometry for massless Dirac fermions---in other words, we find the eigenstates and eigenenergies for massless Dirac fermions, confined to the surface of a sphere with radius [special characters omitted] in the presence of a magnetic monopole of strength Q. Previously, effective Hamiltonians were constructed in the planar geometry and were only approximate when used on finite spheres and exact only in the thermodynamic limit. In this thesis, we describe how to construct effective Hamiltonians that include the important effects of Landau level mixing that directly correspond to a finite sphere with any monopole strength Q. In particular, we provide Haldane pseudopotentials characterizing the effective Hamiltonians for Q=4.5, 6, and 9 in the lowest two Landau levels. These results will serve as a starting point for numerical studies of the fractional quantum Hall effect in graphene under the realistic condition of Landau level mixing.