Landau level quantization for massless Dirac fermions in the spherical geometry: graphene fractional quantum Hall effect on the Haldane sphere

TL;DR

This paper derives the eigenenergies and eigenfunctions for massless Dirac fermions on a sphere with a magnetic monopole, enabling detailed study of the fractional quantum Hall effect in graphene within spherical geometry.

Contribution

It provides a formalism for Landau level quantization of Dirac fermions on the Haldane sphere, including pseudopotentials and Landau level mixing Hamiltonians, extending planar results to spherical geometry.

Findings

Derived eigenenergies and eigenfunctions for Dirac fermions on the sphere.

Calculated Haldane pseudopotentials for Coulomb interactions.

Established a framework for studying fractional quantum Hall effect in graphene on the Haldane sphere.

Abstract

We derive the single-particle eigenenergies and eigenfunctions for massless Dirac fermions confined to the surface of a sphere in the presence of a magnetic monopole, i.e., we solve the Landau level problem for electrons in graphene on the Haldane sphere. With the single-particle eigenfunctions and eigenenergies we calculate the Haldane pseudopotentials for the Coulomb interaction in the second Landau level and calculate the effective pseudopotentials characterizing an effective Landau level mixing Hamiltonian entirely in the spherical geometry to be used in theoretical studies of the fractional quantum Hall effect in graphene. Our treatment is analogous to the formalism in the planar geometry and reduces to the planar results in the thermodynamic limit.