Anomalous Quantum Hall Effect on Sphere

TL;DR

This paper investigates the anomalous quantum Hall effect on a spherical surface with a magnetic monopole, establishing connections between Dirac and Landau operators, and proposing wavefunctions for various filling factors, with applications to graphene.

Contribution

It introduces a new approach linking Dirac and Landau operators on a sphere and proposes SU(N) wavefunctions for different filling factors, advancing understanding of quantum Hall physics on curved surfaces.

Findings

Derived Dirac eigenvalues and eigenfunctions from Landau operators.

Identified composite fermion behavior with two effective magnetic fields.

Proposed SU(N) wavefunctions capturing symmetries at various filling factors.

Abstract

We study the anomalous quantum Hall effect exhibited by the relativistic particles living on two-sphere S^2 and submitted to a magnetic monopole. We start by establishing a direct connection between the Dirac and Landau operators through the Pauli--Schr\"odinger Hamiltonian H_{s}^{SP}. This will be helpful in the sense that the Dirac eigenvalues and eigenfunctions will be easily derived. In analyzing H_{s}^{SP} spectrum, we show that there is a composite fermion nature supported by the presence of two effective magnetic fields. For the lowest Landau level, we argue that the basic physics of graphene is similar to that of two-dimensional electron gas, which is in agreement with the planar limit. For the higher Landau levels, we propose a SU(N) wavefunction for different filling factors that captures all symmetries. Focusing on the graphene case, i.e. N=4, we give different configurations those allowed to recover some known results.