Landau levels of a Dirac electron in graphene from non-uniform magnetic fields

TL;DR

This paper explores how specific non-uniform magnetic fields can produce Landau levels in graphene, similar to uniform fields, by using isospectral deformations of the magnetic field in the Dirac Hamiltonian framework.

Contribution

It introduces explicit analytical forms of non-uniform magnetic fields that are isospectral to uniform fields, demonstrating the persistence of Landau levels in graphene.

Findings

Non-uniform magnetic fields can produce Landau levels in graphene.

Explicit analytical expressions for isospectral magnetic fields are derived.

Landau levels are preserved under certain non-uniform magnetic field configurations.

Abstract

The occurrence of Landau levels in quantum mechanics when a charged particle is subjected to a uniform magnetic field is well known. Considering the recent interest in the electronic properties of graphene, which admits a dispersion relation which is linear in the momentum near the Dirac points, we revisit the problem of Landau levels in the spirit of the Dirac Hamiltonian and ask if there are certain non-uniform magnetic fields which also lead to a spectrum consisting of the Landau levels. The answer, as we show, is in the affirmative. In particular, by considering isospectral deformations of the uniform magnetic field, we present explicit analytical expressions for non-uniform magnetic fields that are strictly isospectral to their uniform counterpart, thus supporting the Landau levels.