Relativistic two-dimensional hydrogen-like atom in a weak magnetic field

TL;DR

This paper analytically calculates the relativistic energy corrections for a 2D hydrogen-like atom in a weak magnetic field, revealing that its magnetizability can be expressed with elementary functions, unlike in 3D cases.

Contribution

It provides closed-form expressions for Zeeman energy corrections and magnetizability of a 2D relativistic atom, using Sturmian expansion of the Green function, extending previous 3D results.

Findings

Energy corrections are derived analytically for arbitrary states.

Magnetizability in 2D is expressible via elementary functions.

Results relate to Coulomb impurity problems in graphene.

Abstract

A two-dimensional (2D) hydrogen-like atom with a relativistic Dirac electron, placed in a weak, static, uniform magnetic field perpendicular to the atomic plane, is considered. Closed forms of the first- and second-order Zeeman corrections to energy levels are calculated analytically, within the framework of the Rayleigh-Schr\"odinger perturbation theory, for an arbitrary electronic bound state. The second-order calculations are carried out with the use of the Sturmian expansion of the two-dimensional generalized radial Dirac-Coulomb Green function derived in the paper. It is found that, in contrast to the case of the three-dimensional atom [P. Stefa\'nska, Phys. Rev. A 92 (2015) 032504], in two spatial dimensions atomic magnetizabilities (magnetic susceptibilities) are expressible in terms of elementary algebraic functions of a nuclear charge and electron quantum numbers. The problem considered here is related to the Coulomb impurity problem for graphene in a weak magnetic field.