Orthogonal localized wave functions of an electron in a magnetic field

TL;DR

This paper constructs a set of localized magnetic Wannier orbitals on an infinite plane, characterized by their Gaussian core and power-law tails, providing a useful basis for electron problems in magnetic fields.

Contribution

It proves the existence of two-scale magnetic Wannier orbitals with specific localization properties and a well-defined basis for electron studies in magnetic fields.

Findings

Orbitals are localized with Gaussian core and power-law tail.

The orbitals form a basis suitable for electron problems in magnetic fields.

The core region dominates the normalization integral.

Abstract

We prove the existence of a set of two-scale magnetic Wannier orbitals w_{m,n}(r) on the infinite plane. The quantum numbers of these states are the positions {m,n} of their centers which form a von Neumann lattice. Function w_{00}localized at the origin has a nearly Gaussian shape of exp(-r^2/4l^2)/sqrt(2Pi) for r < sqrt(2Pi)l,where l is the magnetic length. This region makes a dominating contribution to the normalization integral. Outside this region function, w_{00}(r) is small, oscillates, and falls off with the Thouless critical exponent for magnetic orbitals, r^(-2). These functions form a convenient basis for many electron problems.