On edge states in semi-infinite quantum Hall systems

TL;DR

This paper demonstrates that in semi-infinite quantum Hall systems, edge states inevitably occur due to the spectral properties of electrons confined by a smooth potential wall, even with weak impurity potentials.

Contribution

It proves the existence of edge states in semi-infinite quantum Hall systems under weak impurity potentials without requiring decay conditions.

Findings

Edge states occur in semi-infinite quantum Hall systems.

Continuous spectrum appears between Landau levels due to impurities.

Edge states are guaranteed by the spectral analysis of the Hamiltonian.

Abstract

We consider an electron in two dimensions submitted to a magnetic field and to the potential of impurities. We show that when the electron is confined to a half-space by a planar wall described by a smooth increasing potential, the total Hamiltonian necessarily has a continuous spectrum in some intervals in-between the Landau levels provided that both the amplitude and spatial variation of the impurity potential are sufficiently weak. The spatial decay of the impurity potential is not needed. In particular this proves the occurence of edge states in semi-infinite quantum Hall systems.