Spectral flow and level spacing of edge states for quantum Hall hamiltonians

TL;DR

This paper analyzes the spectral flow and level spacing of edge states in quantum Hall Hamiltonians, revealing strong level repulsion linked to topological properties, with implications for the statistical behavior of these states.

Contribution

It provides a rigorous lower bound on the level spacing of edge states in quantum Hall systems, connecting spectral properties to topological invariants.

Findings

Level spacing between edge states is bounded below by a term proportional to L^{-1}.

Level repulsion of edge states exceeds that of typical Anderson extended states.

Spectral flow analysis reveals topological connections influencing level statistics.

Abstract

We consider a non relativistic particle on the surface of a semi-infinite cylinder of circumference $L$ submitted to a perpendicular magnetic field of strength $B$ and to the potential of impurities of maximal amplitude $w$. This model is of importance in the context of the integer quantum Hall effect. In the regime of strong magnetic field or weak disorder $B>>w$ it is known that there are chiral edge states, which are localised within a few magnetic lengths close to, and extended along the boundary of the cylinder, and whose energy levels lie in the gaps of the bulk system. These energy levels have a spectral flow, uniform in $L$, as a function of a magnetic flux which threads the cylinder along its axis. Through a detailed study of this spectral flow we prove that the spacing between two consecutive levels of edge states is bounded below by $2\pi\alpha L^{-1}$ with $\alpha>0$, independent of $L$, and of the configuration of impurities. This implies that the level repulsion of the chiral edge states is much stronger than that of extended states in the usual Anderson model and their statistics cannot obey one of the Gaussian ensembles. Our analysis uses the notion of relative index between two projections and indicates that the level repulsion is connected to topological aspects of quantum Hall systems.