Semiclassical quantization of skipping orbits

TL;DR

This paper introduces a semiclassical quantization approach for edge states in the quantum Hall effect, accurately matching the exact spectrum and analyzing the effects of orbit-edge interactions and diffraction.

Contribution

It presents a simple semiclassical model for edge state spectra in quantum Hall systems, including orbit-edge interactions and diffraction effects, with continuous gamma evolution.

Findings

Excellent quantitative agreement with exact spectrum

Gamma parameter varies from 1/2 to 3/4 near edges

Energy-dependent drift velocities calculated

Abstract

We propose a simple description of the spectrum of edge states in the quantum Hall regime, in terms of semiclassical quantization of skipping orbits along hard wall boundaries, ${\cal A}=2 \pi (n+\gamma) \ell_B^2$, where ${\cal A}$ is the area enclosed between a skipping orbit and the wall and $\ell_B$ is the magnetic length. Remarkably, this description provides an excellent quantitative agreement with the exact spectrum. We discuss the value of $\gamma$ when the skipping orbits touch one or two edges, and its variation when the orbits graze the edges and the semiclassical quantization has to be corrected by diffraction effects. The value of $\gamma$ evolves continuously from $1/2$ to $3/4$. We calculate the energy dependence of the drift velocity along the different Landau levels. We compare the structure of the semiclassical cyclotron orbits, their position with respect to the edge, to the wave function of the corresponding eigenstates.