Physics on the edge: contour dynamics, waves and solitons in the quantum Hall effect

TL;DR

This paper develops a non-linear contour dynamics framework to analyze edge excitations in the quantum Hall effect, revealing soliton solutions and a modified Korteweg-de Vries equation governing curvature evolution.

Contribution

It introduces a non-linear analysis of edge shape deformations in quantum Hall systems, deriving a novel PDE for curvature and identifying soliton solutions.

Findings

Soliton solutions propagate without distortion.

Edge dynamics described by a modified Korteweg-de Vries equation.

Comparison with vortex patches in hydrodynamics enhances understanding.

Abstract

We present a theoretical study of the excitations on the edge of a two-dimensional electron system in a perpendicular magnetic field in terms of a contour dynamics formalism. In particular, we focus on edge excitations in the quantum Hall effect. Beyond the usual linear approximation, a non-linear analysis of the shape deformations of an incompressible droplet yields soliton solutions which correspond to shapes that propagate without distortion. A perturbative analysis is used and the results are compared to analogous systems, like vortex patches in ideal hydrodynamics. Under a local induction approximation we find that the contour dynamics is described by a non-linear partial differential equation for the curvature: the modified Korteweg-de Vries equation. PACS number(s): 73.40.Hm, 02.40.Ma, 03.40.Gc, 11.10.Lm