Effective field theory and tunneling currents in the fractional quantum Hall effect

TL;DR

This paper reviews the effective field theory for abelian quantum Hall fluids, focusing on tunneling currents between edges, and analyzes interference effects in different interferometer geometries.

Contribution

It provides a detailed calculation of tunneling currents in quantum Hall systems using Chern-Simons theory, including interference effects in Fabry-Perot and Mach-Zehnder geometries.

Findings

Interference term absent when one edge hole lies inside the interference contour.

Tunneling current depends on the quantum Hall state and magnetic flux.

Analysis of edge mode speeds and their impact on tunneling phenomena.

Abstract

We review the construction of a low-energy effective field theory and its state space for "abelian" quantum Hall fluids. The scaling limit of the incompressible fluid is described by a Chern-Simons theory in 2+1 dimensions on a manifold with boundary. In such a field theory, gauge invariance implies the presence of anomalous chiral modes localized on the edge of the sample. We assume a simple boundary structure, i.e., the absence of a reconstructed edge. For the bulk, we consider a multiply connected planar geometry. We study tunneling processes between two boundary components of the fluid and calculate the tunneling current to lowest order in perturbation theory as a function of dc bias voltage. Particular attention is paid to the special cases when the edge modes propagate at the same speed, and when they exhibit two significantly distinct propagation speeds. We distinguish between two "geometries" of interference contours corresponding to the (electronic) Fabry-Perot and Mach-Zehnder interferometers, respectively. We find that the interference term in the current is absent when exactly one hole in the fluid corresponding to one of the two edge components involved in the tunneling processes lies inside the interference contour (i.e., in the case of a Mach-Zehnder interferometer). We analyze the dependence of the tunneling current on the state of the quantum Hall fluid and on the external magnetic flux through the sample.