The Electromagnetic Interactions of Electrons in the Lowest Landau Level

TL;DR

This paper derives an effective gauge-invariant Lagrangian for electrons in the lowest Landau level, revealing the importance of edge states and surface vibrations in the quantum Hall effect.

Contribution

It introduces a gauge-invariant effective Lagrangian splitting into surface and interior components, highlighting the role of edge states in quantum Hall systems.

Findings

Effective Lagrangian for L.L.L. electrons derived

Edge states are essential for gauge invariance

Surface vibrations correspond to bosonized edge modes

Abstract

Starting from a system of planar electrons in a strong magnetic field normal to the plane, interacting with perturbing electromagnetic fields, an effective Lagrangian for the fermions in the lowest Landau level (L.L.L.) has been derived. By choosing a suitable background electrostatic potential, an incompressible droplet of these electrons is constructed. The gauge invariant effective Lagrangian for the electrons in the L.L.L. is shown to split naturally into a $1+1$ dimensional Lagrangian for the electrons on the surface of the droplet and into a $2+1$ dimensional gauge-field Lagrangian representing the contribution of the interior of the droplet. Upon bosonization, the former represents the surface vibrations of the droplet. Individually neither of these two actions is gauge invariant, but it is shown that the gauge dependence from the two pieces cancels out. This demonstrates that the edge degrees of freedom are essential for maintaining gauge invariance.