The Constraint for the Lowest Landau Level and the Chern-Simons Field Theory Approach for the Fractional Quantum Hall Effect: Infinite and Finite Systems

TL;DR

This paper incorporates the lowest Landau level constraint into the Chern-Simons theory for fractional quantum Hall systems, deriving hierarchical states, edge excitations, and explicit drift velocities.

Contribution

It introduces a method to transmit the Landau level constraint through hierarchical states and analytically derives edge excitation properties and drift velocities.

Findings

Hierarchical states can be connected via the Landau level constraint.

Edge and bulk actions can be separated in finite systems.

Explicit formulas for edge excitation drift velocities are obtained.

Abstract

We build the constraint that all electrons are in the lowest Landau level into the Chern-Simons field theory approach for the fractional quantum Hall system. We show that the constraint can be transmitted from one hierarchical state to the next. As a result, we derive in generic the equations of the fractionally charged vortices ( quasi-particles ) for arbitrary hierarchy filling. For a finite system, we show that the action for each hierarchical state can be divided into two parts: the surface part provides the action for the edge excitations while the remaining bulk part is exactly the action for the next hierarchical states. In particular, we not only show that the surface action for the edge excitations would be decoupled from the bulk at each hierarchy filling, but also derive the explicit expressions analytically for the drift velocities of the hierarchical edge excitations.