Chern-Simons theory of multi-component quantum Hall systems

TL;DR

This paper generalizes the Chern-Simons approach to multi-component quantum Hall systems, providing a framework to understand fractional quantum Hall states with multiple degrees of freedom.

Contribution

It extends Chern-Simons transformations to systems with any number of components, linking the Hamiltonian to Halperin wave functions through a harmonic oscillator analogy.

Findings

The Hamiltonian becomes quadratic and solvable as a harmonic oscillator.

Conditions for the validity of the model are identified.

Analysis of symmetric states and singular cases.

Abstract

The Chern-Simons approach has been widely used to explain fractional quantum Hall states in the framework of trial wave functions. In the present paper, we generalise the concept of Chern-Simons transformations to systems with any number of components (spin or pseudospin degrees of freedom), extending earlier results for systems with one or two components. We treat the density fluctuations by adding auxiliary gauge fields and appropriate constraints. The Hamiltonian is quadratic in these fields and hence can be treated as a harmonic oscillator Hamiltonian, with a ground state that is connected to the Halperin wave functions through the plasma analogy. We investigate several conditions on the coefficients of the Chern-Simons transformation and on the filling factors under which our model is valid. Furthermore, we discuss several singular cases, associated with symmetric states.