On the Chern-Simons Gauge Theory of Anyons in the Fractional Quantum Hall Effect

TL;DR

This paper clarifies the role of Chern-Simons gauge theory in describing anyons within the fractional quantum Hall effect, emphasizing the physical significance of gauge invariance and the nature of the anyon condensate.

Contribution

It provides a new perspective on the physical states of anyons, emphasizing gauge invariance and challenging the necessity of Ginzburg-Landau theory in this context.

Findings

Bosonized electrons are unphysical in Chern-Simons gauge theory.

The true physical state is a liquid of degenerate anyons.

Gauge invariance is crucial in composite-fermion theories.

Abstract

This short note was born out of discussions on anyons in the FQHE at the YITP workshop "Fundamental problems of quantum field theories" (December 19-21, 2001, YITP, Kyoto). At that time, I felt that there might not be a sound consensus of opinion on the subject. Now, I would like to show my understanding here, the essential part of which is based on hep-th/0110197. The first problem discussed is a notion of ``bosonized electrons (bosonization)", which are unphysical objects from the standpoint of the Chern-Simons gauge invariance. Therefore, their condensate is merely of mathematical concept and the true physical state realized is a liquid-like one made of degeneration of anyons. Based on this recognition, I argue about a mechanism of the degeneration that results from the genuine CS gauge field theory. It is noted that the Ginzburg-Landau effective theory is not necessary. As the last problem, the gauge invariance in the "composite-fermion theories" is discussed.