Algebraic Aspects of the Fractional Quantum Hall Effect

TL;DR

This paper explores the algebraic structures underlying the fractional quantum Hall effect, revealing a $W_{ abla}$ algebra acting on Laughlin wavefunctions and analyzing the effects of two-body interactions.

Contribution

It identifies a $W_{ abla}$ algebra acting on Laughlin states and characterizes the algebraic structure arising from general two-body interactions in FQHE.

Findings

Existence of a $W_{ abla}$ algebra on Laughlin wavefunctions

Representation of algebra generators via fermion and vortex operators

Emergence of algebraic structures from two-body interactions

Abstract

Some algebraic issues of the FQHE are presented. First, it is shown that on the space of Laughlin wavefunctions describing the $\nu =1/m$ FQHE, there is an underlying $W_{\infty}$ algebra, which plays the role of a spectrum generating algebra and expresses the symmetry of the ground state. Its generators are expressed in a second quantized language in terms of fermion and vortex operators. Second, we present the naturally emerging algebraic structure once a general two-body interaction is introduced and discuss some of its properties.