$W_{1+\infty}, Similarity Transformation and Interplay Between Integer and Fractional Quantum Hall Effect

TL;DR

This paper explores the mathematical relationship between integer and fractional quantum Hall effects using similarity transformations of the $W_{1+ abla}$ algebra, connecting different filling fractions through gauge potentials and deriving the Laughlin wavefunction.

Contribution

It introduces a non-unitary similarity transformation linking algebraic representations of quantum Hall states and derives the second-quantized Laughlin wavefunction.

Findings

Unified algebraic framework for integer and fractional quantum Hall states

Connection between gauge potentials and algebraic transformations

Derivation of the second-quantized Laughlin function

Abstract

We consider non-unitary similarity transformation, interconnecting the $W_{1+\infty}$ algebra representations for the fractional $\nu=\frac{1}{2p+1}$ and integer $\nu=1$ filling fractions. This transformation corresponds to the introduction of the complex abelian Chern-Simons gauge potentials, in terms of which the field-theoretic description of FQHE can be developed. The Jain's composite fermion approach and Lopez-Fradkin equivalence assertion are considered from the point of view of unitary and similarity transformations. As an application the second-quantized form of Laughlin function is derived.