On the NCCS model of the quantum Hall fluid

TL;DR

This paper explores how area non-preserving transformations in the non-commutative plane can map integer quantum Hall states to fractional ones, revealing a deep connection via non-unitary transformations in a non-commutative Chern-Simons framework.

Contribution

It introduces a novel class of geometric transformations that relate integer and fractional quantum Hall effects through a non-commutative gauge theory approach.

Findings

Transformations are generated by vector fields satisfying Gauss law.

Reconstruction of the field-theory Lagrangian in the non-commutative setting.

Establishment of a link between integer and fractional QHE via non-unitary similarity transformations.

Abstract

Area non-preserving transformations in the non-commutative plane are introduced with the aim to map the $\nu=1$ integer quantum Hall effect (IQHE) state on the fractional quantum Hall effect (FQHE) $\nu=\frac{1}{2p+1}$ FQHE states. Using the hydrodynamical description of the quantum Hall fluid, it is shown that these transformations are generated by vector fields satisfying the Gauss law in the interacting non-commutative Chern-Simons gauge theory, and the corresponding field-theory Lagrangian is reconstructed. It is demonstrated that the geometric transformations induce quantum-mechanical non-unitary similarity transformations, establishing the interplay between integral and fractional QHEs.