Geometric Transformations and NCCS Theory in the Lowest Landau Level

TL;DR

This paper explores the connection between geometric transformations, non-commutative Chern-Simons theory, and the fractional quantum Hall effect, providing a theoretical framework for understanding Laughlin states through gauge fields and Landau level constraints.

Contribution

It introduces a model linking geometric transformations and non-commutative Chern-Simons theory to the fractional quantum Hall effect and Laughlin states.

Findings

Derivation of a gauge field from area-preserving transformations.

Establishment of constraints corresponding to Gauss law and Landau level conditions.

Identification of physically reasonable solutions as Laughlin states.

Abstract

Chern-Simons type gauge field is generated by the means of the singular area preserving transformations in the lowest Landau level of electrons forming fractional quantum Hall state. Dynamics is governed by the system of constraints which correspond to the Gauss law in the non-commutative Chern-Simons gauge theory and to the lowest Landau level condition in the picture of composite fermions. Physically reasonable solution to this constraints corresponds to the Laughlin state. It is argued that the model leads to the non-commutative Chern-Simons theory of the QHE and composite fermions.