Interactions and non-commutativity in quantum Hall systems

TL;DR

This paper explores how interactions influence the non-commutative geometry in quantum Hall systems, showing that interactions modify the non-commutative parameter and affect filling fractions, with results aligning with known cases.

Contribution

It demonstrates that interactions alter the non-commutative parameter in quantum Hall systems and provides an heuristic method to determine filling fractions under these conditions.

Findings

Interactions renormalize the non-commutative parameter from its non-interacting value.

The effective non-commutative parameter depends on angular momentum.

Filling fractions are renormalized by interactions, consistent with known results.

Abstract

We discuss the role that interactions play in the non-commutative structure that arises when the relative coordinates of two interacting particles are projected onto the lowest Landau level. It is shown that the interactions in general renormalize the non-commutative parameter away from the non-interacting value $\frac{1}{B}$. The effective non-commutative parameter is in general also angular momentum dependent. An heuristic argument, based on the non-commutative coordinates, is given to find the filling fractions at incompressibilty, which are in general renormalized by the interactions, and the results are consistent with known results in the case of singular magnetic fields.