On the modular covariance properties of composite fermions on the torus

TL;DR

This paper demonstrates that the composite fermion construction on the torus maintains modular covariance, both before and after projection, which is essential for the consistency of the theory in quantum Hall systems.

Contribution

It establishes the modular covariance of composite fermion states on the torus, including under recent projection methods, ensuring their proper mathematical and physical consistency.

Findings

Modular covariance holds for CF states on the torus both before and after projection.

Modular properties are preserved under exact and JK projection methods.

Proper CF states must have no holes in the occupied $\\Lambda$-levels.

Abstract

In this work we show that the composite fermion construction for the torus geometry is modular covariant. We show that this is the case both before and after projection, and that modular covariance properties are preserved under both exact projection and under JK projection which was recently introduced by Pu, Wu, and Jain (PRB 96, 195302 (2017)). It is crucial for the modular properties to hold that the CF state is a proper state, i.e. that there are no holes in the occupied $\Lambda$-levels.