Matrix Effective Theories of the Fractional Quantum Hall effect

TL;DR

This paper reviews matrix variable effective theories, specifically Maxwell-Chern-Simons matrix gauge theory, to describe the fractional quantum Hall effect, connecting matrix ground states with Laughlin and Jain states.

Contribution

It demonstrates that matrix gauge theories provide a natural framework for understanding composite fermion states in the fractional quantum Hall effect.

Findings

Matrix ground states correspond to Laughlin and Jain states.

Matrix theory offers a physical limit with commuting matrices within the same phase.

Matrix approach generalizes quantum Hall effect through gauge invariance.

Abstract

The present understanding of nonperturbative ground states in the fractional quantum Hall effect is based on effective theories of the Jain "composite fermion" excitations. We review the approach based on matrix variables, i.e. D0 branes, originally introduced by Susskind and Polychronakos. We show that the Maxwell-Chern-Simons matrix gauge theory provides a matrix generalization of the quantum Hall effect, where the composite-fermion construction naturally follows from gauge invariance. The matrix ground states obtained by suitable projections of higher Landau levels are found to be in one-to-one correspondence with the Laughlin and Jain hierarchical states. The matrix theory possesses a physical limit for commuting matrices that could be reachable while staying in the same phase.