Charge and Current in the Quantum Hall Matrix Model

TL;DR

This paper extends the quantum Hall matrix model to include external fields, analyzing its linear response and demonstrating it captures key features of electron dynamics in the lowest Landau level, despite ordering ambiguities.

Contribution

It introduces couplings to external fields in the quantum Hall matrix model and computes the linear response, including the fractional Hall conductivity and density modulations.

Findings

Correct fractional quantum Hall conductivity obtained

Density modulations in response to magnetic fields demonstrated

Classical model captures key electron dynamics features

Abstract

We extend the quantum Hall matrix model to include couplings to external electric and magnetic fields. The associated current suffers from matrix ordering ambiguities even at the classical level. We calculate the linear response at low momenta -- this is unambigously defined. In particular, we obtain the correct fractional quantum Hall conductivity, and the expected density modulations in response to a weak and slowly varying magnetic field. These results show that the classical quantum Hall matrix models describe important aspects of the dynamics of electrons in the lowest Landau level. In the quantum theory the ordering ambiguities are more severe; we discuss possible strategies, but we have not been able to construct a good density operator, satisfying the pertinent lowest Landau level commutator algebra.