Matrix oscillator and Laughlin Hall states

TL;DR

This paper introduces a quantum matrix oscillator model that directly constructs quantum Hall states, linking it to existing models and demonstrating its equivalence to the Laughlin wave function and Calogero model.

Contribution

It presents a novel quantum matrix oscillator framework for quantum Hall states, connecting it to the Polychronakos matrix model and Calogero model, and analyzing its spectrum.

Findings

The quantum matrix oscillator describes electrons in the lowest Landau level with Laughlin-type ground state.

It establishes the equivalence between the quantum matrix oscillator and the Calogero model.

The spectrum of the quantum matrix oscillator matches that of the finite matrix Chern-Simons model on the singlet sector.

Abstract

We propose a quantum matrix oscillator as a model that provides the construction of the quantum Hall states in a direct way. A connection of this model to the regularized matrix model introduced by Polychronakos is established . By transferring the consideration to the Bargmann representation with the help of a particular similarity transformation, we show that the quantum matrix oscillator describes the quantum mechanics of electrons in the lowest Landau level with the ground state described by the Laughlin-type wave function. The equivalence with the Calogero model in one dimension is emphasized. It is shown that the quantum matrix oscillator and the finite matrix Chern-Simons model have the same spectrum on the singlet state sector.