A Matrix Model for \nu_{k_1k_2}=\frac{k_1+k_2}{k_1 k_2} Fractional Quantum Hall States

TL;DR

This paper introduces a matrix model for fractional quantum Hall states with specific filling factors, utilizing gauge invariance and hierarchy ideas, and interprets electrons as fractional D0-branes.

Contribution

It develops a novel matrix model for FQH states with general filling factors _{k_1k_2}, incorporating gauge invariance and hierarchy concepts, and interprets electrons as fractional D0-branes.

Findings

Constructed vacuum configurations for series _{p_1p_2}

Demonstrated degeneracy lifting due to non-commutative geometry

Illustrated the formalism for =2/5 state

Abstract

We propose a matrix model to describe a class of fractional quantum Hall (FQH) states for a system of (N_1+N_2) electrons with filling factor more general than in the Laughlin case. Our model, which is developed for FQH states with filling factor of the form \nu_{k_1k_2}=\frac{k_1+k_2}{k_1k_2} (k_1 and k_2 odd integers), has a U(N_1)\times U(N_2) gauge invariance, assumes that FQH fluids are composed of coupled branches of the Laughlin type, and uses ideas borrowed from hierarchy scenarios. Interactions are carried, amongst others, by fields in the bi-fundamentals of the gauge group. They simultaneously play the role of a regulator, exactly as does the Polychronakos field. We build the vacuum configurations for FQH states with filling factors given by the series \nu_{p_1p_2}=\frac{p_2}{p_1p_2-1}, p_1 and p_2 integers. Electrons are interpreted as a condensate of fractional D0-branes and the usual degeneracy of the fundamental state is shown to be lifted by the non-commutative geometry behaviour of the plane. The formalism is illustrated for the state at \nu={2/5}.