Quantum Hall Physics = Noncommutative Field Theory

TL;DR

This paper establishes a precise equivalence between a matrix-regularized noncommutative Chern-Simons theory and the Laughlin wavefunctions describing fractional quantum Hall states, confirming a conjecture by Susskind.

Contribution

It provides a complete basis of wavefunctions for the matrix Chern-Simons theory and proves its equivalence to Laughlin states at arbitrary filling fractions in the large N limit.

Findings

Exact wavefunctions match Laughlin states for quantum Hall droplets.

Finite matrix Chern-Simons theory is equivalent to composite fermion theory.

Large N limit confirms the conjecture relating noncommutative gauge theory to quantum Hall physics.

Abstract

In this note, we study a matrix-regularized version of non-commutative U(1) Chern-Simons theory proposed recently by Polychronakos. We determine a complete minimal basis of exact wavefunctions for the theory at arbitrary level k and rank N and show that these are in one-to-one correspondence with Laughlin-type wavefunctions describing excitations of a quantum Hall droplet composed of N electrons at filling fraction 1/k. The finite matrix Chern-Simons theory is shown to be precisely equivalent to the theory of composite fermions in the lowest Landau level, believed to provide an accurate description of the filling fraction 1/k fractional quantum Hall state. In the large N limit, this implies that level k noncommutative U(1) Chern-Simons theory is equivalent to the Laughlin theory of the filling fraction 1/k quantum Hall fluid, as conjectured recently by Susskind.