Quantum hall physics equals noncommutative field theory?

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. We show that these have a form highly reminiscent of Laughlin-type wavefunctions describing excitations of a quantum Hall droplet composed of N electrons at filling fraction 1/k, and demonstrate explicitly that the energy levels and degeneracies of the two theories are identical. Thus, at the level of the Hilbert space, finite matrix Chern-Simons theory is identical 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 provides further evidence 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.