A Quantum Hall Fluid of Vortices

In this note we demonstrate that vortices in a non-relativistic Chern-Simons theory form a quantum Hall fluid. We show that the vortex dynamics is controlled by the matrix mechanics previously proposed by Polychronakos as a description of the quantum Hall droplet. As the number of vortices becomes large, they fill the plane and a hydrodynamic treatment becomes possible, resulting in the non-commutative theory of Susskind. Key to the story is the recent D-brane realisation of vortices and their moduli spaces. The Introduction Chern-Simons theories [1] provide an effective, long distance description of the fractional quantum Hall effect (FQHE). In fact, they provide several such descriptions. The range of models on the market fall roughly into one of two categories depending on the physical interpretation of the vector potential A appearing in the A ∧ F term. In the initial papers on the subject [2, 3], A acts as a statistical gauge field of the type suggested by Wilczek [4]. Its role is to endow the excitations of the model with the charge and statistics appropriate to the quantum Hall system. Later works concentrate on hydrodynamic properties of the quantum Hall fluid in which either A or F are vector fields associated with conserved currents and charge density [5, 6]. More recently, Susskind has suggested that the hydrodynamic properties of the quantum Hall fluid are captured by a Chern-Simons theory at level k, defined on a noncommutative background [9]. The electrons sit at Laughlin filling fraction ν = 1/(k+1) and the fluid fills the infinite plane. Subsequently, Polychronakos proposed a matrix model regularisation of Susskind’s theory in order to describe a finite quantum Hall droplet consisting of N electrons [10]. As N → ∞, the droplet expands to fill the plane and we recover Susskind’s non-commutative dynamics. Several properties of this matrix model have since been explored, including the relationship to Laughlin wavefunctions [11, 12] and the coupling to external electromagnetic fields [13]. In this paper we study a non-relativistic Chern-Simons theory defined on an ordinary, mundane space in which the coordinates commute. The theory does not give an immediate description of a fractional quantum Hall fluid, but rather defines a background into which spin-polarised (i.e. spinless) electrons may be injected. These electrons arise as the vortices of the theory and we show that their quantum dynamics is controlled by the matrix model of Polychronakos. The vortices thus form a fractional quantum Hall droplet. As the number of vortices becomes large, they may be described by Susskind’s hydrodynamic, non-commutative Chern-Simons theory. The key to the connection between vortex dynamics and the FQHE is provided by the recent string theory realisation of vortices and their moduli spaces given in [14]. While [14] considered vortices in the relativistic Maxwell-Higgs theory, we here extend the results to the non-relativistic Chern-Simons case. Rather than present a new Dbrane picture, we instead make use of known connections between vortex dynamics in For readers whose brane activity usually takes place at the Planck scale, introductions to facets of the FQHE may be found in [2, 7, 8].