Collective Coordinate Action for Charged Sigma-Model Vortices in Finite Geometries

In this Letter the method of Lund is applied to formulate a variational principle for the motion of charged vortices in an effective non-linear Schrödinger field theory describing finite size two-dimensional quantum Hall samples under the influence of an arbitrary perpendicular magnetic field. Freezing out variations in the modulus of the effective field yields a U(1) sigma-model. A duality transformation on the sigma-model reduces the problem to finding the Green function for a related electrostatics problem. This duality illuminates the plasma analogy to the Laughlin wave function. ∗ [email protected] The fractional quantum Hall effect shares a number of remarkable features with the macroscopic quantum phenomena of superconductivity and superfluidity. It occurs only at very low temperatures, is dissipationless and is a result of a broken (translational) symmetry. [1] It is reasonable to assume that there should exist an effective field theory [2] for the macroscopic quantum state of fractional quantum Hall systems. As in the effective theories of the other macroscopic quantum phenomena, there are vortex excitations in the effective theory of the quantum Hall effect. However, unlike their better-known cousins, these vortices are point-like (because of the two-dimensional nature of the effect) and play the central role in the physics of this phenomenon. An effective field theory, or Ginzburg-Landau, description of the fractional quantum Hall effect provides a beautiful characterization of the fractionally charged quasiparticles necessary for fractional conductivity as charged vortex excitations of the quantum Hall fluid. The quantum mechanics of a charged vortex yields the lowest Landau level states of a particle of equal charge, [3] even though vortex mechanics is quite different in detail from the usual charged particle mechanics. Because vortices are extended field configurations, the geometry of the sample in which they occur determines the Hamiltonian and hence their dynamics. The purpose of this Letter is to derive the action for arbitrary sample geometry. Vortices in an infinite plane have a pair-wise logarithmic interaction as well as a singlevortex interaction with the background magnetic field. When boundaries are present, there is an additional vortex—boundary interaction which can be written with the help of image vortices in simple geometries. The logarithmic interaction is analogous to vortex-vortex interactions in fluid mechanics. The center of one vortex is carried along with the bulk motion of fluid around another. The external vector potential corresponds to a background motion of the fluid with vorticity density equal to the applied magnetic field. Motion in a uniform field, for instance, is analogous to a uniform rotation about some point. In this Letter, the duality between this viewpoint and an electrostatic description will be used to derive the expression for the collective Hamiltonian in a two-dimensional bounded region for an arbitrary applied magnetic field. The derivation of the collective action for the motion of the vortex centers starts with the elegant idea of F. Lund, [4] which is to substitute the collective vortex ansatz for the field configuration with vortices at X, A = 1, . . . , N , into the effective field theory action and 2 reduce the action to a function of the collective coordinates, SV [{X(t)}] = Seft[ψ(x, t; {X(t)})]. (1) Here we follow Lund’s method but we must be careful in choosing the multi-vortex ansatz, ψ = ψ(x, t; {XA(t)}). The usual ansatz in an infinite region, ψ = √ ρ0 exp {