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arXiv:hep-th/0307175

Noncommuting coordinates in the Hall effect and in vortex dynamics

P. A. Horváthy

Published Jul 18, 2003• 3 citations
arXivHTMLPDF

Abstract

Laughlin's Ansatz to explain the fractional Quantum Hall effect is derived by coupling a particle associated with ``exotic'' the two-fold central extension of the planar Galilei group. The reduced system is identical to the one used to describe the dynamics of vortices in an incompressible planar fluid.
DOI
10.48550/arxiv.hep-th/0307175

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Owing to the two-parameter central extension of the planar Galilei group, anon relativistic particle in the plane admits an extra structure, which yieldsnoncommuting coordinates. For a particle moving in a magnetic fieldperpendicular to the plane, the two parameters combine with the magnetic fieldto provide an effective mass. For vanishing effective mass the phase spaceadmits a two-dimensional reduction, which represents the condensation tocollective ``Hall'' motions and justifies the rule called ``Peierlssubstitution''. Then Geometric Quantization yields the wave functions proposedby Laughlin to describe the Fractional Quantum Hall Effect.
Taking advantage of the two-parameter central extension of the planar Galilei group, we construct a non relativistic particle model in the plane. Owing to the extra structure, the coordinates do not commute. Our model can be viewed as the non-relativistic counterpart of the relativistic anyon considered before by Jackiw and Nair. For a particle moving in a magnetic field perpendicular to the plane, the two parameters combine with the magnetic field to provide an effective mass. For vanishing effective mass the phase space admits a two-dimensional reduction, which represents the condensation to collective “Hall” motions, and justifies the rule called “Peierls substitution”. Quantization yields the wave functions proposed by Laughlin to describe the Fractional Quantum Hall Effect.
Laughlin's quantum Hall wave function is obtained as an exact ground state of an N-particle Hamiltonian, in which the electrons themselves are coupled to the Chern-Simons field. The proof requires quantum field-theoretic methods in the Schr\"odinger picture, and clearly exhibits the relation to conformal field theory since the Knizhnik-Zamolodchikov connection shows up and the Laughlin ground state is recognized as a conformal block. A refined version of this approach can be applied for toroidal boundary conditions, i.e., the electrons are confined to the physical interior of the Hall sample. The result turns out to be a nontrivial modification of the p-fold degenerate Haldane-Rezayi wave function. Furthermore, it is shown that the degeneracy of the ground state is accounted for by the quantum group Uq(su2){U}_{q}{(su}_{2})Uq​(su2​) with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.
Electrons in two dimensions in a high magnetic field can condense into an incompressible quantum fluid state when the electron density is equal to certain rational fractions of the flux density. Much progress has been made in understanding the nature of this state based on the ansatz wavefunction of Laughlin. Here, we show that the incompressible nature of the quantum fluid can be derived from a semiclassical evaluation of the imaginary time propagator. The dominant classical paths consist of cooperative ring exchange events. When these condense, they mediate a logarithmic interaction between density fluctuations, and produce a precise fractional quantization condition for the quasi-particle charges. We further show that the essential long wavelength properties of Laughlin's ground state and quasi-particle wavefunctions can be derived constructively from this approach. We conclude with some speculations concerning the nature of the off-diagonal order in the quantum Hall liquid.
We present an application of collective field theory to the fractional quantum Hall effect (FQHE). We first express the condition, that the electrons are all in the lowest Landau level, as a constraint equation for the state functional. We then derive the fractional filling factor from this equation together with the no-free-vortex assumption. A hierarchy of filling factors is derived by using the particle-vortex dual transformations. In the final section we discuss an attempt at a dynamical theory of FQHE, which would justify the no-free-vortex assumption. A derivation of Laughlin’s wave function with and without quasi-hole excitations is also given.
Through the calculations of the diagonal and off-diagonal elements of the full NNN-particle density matrix we show that all essential features of the Laughlin variational wave functions can be derived in a generalized cooperative-ring-exchange theory of the fractional quantum Hall fluid.