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Physical review. B, Condensed matter

Many-particle exchanges and the fractional quantized Hall effect

S. T. Chui

Published Dec 15, 1985• 15 citations
Publisher

Abstract

Evidence is presented which suggests that many-particle exchanges stabilizes the ground state at particular filling factors. Common features are seen in the Laughlin trial wave function, in charge-density-wave wave functions as discussed by Tosatti and Parrinello and more recently by Chui Ma, and Hakim, as well as in the eigenstates of small clusters of particles.

DOI
10.1103/physrevb.32.8438

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Physical review. B, Condensed matter•Feb 15, 2000•1 citation
Evidence is presented which suggests that many-particle exchanges stabilizes the ground state at particular filling factors. Common features are seen in the Laughlin trial wave function, in charge-density-wave wave functions as discussed by Tosatti and Parrinello and more recently by Chui Ma, and Hakim, as well as in the eigenstates of small clusters of particles.
The many-particle exchange integral for a class of correlated Gaussian wave functions for the fractionally quantized Hall effect are discussed. When the correlation is of a long-range nature, the evaluation of this exchange can be cast into languages involving the study of the energies and fluctuations of grain boundaries in two dimensions. For a class of trial wave functions studied recently by Chui, Ma, and Hakim, exchanges involving a large number of particles are significant in that, for a loop that consists of two long parallel boundaries, the overlap integral actually increases as the number of particles involved in the exchange is increased. The question of a collection of boundaries is discussed. We find that there is a repulsion between the boundaries of a range proportional to the size of the boundary.
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.