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Journal of Physics A Mathematical and General

Fractional quantum Hall effect on the 2-sphere: a case study of symmetry reduction

Roger Sollie, Kaare Olaussen

Published Jan 21, 1990• 3 citations
Publisher

Abstract

The authors make a case study of how a standard class of Hamiltonians for the fractional quantum Hall effect (modelled in a spherical geometry) can be explicitly symmetry reduced with respect to the full rotation group, and perform finite-size calculations for small particle numbers.
DOI
10.1088/0305-4470/23/2/015

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The authors study a class of models for the fractional quantum Hall effect, where the system is mapped onto the surface of a sphere around a strong magnetic monopole. When the magnetic charge of the monopole is half-integer (in units of h(cross)/e), there exists an SU(2) quasispin group operating on the Fock space of these models. This group commutes with the group of rotations on the sphere, but not with the particle number operator N. This fact may be utilized to easily construct states of definite angular momentum at high particle numbers from states of the same angular momentum at lower particle numbers. Using the Wigner-Eckart theorem the matrix elements of the Hamiltonian between these states can also be constructed. For a large class of pair interactions this leads to a computational method of finding exact eigenstates and eigenvalues of the Hamiltonian.
With the use of Haldane's spherical model, the fractional quantum Hall effect is studied by means of large-scale configuration-interaction calculations. The exact wave function and the ground-state energy are computed numerically up to ten electrons for filling \ensuremath{\nu}=(1/3), and up to seven electrons for filling \ensuremath{\nu}=(1/5). Quasiparticle, quasihole, and exciton energies are computed. The mutual relation between wave functions on the sphere and wave functions on the disk is fully analyzed. A wave function originally proposed by Halperin for the disk geometry is tested on the sphere; the numerical results up to six electrons for \ensuremath{\nu}=(2/5) and \ensuremath{\nu}=(2/9) are favorable.
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.