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Physical review. B, Condensed matter

Symmetries of the Yoshioka-Halperin-Lee Hamiltonian for the fractional quantum Hall effect

Myung-Hoon Chung, Jin Hong, Jae-Hoon Kwon

Published Jan 15, 1997• 8 citations
Publisher

Abstract

A many-particle Hamiltonian is considered in connection with the fractional quantum Hall effect for fractional filling factors \ensuremath{\nu}1. Symmetries of the Hamiltonian are found, and representations of the corresponding symmetry group are studied. By using the characters in the representations, the Hilbert space is decomposed. The decompositions of the Hilbert space are compared with numerical works done with a small number of electrons.

DOI
10.1103/physrevb.55.2249

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Physical review. B, Condensed matter•Feb 15, 2000•1 citation
A many-particle Hamiltonian is considered in connection with the fractional quantum Hall effect for fractional filling factors \ensuremath{\nu}1. Symmetries of the Hamiltonian are found, and representations of the corresponding symmetry group are studied. By using the characters in the representations, the Hilbert space is decomposed. The decompositions of the Hilbert space are compared with numerical works done with a small number of electrons.
The eigenstates of interacting electrons in the fractional quantum Hall phase typically form fairly well-defined bands in the energy space. We study systems of six and eight electrons for filling factor 3/7>\ensuremath{\nu}> 2) / 7 and show that composite fermion theory gives insight into the origin of these bands and provides an accurate and complete microscopic description of the strongly correlated many-body states in the low-energy bands. This implies that, somewhat like in Landau's Fermi liquid theory, there is a one-to-one analogy between the low-energy Hilbert space of strongly interacting electrons in the fractional quantum Hall effect and that of weakly interacting electrons in the integer quantum Hall effect.
We study fractional quantum Hall states at filling fractions in the Jain sequences using the framework of composite Dirac fermions. Synthesizing previous work, we write an effective field theory consistent with all symmetry requirements, including Galilean invariance and particle-hole symmetry. Employing a Fermi-liquid description, we demonstrate the appearance of the Girvin-Macdonald-Platzman algebra and compute the dispersion relation of neutral excitations and various response functions. Our results satisfy requirements of particle-hole symmetry. We show that while the dispersion relation obtained from the modified random-phase approximation (MRPA) of the Halperin-Lee-Read (HLR) theory is particle-hole symmetric, correlation functions obtained from this scheme are not. The results of the Dirac theory are shown to be consistent with the Haldane bound on the projected structure factor, while those of the MPRA of the HLR theory violate it.
The second-quantized form of the Laughlin states for the fractional quantum Hall effect is discussed by decomposing the Laughlin wavefunctions into the N-particle Slater basis. A general formula is given for the expansion coefficients in terms of the characters of the symmetric group, and the expansion coefficients are shown to possess numerous interesting symmetries. For expectation values of the density operator it is possible to identify individual dominant Slater states of the correct uniform bulk density and filling fraction in the physically relevant N→∞ limit.
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.