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arXiv:cond-mat/9703217 · International Journal of Modern Physics B

Fractional Quantum Hall Effect States as Exact Ground States

Ranjan Kumar Ghosh, Sumathi Rao

Published May 10, 1998• 4 citations
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Abstract

We construct many particle Hamiltonians for which the Laughlin and Jain wavefunctions are exact ground states. The Hamiltonians involve fermions in a magnetic field and with inter-particle interactions. For the Laughlin wave-functions, the interactions involve two- and three-body correlations similar to the Chern–Simons interactions, whereas for the projected Jain wave-functions, N-body interactions (which cannot be explicitly written down in general) are involved.

DOI
10.1142/s0217979298000612

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We construct many particle Hamiltonians for which the Laughlin and Jain wavefunctions are exact ground states. The Hamiltonians involve fermions in a magnetic field and with inter-particle interactions. For the Laughlin wave-functions, the interactions involve two- and three-body correlations similar to the Chern–Simons interactions, whereas for the projected Jain wave-functions, N-body interactions (which cannot be explicitly written down in general) are involved.
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.
Laughlin's wave functions, which describe the fractional quantum Hall effect at filling factors $\ensuremath{\nu}=1/(2k+1)$, can be obtained as correlation functions in a conformal field theory, and recently, this construction was extended to Jain's composite fermion wave functions at filling factors $\ensuremath{\nu}=n/(2kn+1)$. Here, we generalize this latter construction and present ground state wave functions for all quantum Hall hierarchy states that are obtained by successive condensation of quasielectrons (as opposed to quasiholes) in the original hierarchy construction. By considering these wave functions on a cylinder, we show that they approach the exact ground states, which are the Tao-Thouless states, when the cylinder becomes thin. We also present wave functions for the multihole states, make the connection to Wen's general classification of Abelian quantum Hall fluids, and discuss whether the fractional statistics of the quasiparticles can be analytically determined. Finally, we discuss to what extent our wave functions can be described in the language of composite fermions.
It has long been known that Laughlin's wave functions, describing the fractional quantum Hall effect at filling fractions ν=1/(2k+1), can be obtained as correlation functions in conformal field theory. We show how to generalize this approach to construct explicit trial wave functions for all states in the quantum Hall hierarchy corresponding to quasiparticle (as opposed to quasihole) condensates, including the recently observed state at ν= 4/11. At the filling fractions ν= n/(2np+1) this construction exactly reproduces Jain's composite fermion wave functions. An explicit connection is made to Wen's topological classification of FQH states.
The present understanding of nonperturbative ground states in the fractional quantum Hall effect is based on effective theories of the Jain ‘composite fermion’ excitations. We review the approach based on matrix variables, i.e. D0 branes, originally introduced by Susskind and Polychronakos. We show that the Maxwell–Chern–Simons matrix gauge theory provides a matrix generalization of the quantum Hall effect, where the composite-fermion construction naturally follows from gauge invariance. The matrix ground states obtained by suitable projections of higher Landau levels are found to be in one-to-one correspondence with the Laughlin and Jain hierarchical states. The matrix theory possesses a physical limit for commuting matrices that could be reachable while staying in the same phase.
We introduce one-dimensional lattice models with exact matrix-product ground states describing the fractional quantum Hall (FQH) states in Laughlin series (given by filling factors $\ensuremath{\nu}=1/q$) on torus geometry. Surprisingly, the exactly solvable Hamiltonian has the same mathematical structure as that of the pseudopotential for the Laughlin wave function, and naturally derives the general properties of the Laughlin wave function such as the Z2{Z}_{2}Z2​ properties of the FQH states and the fermion-boson relation. The obtained exact ground states have high overlaps with the Laughlin states and well describe their properties. Using the matrix product method, density functions and correlation functions are calculated analytically. Especially, obtained entanglement spectra reflect gapless edge states as was discussed by Li and Haldane.