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arXiv:cond-mat/0208429 · Physical review. B, Condensed matter

Conformal field theory approach to bulk wave functions in the fractional quantum Hall effect

Michael Flohr, Klaus Osterloh

Published Jun 18, 2003• 10 citations
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Abstract

We propose to describe bulk wave functions of fractional quantum Hall states in terms of correlators of nonunitary $b/c$-spin systems. These yield a promising conformal field-theory analogon of the composite fermion picture of Jain. Fractional statistics are described by twist fields which naturally appear in the $b/c$-spin systems. We provide a geometrical interpretation of our approach in which bulk wave functions are seen as holomorphic functions over a ramified covering of the complex plane, where the ramification precisely resembles the fractional statistics of the quasiparticle excitations in terms of branch points on the complex plane. To extend Jain's main series, we use the concept of composite fermions pairing to spin singlets, which enjoys a natural description in terms of the particular $c=\ensuremath{-}2$ $b/c$-spin system as known from the Haldane-Rezayi state. In this way we derive conformal field-theory proposals for lowest Landau-level bulk wave functions for more general filling fractions. We obtain a natural classification of the experimentally confirmed filling fractions, which does not contain prominent unobserved fillings. Furthermore, our scheme fits together with classifications in terms of K matrices of effective multilayer theories leading to striking restrictions of these coupling matrices.

DOI
10.1103/physrevb.67.235316

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We propose to describe bulk wave functions of fractional quantum Hall states in terms of correlators of nonunitary b/cb/cb/c-spin systems. These yield a promising conformal field-theory analogon of the composite fermion picture of Jain. Fractional statistics are described by twist fields which naturally appear in the b/cb/cb/c-spin systems. We provide a geometrical interpretation of our approach in which bulk wave functions are seen as holomorphic functions over a ramified covering of the complex plane, where the ramification precisely resembles the fractional statistics of the quasiparticle excitations in terms of branch points on the complex plane. To extend Jain's main series, we use the concept of composite fermions pairing to spin singlets, which enjoys a natural description in terms of the particular c=\ensuremath−2c=\ensuremath{-}2c=\ensuremath−2 b/cb/cb/c-spin system as known from the Haldane-Rezayi state. In this way we derive conformal field-theory proposals for lowest Landau-level bulk wave functions for more general filling fractions. We obtain a natural classification of the experimentally confirmed filling fractions, which does not contain prominent unobserved fillings. Furthermore, our scheme fits together with classifications in terms of K matrices of effective multilayer theories leading to striking restrictions of these coupling matrices.
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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.
Construction of a series of topologically distinct ν=2/5\nu=2/5ν=2/5 fractional quantum Hall wave functions by conformal field theory
In this paper, a series of ν=2/5\nu=2/5ν=2/5 fractional quantum Hall wave functions are constructed from conformal field theory(CFT). They share the same topological properties with states constructed by Jain's composite fermion approach. Upon exact lowest Landau level(LLL) projection, some of Jain's composite fermion states would not survive if constraints on Landau level indices given in appendices of this paper are not satisfied. By contrast, states constructed from CFT always stay in LLL. These states are characterized by different multi-body relative angular momenta and topological shifts, thus belong to different topological sectors. As a by-product, in appendices we prove the necessary conditions for general $ \nu=p/(2p+1) $ composite fermion states to have non-vanishing LLL-projection.
We show that the quantum Hall wave functions for the ground states in the Jain series nu=n/(2np+1) can be exactly expressed in terms of correlation functions of local vertex operators Vn corresponding to composite fermions in the nth composite-fermion (CF) Landau level. This allows for the powerful mathematics of conformal field theory to be applied to the successful CF phenomenology. Quasiparticle and quasihole states are expressed as correlators of anyonic operators with fractional (local) charge, allowing a simple algebraic understanding of their topological properties that are not manifest in the CF wave functions. Moreover, our construction shows how the states in the nu=n/(2np+1) Jain sequence may be interpreted as condensates of quasiparticles.