We review the main results of the effective description of the Quantum Hall fluid for the Jain fillings, nu=m/2pm+1, and the non-standard ones nu=m/pm+2 by a conformal field theory (CFT) in two dimensions. It is stressed the unifying character of the m-reduction procedure to construct appropriate twisted CFT models, called Twisted Models (TM), which by construction reproduce the Quantum Hall topological properties at those fillings. Indeed for the Jain plateaux we find that the different descriptions given in the literature fall into different sectors of the TM for the torus topology. Other interesting aspects are explicitly seen for the m=2 non standard filling nu=1/p+1 (the pairing case) as the merging of non-Abelian statistics or the instability of the TM model (c=2) versus the Moore-Read one (c=3/2). Furthermore by using Boundary CFT techniques the presence of localized impurities and/or dissipation is shown to be closely connected with the twisted sector of the TM, whose presence assures the consistency of the construction and whose role in describing non trivial global properties of 2D quantum condensed matter systems is still under study.
It is known that a subset of fractional quantum Hall wave functions has been expressed as conformal field theory (CFT) correlators, notably the Laughlin wave function [Phys. Rev. Lett. 50, 1395 (1983)] at filling factor $\ensuremath{\nu}=1∕m$ (
m odd) and its quasiholes, and the Pfaffian wave function at $\ensuremath{\nu}=1∕2$ and its quasiholes. We develop a general scheme for constructing composite-fermion (CF) wave functions from conformal field theory. Quasiparticles at $\ensuremath{\nu}=1∕m$ are created by inserting anyonic vertex operators,
P1∕m(z), that replace a subset of the electron operators in the correlator. The one-quasiparticle wave function is identical to the corresponding CF wave function, and the two-quasiparticle wave function has correct fractional charge and statistics and is numerically almost identical to the corresponding CF wave function. We further show how to exactly represent the CF wave functions in the Jain series $\ensuremath{\nu}=s∕(2sp+1)$ [Phys. Rev. Lett. 63, 199 (1989); Composite Fermions (Cambridge University Press, Cambridge, 2007)] as the CFT correlators of a new type of fermionic vertex operators,
Vp,n(z), constructed from
n free compactified bosons; these operators provide the CFT representation of composite fermions carrying
2p flux quanta in the
nth CF Landau level. We also construct the corresponding quasiparticle and quasihole operators and argue that they have the expected fractional charge and statistics. For filling fractions
2∕5 and
3∕7, we show that the chiral CFTs that describe the bulk wave functions are identical to those given by Wen's general classification [Int. J. Mod. Phys. B 6, 1711 (1992); Adv. Phys. 44, 405 (1995)] of quantum Hall states in terms of
K matrices and
l and
t vectors, and we propose that to be generally true. Our results suggest a general procedure for constructing quasiparticle wave functions for other fractional Hall states, as well as for constructing ground states at filling fractions not contained in the principal Jain series.
The Pfaffian quantum Hall states, which can be viewed as involving pairing either of spin-polarized electrons or of composite fermions, are generalized by finding the exact ground states of certain Hamiltonians with
k+1-body interactions, for all integers
k> 1. The remarkably simple wave functions of these states involve clusters of k particles, and are related to correlators of parafermion currents in two-dimensional conformal field theory. The
k=2 case is the Pfaffian. For
k> 2, the quasiparticle excitations of these systems are expected to possess non-Abelian statistics, like those of the Pfaffian. For
k=3, these ground states have large overlaps with the ground states of the (two-body) Coulomb-interaction Hamiltonian for electrons in the first excited Landau level at total filling factors $\ensuremath{\nu}=2+3/5,2+2/5.$
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.
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.
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.