We propose trial wave functions for quasiparticle and exciton excitations of the Moore-Read Pfaffian fractional quantum Hall states, both for bosons and for fermions, and study these numerically. Our construction of trial wave functions employs a picture of the bosonic Moore-Read state as a symmetrized double layer composite fermion state. We obtain the number of independent angular momentum multiplets of quasiparticle and exciton trial states for systems of up to 20 electrons. We find that the counting for quasielectrons at large angular momentum on the sphere matches that expected from the conformal field theory (CFT) that describes the Moore-Read state's boundary theory. In particular, the counting for quasielectrons is the same as for quasiholes, in accordance with the idea that the CFT describing both sides of the Fractional Quantum Hall plateau should be the same. We also show that our trial wave functions have good overlaps with exact wave functions obtained using various interactions, including second Landau level Coulomb interactions and the three-body delta interaction for which the Pfaffian states and their quasiholes are exact ground states. We discuss how these results relate to recent work by Sreejith et al. on a similar set of trial wave functions for excitations over the Pfaffian state as well as to earlier work by Hansson et al., which has produced trial wave functions for quasiparticles based on conformal field theory methods and by Bernevig and Haldane, which produced trial wave functions based on clustering properties and ``squeezing.''
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 $\ensuremath{\nu}=2+3/5,2+2/5.$
We propose trial wave functions for quasiparticle and exciton excitations of the Moore-Read Pfaffian
fractional quantum Hall states, both for bosons and for fermions, and study these numerically.
Our construction of trial wave functions employs a picture of the bosonic Moore-Read state as a
symmetrized double layer composite fermion state. We obtain the number of independent angular
momentum multiplets of quasiparticle and exciton trial states for systems of up to 20 electrons. We
find that the counting for quasielectrons at large angular momentum on the sphere matches that
expected from the CFT which describes the Moore-Read state's boundary theory. In particular,
the counting for quasielectrons is the same as for quasiholes, in accordance with the idea that the
CFT describing both sides of the FQH plateau should be the same. We also show that our trial
wave functions have good overlaps with exact wave functions obtained using various interactions,
including second Landau level Coulomb interactions and the 3-body delta interaction for which the
Pfaffian states and their quasiholes are exact ground states. We discuss how these results relate
to recent work by Sreejith et al. on a similar set of trial wave functions for excitations over the
Pfaffian state as well as to earlier work by Hansson et al., which has produced trial wave functions
for quasiparticles based on conformal field theory methods and by Bernevig and Haldane, which
produced trial wave functions based on clustering properties and 'squeezing'.
We study the non-Abelian statistics of quasiparticles in the Ising-type quantum Hall states which are likely candidates to explain the observed Hall conductivity plateaus in the second Landau level, most notably the one at filling fraction ν = 5/2.We complete the program started in V. Gurarie and C. Nayak, [Nucl. Phys. B 506, 685 (1997)].and show that the degenerate four-quasihole and six-quasihole wave functions of the Moore-Read Pfaffian state are orthogonal with equal constant norms in the basis given by conformal blocks in a c = 1 + 1 2 conformal field theory.As a consequence, this proves that the non-Abelian statistics of the excitations in this state are given by the explicit analytic continuation of these wave functions.Our proof is based on a plasma analogy derived from the Coulomb gas construction of Ising model correlation functions involving both order and (at most two) disorder operators.We show how this computation also determines the non-Abelian statistics of collections of more than six quasiholes and give an explicit expression for the corresponding conformal block-derived wave functions for an arbitrary number of quasiholes.Our method also applies to the anti-Pfaffian wave function and to Bonderson-Slingerland hierarchy states constructed over the Moore-Read and anti-Pfaffian states.
We study the non-Abelian statistics of quasiparticles in the Ising-type quantum Hall states which are likely candidates to explain the observed Hall conductivity plateaus in the second Landau level, most notably the one at filling fraction $\ensuremath{\nu}=5/2$. We complete the program started in V. Gurarie and C. Nayak, [Nucl. Phys. B 506, 685 (1997)]. and show that the degenerate four-quasihole and six-quasihole wave functions of the Moore-Read Pfaffian state are orthogonal with equal constant norms in the basis given by conformal blocks in a
c=1+21 conformal field theory. As a consequence, this proves that the non-Abelian statistics of the excitations in this state are given by the explicit analytic continuation of these wave functions. Our proof is based on a plasma analogy derived from the Coulomb gas construction of Ising model correlation functions involving both order and (at most two) disorder operators. We show how this computation also determines the non-Abelian statistics of collections of more than six quasiholes and give an explicit expression for the corresponding conformal block-derived wave functions for an arbitrary number of quasiholes. Our method also applies to the anti-Pfaffian wave function and to Bonderson-Slingerland hierarchy states constructed over the Moore-Read and anti-Pfaffian states.
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.