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.
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.
Abstract We present a novel way to organise the finite size spectra of a class of conformal field theories (CFT) with = 2 or (nonlinear) = 4 superconformal symmetry. Generalising the spinon basis of the SU ( n ) 1 WZW theories, we introduce supersymmetric spinons ( ϕ − , ϕ + ) , which form a representation of the supersymmetry algebra. In each case, we show how to construct a multi-spinon basis of the chiral CFT spectra. The multi-spinon states are labelled by a collection { n j } of (discrete) momenta. The state-content for given choice of { n j } is determined through a generalised exclusion principle, similar to Haldane's ‘motif’ rules for the SU ( n ) 1 theories. In the simplest case, which is the = 2 superconformal theory with central charge c = 1, we develop an algebraic framework similar to the Yangian symmetry of the SU ( 2 ) 1 theory. It includes an operator H 2 , akin to a CFT Haldane–Shastry Hamiltonian, which is diagonalised by multi-spinon states. In all cases studied, we obtain finite partition sums by capping the spinon-momenta to some finite value. For the = 2 superconformal CFTs, this finitisation precisely leads to the so-called M k supersymmetric lattice models with characteristic order- k exclusion rules on the lattice. Finitising the c = 2 CFT with nonlinear = 4 superconformal symmetry similarly gives lattice model partition sums for spin-full Fermions with on-site and nearest neighbour exclusion.