Partition functions of edge excitations are obtained for non-Abelian Hall states in the second Landau level, such as the anti-Read–Rezayi state, the Bonderson–Slingerland hierarchy and the Wen non-Abelian fluid, as well as for the non-Abelian spin-singlet state. The derivation is straightforward and unique starting from the non-Abelian conformal field theory data and solving the modular invariance conditions. The partition functions provide a complete account of the excitation spectrum and are used to describe experiments of Coulomb blockade and thermopower.
We propose field theories for the bulk and edge of a quantum Hall state in the universality class of the Haldane-Rezayi wavefunction. The bulk theory is associated with the c = −2 conformal field theory. The topological properties of the state, such as the quasiparticle braiding statistics and ground state degeneracy on a torus, may be deduced from this conformal field theory. The 10-fold degeneracy on a torus is explained by the existence of a logarithmic operator in the c = −2 theory; this operator corresponds to a novel bulk excitation in the quantum Hall state. We argue that the edge theory is the c = 1 chiral Dirac fermion, which is related in a simple way to the c = −2 theory of the bulk. This theory is reformulated as a truncated version of a
A (1 + 1)D unitary bosonic rational conformal field theory (RCFT) may be organized according to its genus, a tuple (c,C) consisting of its central charge c and a unitary modular tensor category C which describes the (2 + 1)D topological quantum field theory for which its maximally extended chiral algebra forms a holomorphic boundary condition. We establish a number of results pertaining to RCFTs in “small” genera, by which we informally mean genera with the central charge c and the number of primary operators rank(C) both not too large. We start by completely solving the modular bootstrap problem for theories with at most four primary operators. In particular, we characterize, and provide an algorithm which efficiently computes, the function spaces to which the partition function of any bosonic RCFT with rank(C)≤4 must belong. Using this result, and leveraging relationships between RCFTs and holomorphic vertex operator algebras which come from “gluing” and cosets, we rigorously enumerate all bosonic theories in 95 of the 105 genera (c,C) with c ≤ 24 and rank(C)≤4. This includes as (new) special cases the classification of chiral algebras with three primaries and c < 120/7 ∼ 17.14, and the classification of chiral algebras with four primaries and c < 62/3 ∼ 20.67. We then study two applications of our classification. First, by making use of chiral versions of bosonization and fermionization, we obtain the complete list of purely left-moving fermionic RCFTs with c < 23 as a corollary of the results of the previous paragraph. Second, using a (conjectural) concept which we call “symmetry/subalgebra duality,” we precisely relate our bosonic classification to the problem of determining certain generalized global symmetries of holomorphic vertex operator algebras.
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.
Key to the exact solubility of the unitary minimal models in two-dimensional conformal field theory is the organization of their Hilbert space into Verma modules, whereby all eigenstates of the Hamiltonian are obtained by the repeated action of Virasoro lowering operators onto a finite set of highest-weight states. The usual representation-theoretic approach to removing from all modules zero-norm descendant states generated in such a way is based on the assumption that those states form a nested sequence of Verma submodules built upon singular vectors, i.e., descendant highest-weight states. We show that this fundamental assumption breaks down for the Ramond-sector Verma module with highest weight
c/24 in the even series of
N=1 superconformal minimal models with central charge
c. To resolve this impasse, we conjecture, and prove at low orders, the existence of a nested sequence of linear-dependence relations that enables us to compute the character of the irreducible
c/24 module. Based on this character formula, we argue that imposing modular invariance of the torus partition function requires the introduction of a non-null odd-parity Ramond-sector ground state. This symmetrization of the ground-state manifold allows us to uncover a set of conformally invariant boundary conditions not previously discussed and absent in the odd series of superconformal minimal models, and to derive for the first time a complete set of fusion rules for the even series of those models.
We show that all lowest Landau-level projected and unprojected chiral parton type fractional quantum Hall ground and edge-state trial wave functions, which take the form of products of integer quantum Hall wave functions, can be expressed as conformal field theory (CFT) correlation functions, where we can associate a chiral algebra to each parton state such that the CFT defined by the algebra is the “smallest” such CFT that can generate the corresponding ground and edge-state trial wave functions (assuming that the corresponding chiral algebra does indeed define a physically “sensible” CFT). A field-theoretic generalization of Laughlin's plasma analogy, known as generalized screening, is formulated for these states. If this holds, along with an additional assumption, we argue that the inner products of edge-state trial wave functions, for parton states where the “densest” trial wave function is unique, can be expressed as matrix elements of an exponentiated local action operator of the CFT, generalizing the result of Dubail [], which implies the equality between edge-state and entanglement level counting to state counting in the corresponding CFT. We numerically test this result in the case of the unprojected ν=2/5 composite fermion state and the bosonic ν=1ϕ22 parton state. We discuss how Read's arguments [] still apply, implying that conformal blocks of the CFT defined by the corresponding chiral algebra are valid quasihole trial wave functions, with the adiabatic braiding statistics given by the monodromy of these functions, assuming the existence of a quasiparticle trapping Hamiltonian. Generalizations of these constructions are discussed, with particular attention given to simple current constructions. It is shown that all chiral composite fermion wave functions can be expressed as CFT correlation functions without explicit symmetrization or antisymmetrization and that the ground, edge, and certain quasihole trial wave functions of the ϕnm parton states can be expressed as the conformal blocks of the U(1)⊗SU(n)m WZW models. Finally, we discuss the relation of the ϕ2k series with the Read-Rezayi series, where several examples of quasihole braiding statistics calculations are given. Published by the American Physical Society 2024