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arXiv:hep-th/9202027

A Conformal Field Theory for the Quantum Hall Effect

Greg Nagao

Published Feb 7, 1992• 0 citations
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Abstract

The QHE is studied in the context of a CFT. An effective field of $N$ ``spins" associated with the cyclotron motion of particles is taken as an order parameter from which an effective Hamiltonian may be defined. This effective Hamiltonian describes the COM motion of the $N$ particles (with coupling $κ_0$) together with a current-current interaction (of strength $κ_1$). Such a system gives rise to a CFT in the large $N$ limit when $κ_0 = κ_1$. The Laughlin wavefunction is derived from this CFT as an $N'$-point correlation function of winding state vertex operators.

DOI
10.48550/arxiv.hep-th/9202027

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The QHE is studied in the context of a CFT. An effective field of NNN ``spins" associated with the cyclotron motion of particles is taken as an order parameter from which an effective Hamiltonian may be defined. This effective Hamiltonian describes the COM motion of the NNN particles (with coupling κ0κ_0κ0​) together with a current-current interaction (of strength κ1κ_1κ1​). Such a system gives rise to a CFT in the large NNN limit when κ0=κ1κ_0 = κ_1κ0​=κ1​. The Laughlin wavefunction is derived from this CFT as an N′N'N′-point correlation function of winding state vertex operators.
An effective Hamiltonian for the study of the quantum Hall effect is proposed. This Hamiltonian, which includes a "current-current" interaction has the form of a Hamiltonian for a conformal field theory in the large N limit. An order parameter is constructed from which the Hamiltonian may be derived. This order parameter may be viewed as either a collective coordinate for a system of N charged particles in a strong magnetic field; or as a field of spins associated with the cyclotron motion of these particles.
Laughlin's quantum Hall wave function is obtained as an exact ground state of an N-particle Hamiltonian, in which the electrons themselves are coupled to the Chern-Simons field. The proof requires quantum field-theoretic methods in the Schr\"odinger picture, and clearly exhibits the relation to conformal field theory since the Knizhnik-Zamolodchikov connection shows up and the Laughlin ground state is recognized as a conformal block. A refined version of this approach can be applied for toroidal boundary conditions, i.e., the electrons are confined to the physical interior of the Hall sample. The result turns out to be a nontrivial modification of the p-fold degenerate Haldane-Rezayi wave function. Furthermore, it is shown that the degeneracy of the ground state is accounted for by the quantum group Uq(su2){U}_{q}{(su}_{2})Uq​(su2​) with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.
We examine the application of c=1c=1c=1 conformal field theory to the description of the fractional quantum Hall effect (FQHE). It is found that the Gaussian model together with an appropriate boundary condition for the order parameter furnishes an effective theory for the Laughlin type FQHE. The plateau formation condition corresponds to taking the {\em chiral} portion of the theory.
Certain fractional quantum Hall wavefunctions -- particularly including the Laughlin, Moore-Read, and Read-Rezayi wavefunctions -- have special structure that makes them amenable to analysis using an exeptionally wide range of techniques including conformal field theory (CFT), thin cylinder or torus limit, study of symmetric polynomials and Jack polynomials, and so-called ``special" parent Hamiltonians. This review discusses these techniques as well as explaining to what degree some other quantum Hall wavefunctions share this special structure. Along the way we will explore the physics of quantum Hall edges, entanglement spectra, quasiparticles, nonabelian braiding statistics, and Hall viscosity, among other topics. As compared to a number of other recent reviews, most of this review is written so as to {\it not} rely on results from conformal field theory -- although a short discussion of a few key relations to CFT are included near the end.
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
More than 30 years after its surprising experimental discovery, the quantum Hall effect remains one of the most active and interesting fields of research in condensed matter physics. The theory pertaining to the phenomenon comprises a hugely varied and fascinating body of work, incorporating frameworks such as variational trial wavefunctions, topological phases, conformal field theory and more. There are reasons to expect that some phases of the system harbour nonabelian excitations: particles whose interchange affects the state of the whole system in a way depending on the order of exchange. This has been proposed as a suitable basis for noise resistant quantum computing. The excitations are an emergent property of the totality of the system, whose main active constituent is the well understood electron. The electrons act together in such a way as to manifest exotic quasiparticles; these kinds of strong correlations are a general feature of the system. One can argue that the quantum Hall effect shows the limits of reductionism. In this work we focus on trial wavefunctions as descriptions of the phenomenon. Although highly succesful in the past their evaluation is often impeded by a feature known as lowest Landau level projection. We approach the projection from several angles and in this context introduce an effective and general technique termed energy projection. Using this we examine several trial wavefunctions that have been difficult to deal with in the past. The quantity known as the pair correlation function is an important tool for the analysis of wavefunctions. Its study, however, has often been mainly qualitative in the literature. We construct an expansion useful for exact characterisation and comparison of pair correlations and show that it has desirable properties as compared to a similar preexisting expansion. This is then used to scale pair correlation functions to macroscopic sizes.