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International Journal of Modern Physics B

COLLECTIVE FIELD THEORY APPLIED TO THE FRACTIONAL QUANTUM HALL EFFECT

BUNJI SAKITA, Dong-Ning Sheng, Zhao-Bin Su

Published Jan 1, 1991• 0 citations
Publisher

Abstract

We present an application of collective field theory to the fractional quantum Hall effect (FQHE). We first express the condition, that the electrons are all in the lowest Landau level, as a constraint equation for the state functional. We then derive the fractional filling factor from this equation together with the no-free-vortex assumption. A hierarchy of filling factors is derived by using the particle-vortex dual transformations. In the final section we discuss an attempt at a dynamical theory of FQHE, which would justify the no-free-vortex assumption. A derivation of Laughlin’s wave function with and without quasi-hole excitations is also given.

DOI
10.1142/s0217979291000262

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We present an application of collective field theory to the fractional quantum Hall effect (FQHE). We first express the condition, that the electrons are all in the lowest Landau level, as a constraint equation for the state functional. We then derive the fractional filling factor from this equation together with the no-free-vortex assumption. A hierarchy of filling factors is derived by using the particle-vortex dual transformations. In the final section we discuss an attempt at a dynamical theory of FQHE, which would justify the no-free-vortex assumption. A derivation of Laughlin’s wave function with and without quasi-hole excitations is also given.
In the framework of collective field theory, We apply the Chern-Simon field theory treatment to the constraint equation for the lowest Landau level to investigate the generic properties for the quasi-particles of the FQH system. It shows a transparent connection to the Laughlin's wave functions. If we take an average over the wave functional for the constraint equation, the resulted equation can be interpreted as the vortex equation for the fractionally charged quasi-particles. Introducing a generalized ρ (density)-ϑ (phase) transformation, not only the fractional statistics and the hierarchy scheme can be drawn from the constraint equation, it also gives rise an interesting picture that vortices condense as a Halperin like wave fuction on a Laughlin like background condensate of ν=1/m.
A recently developed hierarchy theory of the fractional quantum Hall effect (FQHE) is reviewed in this article. We use a many quasiparticle (QP) wave function to obtain the effective partition function of a FQHE system in the path-integral representation, which has been shown to be formally equivalent to an anyon system confined to the lowest Landau level. However the two systems are different in the dimension of the Hilbert space, which implies that the standard hierarchy theory is invalid beyond second hierarchical states. The only way to overcome the difficulty is to place anyons in higher Landau levels. By making a mean-field approximation to the statistical gauge field, the QPs are converted to bosons and a constraint boson system is derived. We find a new type of condensate of the bosonic QPs, which results in a new type of incompressible states of the electrons. These states which involve only two hierarchies should have higher stability over others. That explains why some sequences of filling fractions occur prominently in experiment. Finally, we show that there is a one-to-one correspondence between the new hierarchy theory and the composite fermion theory proposed by Jain.
We study two-dimensional bosons and fermions in a magnetic field with vanishing range interactions. This model was previously proposed for fermions to illustrate the exactness and uniqueness of Laughlin's wave function as a ground state within the lowest Landau level. We show that the restriction to the lowest Landau level is not a valid approximation of these models for arbitrarily high (but finite) cyclotron frequency, since it predicts fractional quantum Hall effect (FQHE) at an incorrect filling factor. In particular, in contrast to the lowest-Landau-level theory, hard-core bosons are shown not to exhibit any FQHE. However, the set of filling factors for the FQHE are identical in the two cases, thus reaffirming the universality of the FQHE.
In the hierarchical theory of the fractional quantum Hall effect, the low--energy behaviour of a daughter state in the next level of the hierarchy is described by an interacting system of quasiparticles of the parent state. Taking the filled lowest Landau level as the parent state, we examine analytically the quantitative consequences of this approach for electrons interacting via a pseudopotential interaction. It is shown that the ground state energy per particle in the daughter state at a filling factor 2/32/32/3 is exactly equal to that of a system of quasiholes in the parent state with half filling, precisely as predicted by the hierarchical approach. This is achieved with {\em only up to two--particle interactions} in the effective Hamiltonian for the quasiholes. Their single particle energy and two--particle interaction are derived. The results are generalized to the other filling factors attainable from the filled Landau level.
The formalism introduced in a previous paper is used for discussing the Coulomb interaction of many electrons moving in two space-dimensions in the presence of a strong magnetic field. The matrix element of the Coulomb interaction is evaluated in the new basis, whose states are invariant under discrete translations (up to a gauge transformation). This paper is devoted to the case of low filling factor, thus we limit ourselves to the lowest Landau level and to spins all oriented along the magnetic field. For the case of filling factor νf = 1/u we give an Ansatz on the state of many electrons which provides a good approximated solution of the Hartree–Fock equation. For general filling factor νf = u′/u a trial state is given which converges very rapidly to a solution of the self-consistent equation. We generalize the Hartree–Fock equation by considering some correlation: all quantum states are allowed for the u′ electrons with the same translation quantum numbers. Numerical results are given for the mean energy and the energy bands, for some values of the filling factor (νf = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5). Our results agree numerically with the Charge Density Wave approach. The boundary conditions are shown to be very important: only large systems (degeneracy of Landau level over 200) are not affected by the boundaries. Therefore results obtained on small scale systems are somewhat unreliable. The relevance of the results for the Fractional Quantum Hall Effect is briefly discussed.
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.