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Physical review. B, Condensed matter

Many-particle exchanges in a class of correlated Gaussian wave functions for the fractionally quantized Hall effect

S. T. Chui

Published Jul 15, 1986• 6 citations
Publisher

Abstract

The many-particle exchange integral for a class of correlated Gaussian wave functions for the fractionally quantized Hall effect are discussed. When the correlation is of a long-range nature, the evaluation of this exchange can be cast into languages involving the study of the energies and fluctuations of grain boundaries in two dimensions. For a class of trial wave functions studied recently by Chui, Ma, and Hakim, exchanges involving a large number of particles are significant in that, for a loop that consists of two long parallel boundaries, the overlap integral actually increases as the number of particles involved in the exchange is increased. The question of a collection of boundaries is discussed. We find that there is a repulsion between the boundaries of a range proportional to the size of the boundary.

DOI
10.1103/physrevb.34.1130

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The many-particle exchange integral for a class of correlated Gaussian wave functions for the fractionally quantized Hall effect are discussed. When the correlation is of a long-range nature, the evaluation of this exchange can be cast into languages involving the study of the energies and fluctuations of grain boundaries in two dimensions. For a class of trial wave functions studied recently by Chui, Ma, and Hakim, exchanges involving a large number of particles are significant in that, for a loop that consists of two long parallel boundaries, the overlap integral actually increases as the number of particles involved in the exchange is increased. The question of a collection of boundaries is discussed. We find that there is a repulsion between the boundaries of a range proportional to the size of the boundary.
It is shown that the wave function proposed by Tao and Thouless to explain the fractional quantization of the Hall current implies long-range correlations between the electrons that are very different from the properties expected from Laughlin's wave function. Calculations for a small number of electrons have shown that Laughlin's wave function is an excellent approximation to the true ground state, so it must be concluded that the Tao-Thouless wave function is not a good approximation. There is some discussion of the question of broken symmetry.
We analyze properties of trial wave functions in fractional quantum Hall effect, generated by the "short-range", three-body repulsion and its mean field approximation.Ground states of electron repulsion Hamiltonians at filing factor ν = 1/2 are evaluated as a description of physically observed state ν = 5/2.We analyze overlaps between the Moore-Read state and its mean field approximation for different number of particles and compare both states with ground state of the Coulomb interaction in the first excited Landau level.Our study also includes examination of electron-electron correlation functions and electron densities of excited states for both interactions.
We consider spaces of trial wave functions for ground states and edge excitations in the fractional quantum Hall effect that can be obtained in various ways. In one way, functions are obtained by analyzing the entanglement of the ground-state wave function, partitioned into two parts. In another, functions are defined by the way in which they vanish as several coordinates approach the same value, or by a projection-operator Hamiltonian that enforces those conditions. In a third way, functions are given by conformal blocks from a conformal field theory (CFT). These different spaces of functions are closely related. The use of CFT methods permits an algebraic formulation to be given for all of them. In some cases, we can prove that there is a ground state, a Hamiltonian, and a CFT such that, for any number of particles, all of these spaces are the same. For such cases, this resolves several questions and conjectures: it gives a finite-size bulk-edge correspondence, and we can use the analysis of functions to construct a projection-operator Hamiltonian that produces those functions as zero-energy states. For a model related to the N=1\mathcal{N}=1N=1 superconformal algebra, the corresponding Hamiltonian imposes vanishing properties involving only three particles; for this we determine all the wave functions explicitly. We do the same for a sequence of models involving the M(3,p)M(3,p)M(3,p) Virasoro minimal models that has been considered previously, using results from the literature. We exhibit the Hamiltonians for the first few cases of these. The techniques we introduce can be applied in numerous examples other than those considered here.
The two-dimensional Wigner crystals are studied with the variational quantum Monte Carlo method. The close relationship between the ground-state wave function and the collective excitations in the system is illustrated, and used to guide the construction of the ground-state wave function of the strongly correlated solid. Exchange, correlation, and magnetic-field effects all give rise to distinct physical phenomena. In the absence of any external magnetic field, interesting spin orderings are observed in the ground state of the electron crystal in various two-dimensional lattices. In particular, two-dimensional bipartite lattices are shown not to lead necessarily to an antiferromagnetic ground state. In the quantum Hall effect regime, a strong magnetic field introduces energy and length scales. The magnetic field quenches the kinetic energy and poses constraints on how the electrons may correlate with each other. Care is taken to ensure the appropriate translational properties of the wave function when the system is in a uniform magnetic field. We have examined the exchange, intra-Landau-level correlation as well as Landau-level-mixing effects with various variational wave functions. We also determine their dependences on the experimental parameters such as the carrier effective mass at a modulation-doped semiconductor heterojunction. Our results, when combined with some recent calculations for the energy of the fractional quantum Hall liquid including Landau-level-mixing, show quantitatively that in going from n doping to p doping in GaAs/Alx{\mathrm{Al}}_{\mathit{x}}Alx​${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As heterojunction systems, the crossover filling factor from the fractional quantum Hall liquid to the Wigner crystal changes from filling factor \ensuremath{\nu}\ensuremath{\sim}1/5 to \ensuremath{\nu}\ensuremath{\sim}1/3. This lends strong support to the claim that the observed reentrant insulating phases around \ensuremath{\nu}=1/5 for n-doped and around \ensuremath{\nu}=1/3 for p-doped high-mobility samples are primarily caused by electron-electron interaction effects. We discuss the possible implications of our theoretical results for some recent experiments carried out in the quantum Hall regime in search of the electron solid.