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Progress of Theoretical Physics

A Mean Field Theory for the Quantum Hall Liquid. I

K. Ishikawa

Published Nov 1, 1992• 1 citation
Publisher

Abstract

A mean field theory for the quantum Hall liquid based on a bi-local collective field is presented. A self-consistent mean field and fluctuations due to low spin fields around the mean field are investigated. We find that the mean field energy is negative and its correction due to the fluctuations of the density field is finite and negative and the correction due to the vector fields cancells completely in the lowest order. The finite Hall conductance is computed and the massive Chern-Simons gauge theory is obtained as an effective long wave length theory.

DOI
10.1143/ptp.88.881

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A mean field theory for the quantum Hall liquid based on a bi-local collective field is presented. A self-consistent mean field and fluctuations due to low spin fields around the mean field are investigated. We find that the mean field energy is negative and its correction due to the fluctuations of the density field is finite and negative and the correction due to the vector fields cancells completely in the lowest order. The finite Hall conductance is computed and the massive Chern-Simons gauge theory is obtained as an effective long wave length theory.
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We present a quantitative study of the quasiparticle properties of a three-dimensional electron Fermi liquid. Our approach is based on the theory of the many-body local field factors which we use to include vertex corrections associated with charge and spin fluctuations. Extensive use is made of the results of recent quantum Monte Carlo calculations. Several models for the wave-vector dependence of the spin-antisymmetric many-body local field factor ${G}_{\ensuremath{-}}$, for which no numerical results are currently available, are discussed and compared. Both the real and imaginary parts of the self-energy as well as the quasiparticle renormalization constant and the enhancement of the effective mass are calculated in the experimentally accessible range of electron densities. The results obtained by means of the on-shell approximation are critically compared with those given by a self-consistent solution of the Dyson equation. An ultraviolet catastrophe in the Coulomb-hole part of the self-energy is identified and a satisfactory resolution of this impasse is presented. The same many-body local field factors are also used to obtain the Landau interaction function. This allows us to obtain both the spin susceptibility and the proper compressibility of the system.
Fradkin's formulation of statistical field theory is applied to the Coulomb interacting electron gas in a magnetic field. The electrons are confined to a plane in normal three-dimensional space and also interact with the physical three-dimensional electromagnetic field. The magnetic-translation-group Ward identities are derived. By using them, it is shown that the exact electron propagator is diagonalized in the basis of the wave functions of the free electron in a magnetic field whenever the magnetic-translation-group symmetry is unbroken. The general tensor structure of the polarization operator is obtained and used to show that the Chern-Simons action always describes the Hall-effect properties of the system. A general proof of the Str\^eda formula for the Hall conductivity is presented. It follows that the coefficient of the Chern-Simons terms in the long-wavelength approximation is exactly given by this relation. Such a formula, expressing the Hall conductivity as a simple derivative, in combination with a diagonal form of the full propagator, allows us to obtain a simple expression for the filling factor and the Hall conductivity. Indeed, these results, after assuming that the chemical potential lies in a gap of the density of states, lead to the conclusion that the Hall conductivity is given without corrections by ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{y}}$=\ensuremath{\nu}e2{\mathit{e}}^{2}e2/h, where \ensuremath{\nu} is the filling factor. In addition, it follows that the filling factor is independent of the magnetic field if the value of the chemical potential remains in the gap.
The derivation of the non-magnetic Laughlin state and other macroscopic quantum states in the semi-localized 2D electron system in the network of circular molecular orbits is made by the study of zero-point plasma oscillation. In the imaginary time representation, the electric field is transformed to the vector potential. After the cancellation of the mean-field component of the inter-electron repulsive field with the ion-lattice field, the boson Hamiltonian with respect to the phase-charge fluctuation is obtained using a Chern–Simons gauge field. Based on the resultant boson wave function, the macroscopic quantum state in hole doping is found to lead to a superfluidity that is described by a coherent function when λΘ > λQ, and to the particle-number-definite state described by a Laughlin function when λQ > λΘ, where λΘ is the phase-coherence length and λQ is the incompressibility length.