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Physical review. B, Condensed matter

Long-range order and the fractional quantum Hall effect

D. J. Thouless

Published Jun 15, 1985• 23 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.31.8305

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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.
It is now established that the state proposed by Tao and Thouless for the fractional quantum Hall effect in 1983, shortly after Laughlin's work, is the ground state in the so-called Tao-Thouless limit, and that it is adiabatically connected to the Laughlin state. We review the interesting history of the Tao-Thouless state, and its generalizations, and discuss its relevance and shortcomings. In particular, we calculate the exclusion statistics of quasiparticles in the Tao-Thouless limit and point out a principal difficulty which prevents calculation of the fractional exchange statistics in this limit.
We show that the wave functions of the fluid ground states of fractional quantum Hall systems, in the thermodynamic limit, are universal at long distances and that they have a generalized Laughlin form. This universality is a consequence of the analytic properties of the equal-time density correlation functions at long distances. The correlation functions calculated from the field theoretic approach to the fractional quantum Hall effect have the correct analytic properties and the wave function calculated in the Gaussian approximation becomes exact in the asymptotic limit.
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.
Laughlin's wave functions, which describe the fractional quantum Hall effect at filling factors $\ensuremath{\nu}=1/(2k+1)$, can be obtained as correlation functions in a conformal field theory, and recently, this construction was extended to Jain's composite fermion wave functions at filling factors $\ensuremath{\nu}=n/(2kn+1)$. Here, we generalize this latter construction and present ground state wave functions for all quantum Hall hierarchy states that are obtained by successive condensation of quasielectrons (as opposed to quasiholes) in the original hierarchy construction. By considering these wave functions on a cylinder, we show that they approach the exact ground states, which are the Tao-Thouless states, when the cylinder becomes thin. We also present wave functions for the multihole states, make the connection to Wen's general classification of Abelian quantum Hall fluids, and discuss whether the fractional statistics of the quasiparticles can be analytically determined. Finally, we discuss to what extent our wave functions can be described in the language of composite fermions.
We have formulated a microscopic many-body theory of the electron-hole interaction in the fractional-Hall-effect regime via an analysis based on the Bethe-Salpeter equation. Starting from the noninteracting vacuum suggested by Tao and Thouless we show that the generalized random-phase approximation used by these authors omits important excitonic effects and leads to a poor description of the system. When vertex corrections are included in the ladder approximation the system can undergo an excitonic instability for small values of the single-particle gap. Such an instability is a natural consequence of the fact that the ground state of the system is degenerate. Within our description fractionally charged excitations of the type first introduced by Laughlin can be viewed as domain walls separating different but otherwise equivalent and degenerate ground-state configurations.
The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.