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Physical Review Letters

Generalized Cooperative-Ring-Exchange Theory of the Fractional Quantum Hall Effect

D. H. Lee, G. Baskaran, Steven Allan Kivelson

Published Nov 23, 1987• 25 citations
Publisher

Abstract

Through the calculations of the diagonal and off-diagonal elements of the full $N$-particle density matrix we show that all essential features of the Laughlin variational wave functions can be derived in a generalized cooperative-ring-exchange theory of the fractional quantum Hall fluid.

DOI
10.1103/physrevlett.59.2467

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Generalized Cooperative Ring Exchange Theory of the Fractional Quantum Hall Effect

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Through the calculations of the diagonal and off-diagonal elements of the full NNN-particle density matrix we show that all essential features of the Laughlin variational wave functions can be derived in a generalized cooperative-ring-exchange theory of the fractional quantum Hall fluid.
Electrons in two dimensions in a high magnetic field can condense into an incompressible quantum fluid state when the electron density is equal to certain rational fractions of the flux density. Much progress has been made in understanding the nature of this state based on the ansatz wavefunction of Laughlin. Here, we show that the incompressible nature of the quantum fluid can be derived from a semiclassical evaluation of the imaginary time propagator. The dominant classical paths consist of cooperative ring exchange events. When these condense, they mediate a logarithmic interaction between density fluctuations, and produce a precise fractional quantization condition for the quasi-particle charges. We further show that the essential long wavelength properties of Laughlin's ground state and quasi-particle wavefunctions can be derived constructively from this approach. We conclude with some speculations concerning the nature of the off-diagonal order in the quantum Hall liquid.
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.
We study the transition of the interacting two-dimensional electron gas at high magnetic field from a low-density Wigner crystal to a higher-density correlated state which exhibits the fractionally quantized Hall effect. The phase transition is precipitated by a condensation of ring-exchange processes, which by nature of their sensitivity to enclosed magnetic flux allow for especially low-energy exchange condensates at certain values of the Landau-level filling factor, \ensuremath{\nu}=${\ensuremath{\nu}}_{i}$. In the condensate, the exchanges mediate a logarithmic potential between local-density fluctuations, leading to a cusp in the ground-state energy as a function of \ensuremath{\Vert}\ensuremath{\nu}-${\ensuremath{\nu}}_{i}$\ensuremath{\Vert} in the vicinity of a preferred state. In addition, we derive quasiparticle excitations of sharp fractional charge which are analogous to the excitations derived by Laughlin.
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.
We introduce one-dimensional lattice models with exact matrix-product ground states describing the fractional quantum Hall (FQH) states in Laughlin series (given by filling factors $\ensuremath{\nu}=1/q$) on torus geometry. Surprisingly, the exactly solvable Hamiltonian has the same mathematical structure as that of the pseudopotential for the Laughlin wave function, and naturally derives the general properties of the Laughlin wave function such as the Z2{Z}_{2}Z2​ properties of the FQH states and the fermion-boson relation. The obtained exact ground states have high overlaps with the Laughlin states and well describe their properties. Using the matrix product method, density functions and correlation functions are calculated analytically. Especially, obtained entanglement spectra reflect gapless edge states as was discussed by Li and Haldane.