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Physical review. B, Condensed matter

Computer renormalization-group calculation for the fractionally quantized Hall effect

S. T. Chui

Published Jul 15, 1985• 19 citations
Publisher

Abstract

Results of a computer renormalization-group calculation for electrons in a magnetic field are presented for clusters with up to 16 electrons for the (1/3)-filled case for the first time. The structure factor is computed. We found a ``Bragg-like'' peak at a finite wave vector. The height of this peak increases as the system size is increased. This behavior is different from that of the Laughlin fluid but is much closer to that of a quasisolid.

DOI
10.1103/physrevb.32.1436

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Results of a computer renormalization-group calculation for electrons in a magnetic field are presented for clusters with up to 16 electrons for the (1/3)-filled case for the first time. The structure factor is computed. We found a ``Bragg-like'' peak at a finite wave vector. The height of this peak increases as the system size is increased. This behavior is different from that of the Laughlin fluid but is much closer to that of a quasisolid.
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.
In this paper, the density-matrix renormalization group method is employed to investigate the fractional quantum Hall effect at filling fractions $\ensuremath{\nu}=1/3$ and 5/25/25/2. We first present benchmark results at both filling fractions for large system sizes to show the accuracy as well as the capability of the numerical algorithm. Furthermore, we show that by keeping a large number of basis states, one can also obtain an accurate entanglement spectrum at $\ensuremath{\nu}=5/2$ for large systems with electron numbers up to Ne=34{N}_{e}=34Ne​=34, much larger than systems previously studied. Based on a finite-size scaling analysis, we demonstrate that the entanglement gap defined by Li and Haldane [Phys. Rev. Lett. 101, 010504 (2008)] is finite in the thermodynamic limit, which characterizes the topological order of the fractional quantum Hall effect state.
The incompressible states of interacting two-dimensional electrons in a high magnetic field has been studied by finite-size calculations in the spherical geometry. The excitation spectrum at 13\frac{1}{3}31​ Landau-level filling is described, and the Laughlin-Jastrow character of the Coulomb-interaction ground state is unambiguously confirmed. As the interaction is varied, a transition to a gapless compressible state is observed.
The single-particle spectral function for an incompressible fractional quantum Hall state in the presence of a scalar short-ranged attractive impurity potential is calculated via exact diagonalization within the spherical geometry. In contrast to the noninteracting case, where only a single bound state below the lowest Landau level forms, electron–electron interactions strongly renormalize the impurity potential, effectively giving it a finite range, which can support many quasi-bound states (long-lived resonances). Averaging the spectral weights of the quasi-bound states and extrapolating to the thermodynamic limit, for filling factor ν = 1/3 we find evidence consistent with localized fractionally charged e /3 quasi-particles. For ν = 2/5, the results are slightly more ambiguous, due to finite size effects and possible bunching of Laughlin-quasi-particles.
A scaling procedure on the particle number is developed to construct the wave function of the uniform liquid ground state of the two-dimensional electron system in a strong magnetic field. When the filling factor is equal to the inverse of an odd number, this wave function is the correlated basis function introduced by Laughlin. In more general cases, when the filling factor is equal to a rational number, this wave function also can be regarded as a Laughlin wave function on the scale of a certain cluster. By this procedure, we can prove that an energy gap is always present between such a uniform liquid ground state and the excited states when the filling factor is equal to a rational number.