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Journal of the Physical Society of Japan

Effect of the Landau Level Mixing on the Ground State of Two-Dimensional Electrons

D. Yoshioka

Published Nov 15, 1984• 80 citations
Publisher

Abstract

Two dimensional electrons in a strong magnetic field show the fractional quantum Hall effect at low temperatures. This effect is explained successfully by a discovery of a new liquid type ground state. However the infinitely strong magnetic field has been assumed in existing theories. Here the ground state around one third filling of the lowest Landau level is investigated at a finite magnetic field by numerical diagonalization of the Hamiltonian. When the cyclotron energy is not too small compared to a typical Coulomb energy, no qualitative change of the ground state is found: A natural generalization of the liquid state at the infinite magnetic field describes the ground state. Excitation energies of quasiparticles decrease as the magnetic field decreases.

DOI
10.1143/jpsj.53.3740

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Two dimensional electrons in a strong magnetic field show the fractional quantum Hall effect at low temperatures. This effect is explained successfully by a discovery of a new liquid type ground state. However the infinitely strong magnetic field has been assumed in existing theories. Here the ground state around one third filling of the lowest Landau level is investigated at a finite magnetic field by numerical diagonalization of the Hamiltonian. When the cyclotron energy is not too small compared to a typical Coulomb energy, no qualitative change of the ground state is found: A natural generalization of the liquid state at the infinite magnetic field describes the ground state. Excitation energies of quasiparticles decrease as the magnetic field decreases.
States of strongly interacting particles are of fundamental interest in physics and can produce exotic emergent phenomena and topological structures. We consider here two-dimensional electrons in a magnetic field, and, departing from the standard practice of restricting to the lowest LL, introduce a model short-range interaction that is infinitely strong compared to the cyclotron energy. We demonstrate that this model lends itself to an exact solution for the ground as well as excited states at arbitrary filling factors ν<1/2p and produces a fractional quantum Hall effect at fractions of the form ν=n/(2pn+1), where n and p are integers. The fractional quantum Hall states of our model share many topological properties with the corresponding Coulomb ground states in the lowest Landau level, such as the edge physics and the fractional charge of the excitations.
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.
Finite-size two-dimensional systems are investigated by numerical diagonalization of the Hamiltonian. Both fermion systems and boson systems are investigated. The energy of the liquidlike ground state of the fermion system shows a downward cusp at 13\frac{1}{3}31​ filling of the lowest Landau level. This behavior is consistent with the observed fractional quantum Hall effect. The short-range behavior of the pair distribution function of the ground state is analyzed and good agreement with Laughlin's trial wave function is obtained.
Exact diagonalization of a two-dimensional electron gas in a strong magnetic field in the disk geometry shows that there exists a filling-factor range in the second Landau level where the states significantly differ from those in the lowest Landau level. We show that the difference arises because the interaction between composite fermions is not negligible in higher Landau levels, as indicated by a substantial mixing between composite-fermion Landau-like levels, or $\ensuremath{\Lambda}$ levels. We find that the exact ground state is well reproduced by composite-fermion theory with $\ensuremath{\Lambda}$-level mixing incorporated at the lowest level of approximation. Using the same variational approach in the spherical geometry we estimate the excitation gap at filling 1∕31∕31∕3 in the second Landau level (which corresponds to 7∕37∕37∕3 of experiment).
We study two-dimensional bosons and fermions in a magnetic field with vanishing range interactions. This model was previously proposed for fermions to illustrate the exactness and uniqueness of Laughlin's wave function as a ground state within the lowest Landau level. We show that the restriction to the lowest Landau level is not a valid approximation of these models for arbitrarily high (but finite) cyclotron frequency, since it predicts fractional quantum Hall effect (FQHE) at an incorrect filling factor. In particular, in contrast to the lowest-Landau-level theory, hard-core bosons are shown not to exhibit any FQHE. However, the set of filling factors for the FQHE are identical in the two cases, thus reaffirming the universality of the FQHE.
By exactly solving the effective two-body interaction for two-dimensional electron system with layer thickness and an in-plane magnetic field, we recently found that the effective interaction can be described by the generalized pseudopotentials (PPs) without the rotational symmetry. With this pseudopotential description, we numerically investigate the behavior of the fractional quantum Hall (FQH) states both in the lowest Landau level (LLL) and first Landau level (1LL). The enhancements of the 7/3 FQH state on the 1LL for a small tilted magnetic field are observed when layer thickness is larger than some critical values. While the gap of the 1/3 state in the LLL monotonically reduced with increasing the in-plane field. From the static structure factor calculation, we find that the systems are strongly anisotropic and finally enter into a stripe phase with a large tilting. With considering the Landau level mixing correction on the two-body interaction, we find the strong LL mixing cancels the enhancements of the FQH states in the 1LL.