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Journal of Physics Condensed Matter

Electron correlations and fractional quantum Hall states in a double-layer electron system

N. F. Johnson, Simon C. Benjamin

Published Mar 20, 1995• 2 citations
Publisher

Abstract

An analytically solvable few-electron model of a double-layer electron system is presented which includes mixing with all Landau levels. Inter-layer correlations are found to give rise to novel ground states which differ fundamentally from previously suggested Jastrow-like wavefunctions for double layer N-electron systems. Comparison with experimental data suggests that these states may represent small-N precursors of recently observed fractional quantum Hall states in double-layer (large-N) systems.

DOI
10.1088/0953-8984/7/12/001

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An analytically solvable few-electron model of a double-layer electron system is presented which includes mixing with all Landau levels. Inter-layer correlations are found to give rise to novel ground states which differ fundamentally from previously suggested Jastrow-like wavefunctions for double layer N-electron systems. Comparison with experimental data suggests that these states may represent small-N precursors of recently observed fractional quantum Hall states in double-layer (large-N) systems.
We study the nature of the many-body electron states (and quantum phase transitions between them) in a double-quantum-well structure under a strong external magnetic field as a function of the materials parameters that define the two-layer system, namely, the thickness of individual layers, the separation between them, the individual well potentials confining the electrons, and the potential barrier between them. Motivated by two recent experiments, we consider two different situations, one with (almost) no quantum tunneling between the wells, and the other with substantial interwell quantum tunneling. We use the spherical system finite-size exact diagonalization technique for our calculations. By calculating the overlap of our exact small-system numerical wave functions with various (postulated) analytic wave functions, we comment on the nature of the incompressibility for various Landau-level filling factors (\ensuremath{\nu}). We investigate, in particular details, the possibility of \ensuremath{\nu}=1/2 (i.e., 1/4 average occupancy in each well), and, 1 (i.e., 1/2 occupancy in each well) incompressible states where \ensuremath{\nu} is the total filling factor for the system. We also provide results for the \ensuremath{\nu}=2/3 situation. Our conclusion, based on our use of realistic system parameters in our calculations, is that in both the recent experimental realizations of the \ensuremath{\nu}=1/2 fractional quantum Hall effect, the relevant ground state is the so-called ``331'' state which is stabilized by the competition between intrawell and interwell electron-electron correlations.We calculate for various double-quantum-well structural parameters and magnetic fields the \ensuremath{\nu}=1/2 excitation spectra and the energy gaps, and compare it with the recent experimental data, obtaining excellent quantitative agreement. We provide a detailed quantum phase diagram for the \ensuremath{\nu}=1 state as a function of the interwell Coulomb interaction and the symmetric-antisymmetric single-particle gap, and find that the \ensuremath{\nu}=1 quantum Hall effect observed in the two recent experiments, in contrast to the \ensuremath{\nu}=1/2 effect, belongs in some sense to two different universality classes. We also predict the existence of a reentrant \ensuremath{\nu}=1 quantum Hall effect in double-quantum-well systems.
The authors study the dispersion of collective excitations for fractional quantum Hall states using Feynman-Bijl ansatz wavefunctions. They take the ground-state pair correlation function from direct diagonalisations of the Hamiltonian for a small number of electrons confined to the surface of a sphere, instead of from the simulation of large systems described by a Laughlin-like wavefunction. The results seem remarkably well converged considering the small systems used. They also treat systems with partially filled N=1 Landau levels.
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.
Quantum Hall systems have a one-body energy spectrum consisting of dispersion-less Landau levels. Electron-electron interactions thus dominate in partially filled Landau levels, which exhibit a myriad of strongly correlated phases such as quantum hall ferromagnets and fractional quantum Hall phases. We study two examples of these phenomena. In the first project, we explore the ground state of a system with an interface between two semi-infinite regions with fillings ν= 4 and ν= 3 respectively. The width of the interface can be controlled by varying the background potential, which provides an additional tuning parameter. For a certain range of interaction strengths, the ν= 4 bulk is unpolarized whereas the ν= 3 bulk is fully polarized. In the parameter space spanned by the interaction strength and width of the interface, we find two phases at the interface. Phase A has spin as a good quantum number, and the long-wavelength spin edge excitations are gapped. In phase B, spin rotation symmetry is spontaneously broken at the mean-field level. Using symmetry arguments we find the effective theory near the interface of phase B. This effective theory is known to have gapless long-wavelength spin excitations. In the second project, we study the ground state of a tunnel-decoupled double-layer graphene system when both layers are undoped. We find a simple Hamiltonian in the continuum limit from symmetries of the system. Using the Hartree-Fock approximation we find a state with inter-layer coherence with broken layer U(1) symmetry. This phase becomes magnetized in presence of a non-zero Zeeman field. A first-order phase transition can be driven from the ferromagnetic phase to the magnetized inter-layer coherent phase by increasing the Zeeman field. We predict the number of gapless modes in the bulk.
We generalize the fractional quantum Hall hierarchy picture to apply to arbitrary, possibly non-Abelian, fractional quantum Hall states. Applying this to the $\ensuremath{\nu}=5/2$ Moore-Read state, we construct explicit trial wave functions to describe the fractional quantum Hall effect in the second Landau level. The resulting hierarchy of states, which reproduces the filling fractions of all observed Hall conductance plateaus in the second Landau level, is characterized by electron pairing in the ground state and an excitation spectrum that includes non-Abelian anyons of the Ising type. We propose this as a unifying picture in which ppp-wave pairing characterizes the fractional quantum Hall effect in the second Landau level.