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Physical review. B, Condensed matter

Wigner solid from finite-cluster studies

S. T. Chui, Xiu Qiu

Published Jun 15, 1991• 4 citations
Publisher

Abstract

We perform finite-cluster exact diagonalization studies for electrons in the lowest Landau level for filling factors from 1/3 to 1/11. In contrast to previous studies, we focus on the ground-state eigenfunction. This wave function changes continuously for filling factors from 1/3 to 1/9. At a filling factor of 1/11, there is a dramatic qualitative change in the ground-state wave function. Associated with this change is a peak in the structure factor at a finite wave vector as well as excitation energies that are consistent with gapless excitations at 1/11 filling. These results are consistent with recent experimental findings.

DOI
10.1103/physrevb.43.14293

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arXiv:cond-mat/9505074•Physical review. B, Condensed matter•Aug 15, 1995•58 citations
We perform finite-cluster exact diagonalization studies for electrons in the lowest Landau level for filling factors from 1/3 to 1/11. In contrast to previous studies, we focus on the ground-state eigenfunction. This wave function changes continuously for filling factors from 1/3 to 1/9. At a filling factor of 1/11, there is a dramatic qualitative change in the ground-state wave function. Associated with this change is a peak in the structure factor at a finite wave vector as well as excitation energies that are consistent with gapless excitations at 1/11 filling. These results are consistent with recent experimental findings.
Finite-size exact diagonalization of the Hamiltonian for four, eight and 16 electrons in a half-filled Landau level under periodic boundary conditions in a rectangular geometry was carried out. The magnitude of the peak value of the structure factor is found to increase with the system size to the power 0.36, consistent with a picture of a quasi-solid discussed by Chui. The energy difference of the lowest-energy state of the two different particle-hole symmetry manifolds decreases as the system size is increased, suggestive of broken particle-hole symmetry.
We report finite cluster exact diagonalization studies under periodic boundary conditions of two layers of electrons in a strong magnetic field at a half-filled Landau level. For the system studied by Suen et al., the ground states have more than 97 per cent of its spectral weight in the symmetric single particle levels. The eigenfunctions and eigenvalues up to a constant factor for different aspect ratios are essentially the same as previous numerical results of a single layer of electrons of zero thickness at half-filling.
There exists a widely believed opinion, that the many-body ground state of a two-dimensional electron system at a completely filled lowest Landau level (the filling factor ν=1ν=1ν=1) is described by the so-called Hartree-Fock wave function, and that this solution is the unique, exact eigenstate of the system at ν=1ν=1ν=1. I show that this opinion is erroneous, construct an infinite number of other variational many-body wave functions, and discuss the properties of a few states which have the energy substantially lower than the energy of the Hartree-Fock state.
The ground state at 4/11 filling factor is very well understood, Mukherjee et al. [Phys. Rev. Lett. 112, 016801 (2014)] in terms of the 1/3-filled second effective Landau level of the composite fermions whose correlations resemble that of electrons in the ground state of a two-body Haldane pseudo-potential of relative angular momentum 3, V3{V}_{3}V3​. We here propose a closed-form ground-state wave function for V3{V}_{3}V3​ at 1/3 filling factor. We successfully compare it with the exact wave function for systems with a few electrons by calculating their mutual overlap, pair-correlation function, and entanglement spectra. By numerical exact diagonalization for a few electron systems, we find a window of nonzero V3{V}_{3}V3​ is essential together with V1{V}_{1}V1​ for being 4/11 state incompressible. The constructed wave function for the 4/11 state using this proposed wave function has satisfactorily high overlap with previously studied composite-fermion-diagonalized ground-state wave function.
We report on results of finite-size numerical studies of partially filled lowest Landau level, at low electron filling factors. We find convincing evidence suggesting that electrons form Wigner crystals at sufficiently low filling factors, and the critical filling factor is ${\ensuremath{\nu}}_{c}\ensuremath{\approx}1/7.$ At $\ensuremath{\nu}=1/7$ we find the system undergoes a phase transition from the Wigner crystal to the incompressible Laughlin state when the short-range part of the Coulomb interaction is modified slightly. This transition is either continuous or very weakly first order.
The two-dimensional Wigner crystals are studied with the variational quantum Monte Carlo method. The close relationship between the ground-state wave function and the collective excitations in the system is illustrated, and used to guide the construction of the ground-state wave function of the strongly correlated solid. Exchange, correlation, and magnetic-field effects all give rise to distinct physical phenomena. In the absence of any external magnetic field, interesting spin orderings are observed in the ground state of the electron crystal in various two-dimensional lattices. In particular, two-dimensional bipartite lattices are shown not to lead necessarily to an antiferromagnetic ground state. In the quantum Hall effect regime, a strong magnetic field introduces energy and length scales. The magnetic field quenches the kinetic energy and poses constraints on how the electrons may correlate with each other. Care is taken to ensure the appropriate translational properties of the wave function when the system is in a uniform magnetic field. We have examined the exchange, intra-Landau-level correlation as well as Landau-level-mixing effects with various variational wave functions. We also determine their dependences on the experimental parameters such as the carrier effective mass at a modulation-doped semiconductor heterojunction. Our results, when combined with some recent calculations for the energy of the fractional quantum Hall liquid including Landau-level-mixing, show quantitatively that in going from n doping to p doping in GaAs/Alx{\mathrm{Al}}_{\mathit{x}}Alx​${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As heterojunction systems, the crossover filling factor from the fractional quantum Hall liquid to the Wigner crystal changes from filling factor \ensuremath{\nu}\ensuremath{\sim}1/5 to \ensuremath{\nu}\ensuremath{\sim}1/3. This lends strong support to the claim that the observed reentrant insulating phases around \ensuremath{\nu}=1/5 for n-doped and around \ensuremath{\nu}=1/3 for p-doped high-mobility samples are primarily caused by electron-electron interaction effects. We discuss the possible implications of our theoretical results for some recent experiments carried out in the quantum Hall regime in search of the electron solid.