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arXiv:cond-mat/9812410 · Physical Review Letters

Stability of the Compressible Quantum Hall State around the Half-Filled Landau Level

Kenzo Ishikawa, Nobuki Maeda, Tetsuyuki Ochiai

Published May 24, 1999• 12 citations
arXivHTMLPDF

Abstract

We study the compressible states in the quantum Hall system using a mean field theory on the von Neumann lattice. In the lowest Landau level, a kinetic energy is generated dynamically from Coulomb interaction. The compressibility of the state is calculated as a function of the filling factor $\ensuremath{\nu}$ and the width $d$ of the spacer between the charge carrier layer and dopants. The compressibility becomes negative below a critical value of $d$ and the state becomes unstable at $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$. Within a finite range around $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$, the stable compressible state exists above the critical value of $d$.

DOI
10.1103/physrevlett.82.4292

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Physical Review Letters•Jan 21, 1985•448 citations
We study the compressible states in the quantum Hall system using a mean field theory on the von Neumann lattice. In the lowest Landau level, a kinetic energy is generated dynamically from Coulomb interaction. The compressibility of the state is calculated as a function of the filling factor \ensuremathν\ensuremath{\nu}\ensuremathν and the width ddd of the spacer between the charge carrier layer and dopants. The compressibility becomes negative below a critical value of ddd and the state becomes unstable at \ensuremathν=1/2\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2\ensuremathν=1/2. Within a finite range around \ensuremathν=1/2\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2\ensuremathν=1/2, the stable compressible state exists above the critical value of ddd.
Possible phase transitions between incompressible quantum Hall states and compressible three-dimensional states are discussed for infinite-layer electron systems in strong magnetic field. By variational Monte Carlo calculation, relative stability of some trial states is studied. If the interlayer distance is large enough, the Laughlin state is stabilized for the Landau level filling $\ensuremath{\nu}=1/3$ of each layer. When the interlayer distance is comparable to the magnetic length, the Laughlin state becomes unstable against a different fractional quantum Hall state with interlayer correlation, and/or against a three-dimensional compressible state. It is discussed how the quantum phase transition between them can be controlled in actual systems.
Electron tunneling into a system with strong interactions is known to exhibit an anomaly, in which the tunneling conductance vanishes continuously at low energy due to many-body interactions. Recent measurements have probed this anomaly in a quantum Hall bilayer of the half-filled Landau level, and shown that the anomaly apparently gets stronger as the half-filled Landau level is increasingly spin polarized. Motivated by this result, we construct a semiclassical hydrodynamic theory of the tunneling anomaly in terms of the charge-spreading action associated with tunneling between two copies of the Halperin-Lee-Read state with partial spin polarization. This theory is complementary to our recent work (D. Chowdhury, B. Skinner, and P. A. Lee, arXiv:1709.06091) where the electron spectral function was computed directly using an instanton-based approach. Our results show that the experimental observation cannot be understood within conventional theories of the tunneling anomaly, in which the spreading of the injected charge is driven by the mean-field Coulomb energy. However, we identify a qualitatively new regime, in which the mean-field Coulomb energy is effectively quenched and the tunneling anomaly is dominated by the finite compressibility of the composite Fermion liquid.
We study the effects of disorder on the quantum Hall stripe phases in half-filled high Landau levels using exact numerical diagonalization. We show that, in the presence of weak disorder, a compressible, striped charge-density wave, becomes the true ground state. The projected electron-density profile resembles that of a smectic liquid. With increasing disorder strength W, we find that there exists a critical value, ${W}_{c}\ensuremath{\sim}{0.12e}^{2}/\ensuremath{\epsilon}\mathcal{l},$ where a transition/crossover to an isotropic phase with strong local electron-density fluctuations takes place. The many-body density of states are qualitatively distinguishable in these two phases and help elucidate the nature of the transition.
Quantum Hall plateaus at quarter fillings occur in GaAs wide quantum wells, hole-doped GaAs, and bilayer graphene. However, the interactions favoring incompressible states over compressible composite-Fermi liquids at such fillings are not well understood. We devise a method for the computation of the trial energies for Haldane pseudopotentials via Monte Carlo sampling. Applying it to the quarter-filled lowest Landau level, we find that tuning the third and fifth pseudopotential values can stabilize the anti-Pfaffian, Moore-Read, and fff-wave states. The smallest deviations from pure Coulomb interactions are required by anti-Pfaffian, whose presence is indicated by daughter states in recent experiments of bilayer graphene at $\ensuremath{\nu}=\frac{3}{4}$.
Recent experiments indicate the presence of new anisotropic fractional quantum Hall states at regimes not anticipated before. These experiments raise many fundamental questions regarding the inner nature of the electronic system that leads to such anisotropic states. Interplay between electron mass anisotropy and electron-electron correlation effects in a magnetic field can create a rich variety of possibilities. Several anisotropic electronic states ranging from anisotropic quantum Hall liquids to anisotropic Wigner solids may stabilize due to such effects. The electron mass anisotropy in a two-dimensional electron gas effectively leads to an anisotropic Coulomb interaction potential between electrons. An anisotropic interaction potential may strongly influence the stability of various quantum phases that are close in energy since the overall stability of an electronic system is very sensitive to local order. As a result there is a possibility that various anisotropic electronic phases may emerge even in the lowest Landau level in regimes where one would not expect them. In this work we study the state with filling factor 1/6 in the lowest Landau level, a state which is very close to the critical filling factor where the liquid-solid transition takes place. We investigate whether an anisotropic Coulomb interaction potential is able to stabilize an anisotropic electronic liquid state at this filling factor. We describe such an anisotropic state by means of a liquid crystalline wave function with broken rotational symmetry which can be adiabatically connected to the actual wave function for the corresponding isotropic phase. We perform quantum Monte Carlo simulations in a disk geometry to study the properties of the anisotropic electronic liquid state under consideration. The findings indicate stability of liquid crystalline order in presence of an anisotropic Coulomb interaction potential. The results are consistent with the existence of an anisotropic electronic liquid state in the lowest Landau level.
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.