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arXiv:cond-mat/0112233 · Physical review. B, Condensed matter

Role of disorder in half-filled high Landau levels

D. N. Sheng, Ziqiang Wang, Barry Friedman

Published Oct 29, 2002• 7 citations
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Abstract

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.

DOI
10.1103/physrevb.66.161103

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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, Wc\ensuremath∼0.12e2/\ensuremathϵl,{W}_{c}\ensuremath{\sim}{0.12e}^{2}/\ensuremath{\epsilon}\mathcal{l},Wc​\ensuremath∼0.12e2/\ensuremathϵl, 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.
We investigate the nature of the quantum Hall liquid in a half-filled second Landau level (n=1n=1n=1) as a function of band mass anisotropy using numerical exact diagonalization and density matrix renormalization group methods. We find increasing the mass anisotropy induces a quantum phase transition from the Moore-Read state to a charge density wave state. By analyzing the energy spectrum, guiding center structure factors, and by adding weak pinning potentials, we show that this charge density wave is a unidirectional quantum Hall stripe, which has a periodicity of a few magnetic lengths and survives in the thermodynamic limit. We find smooth profiles for the guiding center occupation function that reveal the strong coupling nature of the array of chiral Luttinger liquids residing at the stripe edges.
We report on numerical studies of two-dimensional electron systems in the presence of a perpendicular magnetic field, with a high Landau level (index $N\ensuremath{\ge}2$) half filled by electrons. Strong and sharp peaks are found in the wave-vector dependence of both the static density susceptibility and the equal-time density-density correlation function, in finite-size systems with up to twelve electrons. Qualitatively different from the partially filled lowest ( N=0N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0N=0) Landau level, these results are suggestive of a tendency toward charge-density-wave ordering in these systems. The ordering wave vector is found to decrease with increasing NNN.
We propose a dynamical theory of the stripe phase arising in a two-dimensional electron liquid near half-integral fillings of high Landau levels. The system is modeled as a novel type of a smectic liquid crystal with Lorentz force dominated dynamics. We calculate the structure factor, the dispersion relation of the collective modes, and their intrinsic attenuation rate. We show that thermal fluctuations cause a strong power-law renormalization of the elastic and dissipative parameters familiar from the conventional smectics but with different dynamical scaling exponents.
The fractional quantum Hall (FQH) effect at the filling number ν=5/2 is a primary candidate for non-Abelian topological order, while the fate of such a state in the presence of random disorder has not been resolved. We address this open question by implementing an unbiased diagnosis based on numerical exact diagonalization. We calculate the disorder averaged Hall conductance and the associated statistical distribution of the topological invariant Chern number, which unambiguously characterize the disorder-driven collapse of the FQH state. As the disorder strength increases towards a critical value, a continuous phase transition is detected based on the disorder configuration averaged wave function fidelity and the entanglement entropy. In the strong disorder regime, we identify a composite Fermi liquid phase with fluctuating Chern numbers, in striking contrast to the well-known ν=1/3 case where an Anderson insulator appears. Interestingly, the lowest Landau level projected a local density profile, the wave function overlap, and the entanglement entropy as a function of disorder strength simultaneously signal an intermediate phase, which may be relevant to the recent proposal of a particle-hole Pfaffian state or Pfaffian-anti-Pfaffian puddle state.
The fate of the current carrying states of a quantum Hall system is considered in the situation when the disorder strength is increased and the transition from the quantum Hall liquid to the Hall insulator takes place. We investigate a two-dimensional lattice model with spatially correlated disorder potentials and calculate the density of states and the localization length either by using a recursive Green-function method or by direct diagonalization in connection with the procedure of level statistics. From the knowledge of the energy and disorder dependence of the localization length and the density of states (DOS) of the corresponding Landau bands, the movement of the current carrying states in the disorder-energy and disorder-filling-factor planes can be traced by tuning the disorder strength. We show results for all subbands, particularly the traces of the Chern and anti-Chern states as well as the peak positions of the DOS. For small disorder strength W we recover the well-known weak levitation of the critical states, but we also reveal, for larger W, the strong levitation of these states across the Landau gaps without merging. We find the behavior to be similar for exponentially, Gaussian, and Lorentzian correlated disorder potentials. Our study resolves the discrepancies of previously published work in demonstrating the conflicting results to be only special cases of a general lattice model with spatially correlated disorder potentials. To test whether the mixing between consecutive Landau bands is the origin of the observed floating, we truncate the Hilbert space of our model Hamiltonian and calculate the behavior of the current carrying states under these restricted conditions.
We investigate the disorder-driven phase transitions in bosonic fractional quantum Hall liquids at filling factors f=1/2f=1/2f=1/2 and f=1f=1f=1 in the lowest Landau level. We use the evolution of ground-state entanglement entropy, fidelity susceptibility, and Hall conductance with increasing disorder strength to identify the underlying phase transitions. The critical disorder strengths obtained from these different quantities are consistent with each other, validating the reliability of our numerical calculations based on exact diagonalization. At f=1/2f=1/2f=1/2, we observe a clear transition from the bosonic Laughlin state to a trivial insulating phase. At f=1f=1f=1, we identify a direct phase transition from the non-Abelian bosonic Moore-Read state to a trivial insulating phase, although some signs of a disorder-induced intermediate fractional quantum Hall phase were recently reported for the f=5/2f=5/2f=5/2 fermionic cousin.