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Journal of Physics Condensed Matter

The effect of disorder on symmetry breaking in quantum Hall systems

Barry Friedman, Ben McCarty

Published Nov 1, 2005• 1 citation
Publisher

Abstract

We have been investigating quantum Hall systems by direct diagonalization and the density matrix renormalization group (DMRG). Calculations of the projected density–density correlation function for 10 electrons in 20 orbitals and 12 electrons in 24 orbitals indicate that disorder makes the correlation function behave more regularly with respect to system size. For these system sizes, randomness tends to enhance symmetry breaking when the wavelength of an additional imposed periodic potential is less than the wavelength of the stripe ordering. To investigate larger system sizes, preliminary calculations have been done using DMRG with momentum conservation in the y-direction. With our implementation of DMRG, promising results were obtained for the N = 0 Landau level. It seems more difficult to obtain reliable results in the N = 2 Landau level.

DOI
10.1088/0953-8984/17/46/017

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We have been investigating quantum Hall systems by direct diagonalization and the density matrix renormalization group (DMRG). Calculations of the projected density–density correlation function for 10 electrons in 20 orbitals and 12 electrons in 24 orbitals indicate that disorder makes the correlation function behave more regularly with respect to system size. For these system sizes, randomness tends to enhance symmetry breaking when the wavelength of an additional imposed periodic potential is less than the wavelength of the stripe ordering. To investigate larger system sizes, preliminary calculations have been done using DMRG with momentum conservation in the y-direction. With our implementation of DMRG, promising results were obtained for the N = 0 Landau level. It seems more difficult to obtain reliable results in the N = 2 Landau level.
The entanglement entropy of the ν=1/3 and ν=5/2 quantum Hall states in the presence of short-range random disorder has been calculated using direct diagonalization. A microscopic model of electron–electron interaction is used and spin-polarized electrons are confined to a single Landau level and interact with long-range Coulomb interaction. In the case of very weak disorder, the values of topological entanglement entropy are roughly consistent with the expected theoretical results. By considering a broad range of disorder strengths, entanglement entropy was studied in an effort to detect quantum phase transitions. In particular, there is a signature of the transition as the function of the disorder strength for the ν=5/2 state. Prospects of using the density matrix renormalization group to compute the entanglement entropy for larger system sizes are discussed.
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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 have numerically studied bosonic off-diagonal long range order, introduced by Read to describe the ordering in ideal quantum Hall states, for noninteracting electrons in random potentials confined to the lowest Landau level. We find that it also describes the ordering in disordered quantum Hall states; the proposed order parameter vanishes in the disordered (${\ensuremath{\sigma}}_{\mathrm{xy}}=0$) phase and increases continuously from zero in the ordered (${\ensuremath{\sigma}}_{\mathrm{xy}}=\frac{{e}^{2}}{h}$) phase. We study the scaling of the order parameter and find that it is consistent with that of the one-electron Green function.
We investigate the boundary effect of the density matrix renormalization group calculation (DMRG), which is an artifactual induction of symmetry-breaking pseudo-long-range order and takes place when the long-range quantum fluctuation cannot be properly included in the variational wave function due to numerical limitation. The open boundary condition often used in DMRG suffers from the boundary effect the most severely, which is directly reflected in the distinct spatial modulations of the local physical quantity. By contrast, the other boundary conditions such as the periodic one or the sin2{\mathrm{sin}}^{2}sin2-deformed interaction [A. Gendiar, R. Krcmar, and T. Nishino, Prog. Theor. Phys.122, 953 (2009)] keep spatial homogeneity, and are relatively free from the boundary effect. By comparing the numerical results of those various boundary conditions, we show that the open boundary condition sometimes gives unreliable results even after the finite-size scaling. We conclude that the examination of the boundary condition dependence is required besides the usual treatment based on the system size or accuracy dependence in cases where the long-range quantum fluctuation is important.