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arXiv:cond-mat/9403094 · Physical Review Letters

Off-Diagonal Long Range Order and Scaling in a Disordered Quantum Hall System

Shivaji L. Sondhi, Martin P. Gelfand

Published Oct 10, 1994• 4 citations
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Abstract

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.

DOI
10.1103/physrevlett.73.2119

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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 (\ensuremathσxy=0{\ensuremath{\sigma}}_{\mathrm{xy}}=0\ensuremathσxy​=0) phase and increases continuously from zero in the ordered (\ensuremathσxy=e2h{\ensuremath{\sigma}}_{\mathrm{xy}}=\frac{{e}^{2}}{h}\ensuremathσxy​=he2​) phase. We study the scaling of the order parameter and find that it is consistent with that of the one-electron Green function.
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The entanglement entropy of ν=1/2ν=1/2ν=1/2 and ν=9/2ν=9/2ν=9/2 quantum Hall states in the presence of short range disorder has been calculated by direct diagonalization. Spin polarized electrons are confined to a single Landau level and interact with long range Coulomb interaction. For ν=1/2ν=1/2ν=1/2 the entanglement entropy is a smooth monotonic function of disorder strength. For ν=9/2ν=9/2ν=9/2 the entanglement entropy is non monotonic suggestive of a solid-liquid phase transition. As a model of the transition at ν=1/2ν=1/2ν=1/2 free fermions with disorder in 2 dimensions were studied. Numerical evidence suggests the entanglement entropy scales as LLL rather than the Lln⁡LL \ln{L}LlnL as in the disorder free case.
A new nonsingular mapping between bosons and fermions is introduced in connection with the fractional quantum Hall effect. The mapping connects a fermion system of filling ${\ensuremath{\nu}}_{\mathit{f}}$ to a boson system of filling ${\ensuremath{\nu}}_{\mathit{b}}$ through the relation ${\ensuremath{\nu}}_{\mathit{f}}^{\mathrm{\ensuremath{-}}1}$=${\ensuremath{\nu}}_{\mathit{b}}^{\mathrm{\ensuremath{-}}1}$+1. The ground state of the boson system is shown to map accurately to that of the fermion system if and only if they are both incompressible quantum Hall states. The accuracy of the mapping is shown to be related to the validity of the mean-field theory. The relevance of this mapping to off-diagonal long-range order is discussed by numerically studying the binding of wave-function zeros to particle positions.
We present a detailed study of the ground-state entanglement in disordered fractional quantum Hall liquids. We consider electrons at various filling fractions fff in the lowest Landau level, with Coulomb interactions. At f=1/3,1/5f=1/3,1/5f=1/3,1/5, and 2/52/52/5 where an incompressible ground-state manifold exists at zero disorder, we observe a pronounced minimum in the derivative of entanglement entropy with respect to disorder. At each filling, the position of this minimum is stable against increasing system size, but its magnitude grows monotonically and appears to diverge in the thermodynamic limit. We consider this behavior of the entropy derivative as a compelling signal of the expected disorder-driven phase transition from a topological fractional quantum Hall phase to an insulating phase. On the contrary, at f=1/2f=1/2f=1/2 where a compressible composite fermion sea is present at zero disorder, the entropy derivative exhibits much greater, almost chaotic, finite-size effects, without a clear phase transition signal for system sizes within our exact diagonalization limit. However, the dependence of entanglement entropy with system size changes with increasing disorder, consistent with the expectation of a phase transition from a composite fermion sea to an insulator. Finally, we consider f=1/7f=1/7f=1/7 where compressible Wigner crystals are quite competitive at zero disorder, and analyze the level statistics of entanglement spectrum at f=1/3f=1/3f=1/3.
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.