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Illustration of the transfer matrix, T, Eq. (53), of the p-q model. Two q-bonds correspond to the first matrix in the product Eq. (53); two p-bonds correspond to the third block-diagonal matrix in the product.
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We study the localization properties of two-dimensional electrons in a weak perpendicular magnetic field. For this purpose we construct weakly chiral network models on the square and triangular lattices, by separating in space the regions with phase action of magnetic field, where it affects interference in course of disorder scattering, and the re...
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Citations
... The Chalker-Coddington network model describes transport in a lattice with fixed (non-random) scattering matrices at nodes and random phase matrices on links. In the past, different kinds of network models have been used successfully to study QH localization-delocalization transitions and their variants [72][73][74][75][76][77][78][79][80][81][82][83]. Network models are particularly useful because they can be used to efficiently compute the localization length on a quasi-1D geometry, i.e., a long cylinder [79,[84][85][86]. ...
Two-dimensional periodically driven systems can host an unconventional
topological phase unattainable for equilibrium systems, termed the Anomalous
Floquet-Anderson insulator (AFAI). The AFAI features a quasi-energy spectrum
with chiral edge modes and a fully localized bulk, leading to non-adiabatic but
quantized charge pumping. Here, we show how such a Floquet phase can be
realized in a driven, disordered Quantum Anomalous Hall insulator, which is
assumed to have two critical energies where the localization length diverges,
carrying states with opposite Chern numbers. Driving the system at a frequency
close to resonance between these two energies localizes the critical states and
annihilates the Chern bands, giving rise to an AFAI phase. We exemplify this
principle by studying a model for a driven, magnetically doped topological
insulator film, where the annihilation of the Chern bands and the formation of
the AFAI phase is demonstrated using the rotating wave approximation. This is
complemented by a scaling analysis of the localization length for two copies of
a quantum Hall network model with a tunable coupling between them. We find that
by tuning the frequency of the driving close to resonance, the driving strength
required to stabilize the AFAI phase can be made arbitrarily small.
... The Chalker-Coddington network model describes transport in a lattice with fixed (non-random) scattering matrices at nodes and random phase matrices on links. In the past, different kinds of network models have been used successfully to study QH localization-delocalization transitions and their variants [71][72][73][74][75][76][77][78][79][80][81][82]. Network models are particularly useful because they can be used to efficiently compute the localization length on a quasi-1D geometry, i.e., a long cylinder [78,[83][84][85]. ...
Two-dimensional periodically driven systems can host an unconventional topological phase unattainable for equilibrium systems, termed the Anomalous Floquet-Anderson insulator (AFAI). The AFAI features a quasi-energy spectrum with chiral edge modes and a fully localized bulk, leading to non-adiabatic but quantized charge pumping. Here, we show how such a Floquet phase can be realized in a driven, disordered Quantum Anomalous Hall insulator, which is assumed to have two critical energies where the localization length diverges, carrying states with opposite Chern numbers. Driving the system at a frequency close to resonance between these two energies localizes the critical states and annihilates the Chern bands, giving rise to an AFAI phase. We exemplify this principle by studying a model for a driven, magnetically doped topological insulator film, where the annihilation of the Chern bands and the formation of the AFAI phase is demonstrated using the rotating wave approximation. This is complemented by a scaling analysis of the localization length for two copies of a quantum Hall network model with a tunable coupling between them. We find that by tuning the frequency of the driving close to resonance, the driving strength required to stabilize the AFAI phase can be made arbitrarily small.
... Here the possibility of observing "floating" of the states is substantially limited by the long localization length for exponentially low temperatures and exponentially small system sizes. Nevertheless, a microscopic model for this transition has recently been proposed, 22 in which different sides in space have magnetic field phase effects that influence interference in multiple scattering events by charge carriers and by the orbital effect of the magnetic field, where the bending of the electron trajectories and the formation of a cyclotron orbits become significant. This model yielded a result that confirms the "floating" scenario and gave a quantitative prediction of doubling of the critical index for the localization length c as the delocalized states "float." ...
The longitudinal ρxx(B,T) and Hall ρxy(B,T) resistances in magnetic fieldsB up to 12 T at temperatures T = 1.8–80 K are studied experimentally in n-In0.2Ga0.8As/GaAs nanostructures with single and double strongly-coupled quantum wells separated by different barrier widths. It is shown that for ωcτ≅1 there is a critical magnetic field near which the scaling relation ρxx∝|B−BC|T−κ, which is indicative of a phase transition from a dielectric state to a quantum hall state, is satisfied. It is found that the critical index κ depends on the width of the barrier between the double quantum wells. The nature of this behavior is discussed.
... Another approach is to study alloy scattering in a 2D system [48]. This method allows useful study of the critical component of the plateauplateau transition [48][49][50][51][52][53][54][55][56][57], which is equivalent to the insulator-quantum Hall transition. The role of disorder seems to play an important role in whether the observed critical exponent is of a universal value (∼0.42) [48][49][50][51][52][53][54][55][56][57][58]. ...
... This method allows useful study of the critical component of the plateauplateau transition [48][49][50][51][52][53][54][55][56][57], which is equivalent to the insulator-quantum Hall transition. The role of disorder seems to play an important role in whether the observed critical exponent is of a universal value (∼0.42) [48][49][50][51][52][53][54][55][56][57][58]. It may be possible if one can observe evidence for modular symmetry in the vicinity of a direct I-QH transition [59][60][61], one may be able to obtain a unifying picture regarding whether the critical exponent is universal in 2D systems. ...
The direct insulator-quantum Hall (I-QH) transition corresponds to a transition from an insulator to a high Landau-level-filling factor ν > 2 QH state, which is characterized by an approximately temperature T -independent longitudinal resistance of a few nm thick electron (or hole) layer. In this paper, we review both the experimental and theoretical results on the direct I-QH transition. In particular, we attempt to address several interesting yet unsettled issues in the field of the direct I-Q transition. We suggest that further studies are required for obtaining a thorough understanding of the direct I-QH transition observed in nano-scale charge layers.
We consider a recently proposed network model of the integer quantum Hall
(IQH) effect in a weak magnetic field. Using a supersymmetry approach, we
reformulate the network model in terms of a superspin ladder. A subsequent
analysis of the superspin ladder and the corresponding supersymmetric nonlinear
sigma model allows to establish the phase diagram of the network model, and the
form of the critical line of the weak-field IQH transition. Our results confirm
the universality of the IQH transition, which is described by the same sigma
model in strong and weak magnetic fields. We apply the suspersymmetry method to
several related network models that were introduced in the literature to
describe the quantum Hall effect in graphene, the spin-degenerate Landau
levels, and localization of electrons in a random magnetic field.
We study numerically the form of the critical line in the disorder–magnetic field phase diagram of the p–q network model, constructed to study the levitation of extended states at weak magnetic fields. We use one-parameter scaling, keeping either q (related to magnetic field) or p (related to energy) constant, to calculate two critical exponents, describing the divergence of the localization length in each case. The ratio of those two exponents defines the form of the critical line close to zero magnetic field.
Two-dimensional electron gas in the integer quantum Hall regime is
investigated numerically by studying the dynamics of an electron hopping on a
square lattice subject to a perpendicular magnetic field and random on-site
energy with white noise distribution. Focusing on the lowest Landau band we
establish an anti-levitation scenario of the extended states: As either the
disorder strength $W$ increases or the magnetic field strength $B$ decreases,
the energies of the extended states move below the Landau energies pertaining
to a clean system. Moreover, for strong enough disorder, there is a disorder
dependent critical magnetic field $B_c(W)$ below which there are no extended
states at all. A general phase diagram in the $W-1/B$ plane is suggested with a
line separating domains of localized and delocalized states.
We discuss various applications of the network models, starting with the original Chalker-Coddington (CC) network model, proposed to describe inter-plateaux transition in the integer quantum Hall effect (IQHE).. We present a semi-classical picture which serves as a basis for the CC model. We then discuss various generalizations of the CC model: a two-channel per link, allowing to include spin or consider two lowest Landau levels; different symmetries of the transfer matrices corresponding to novel symmetry classes of the spin and thermal quantum Hall effects; quantum spin Hall effect with very nontrivial phase diagram. We proceed by presenting two recent network models: a weakly chiral network model constructed on the square lattices to describe levitation of extended states at low magnetic fields, and a triangular network model analogous to the original CC model, and its generalization, also supporting the levitation scenario. We discuss how the critical exponent of the transition depends on the existence of two critical energies. We conclude our review by discussing the generalization of the CC model, describing the effect of nuclear spin on the tunneling of electron, which allows spin-flip scattering.
We consider magnetotransport in high-mobility 2D electron gas in a
non-quantizing magnetic field. We employ a weakly chiral network model to test
numerically the prediction of the scaling theory that the transition from an
Anderson to a quantum Hall insulator takes place when the Drude value of the
non-diagonal conductivity is equal to 1/2. The weaker is the magnetic field the
harder it is to locate a delocalization transition using quantum simulations.
The main idea of the present study is that the position of the transition does
not change when a strong local inhomogeneity is introduced. Since the strong
inhomogeneity suppresses interference, transport reduces to classical
percolation. We show that the corresponding percolation problem is bond
percolation over two sublattices coupled to each other by random bonds.
Simulation of this percolation allows to access the domain of very weak
magnetic fields. Simulation results confirm the criterion \sigma_{xy}=1/2 for
values \sigma_{xx}\sim 10, where they agree with earlier quantum simulation
results. However for larger \sigma_{xx} we find that the transition boundary is
described by \sigma_{xy} \sigma_{xx}^k with k= 0.5, i.e., the transition takes
place at higher magnetic fields. The strong inhomogeneity limit of
magnetotransport in the presence of a random magnetic field, pertinent to
composite fermions, corresponds to a different percolation problem. In this
limit we find for the delocalization transition boundary \sigma_{xy}
\sigma_{xx}^{0.6}.
We study the spin quantum Hall effect and transitions between Hall plateaus
in quasi two-dimensional network models consisting of several coupled layers.
Systems exhibiting the spin quantum Hall effect belong to class C in the
symmetry classification for Anderson localisation, and for network models in
this class there is an established mapping between the quantum problem and a
classical one involving random walks. This mapping permits numerical studies of
plateau transitions in much larger samples than for other symmetry classes, and
we use it to examine localisation in systems consisting of $n$ weakly coupled
layers. Standard scaling ideas lead one to expect $n$ distinct plateau
transitions, but in the case of the unitary symmetry class this conclusion has
been questioned. Focussing on a two-layer model, we demonstrate that there are
two separate plateau transitions, with the same critical properties as in a
single-layer model, even for very weak interlayer coupling.
