Weakly chiral networks and two-dimensional delocalized states in a weak magnetic field

Abstract

We study numerically 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. The prime idea is to separate in space the regions with phase action of magnetic field, where it affects interference in course of multiple disorder scattering, and the regions with orbital action of magnetic field, where it bends electron trajectories. In our models, the disorder mixes counter-propagating channels on the links, while scattering matrices at the nodes describe exclusively the bending of electron trajectories. By artificially introducing a strong spread in the scattering strengths on the links (but keeping the average strength constant), we eliminate the interference and reduce the electron propagation over a network to a classical percolation problem. In this limit we establish the form of the disorder -magnetic field phase diagram. This diagram contains the regions with and without edge states, i.e. the regions with zero and quantized Hall conductivities. Taking into account that, for a given disorder, the scattering strength scales as inverse electron energy, we find agreement of our phase diagram with levitation scenario: energy separating the Anderson and quantum Hall insulating phases floats up to infinity upon decreasing magnetic field. From numerical study, based on the analysis of quantum transmission of the network with random phases on the links, we conclude that the positions of the weak-field quantum Hall transitions on the phase diagram are very close to our classical-percolation results. We checked that, in accord with the Pruisken theory, presence or absence of time reversal symmetry on the links has no effect on the line of delocalization transitions. We also find that floating up of delocalized states in energy is accompanied by doubling of the critical exponent of the localization radius. We establish the origin of this doubling within classical-percolation analysis.

Spin quantum Hall effect and plateau transitions in multilayer network models

“…Alternative types of behaviour have also been proposed, involving direct transitions between phases with Hall conductance differing by multiple quantum units, 9 and there has developed quite an extensive literature on the subject, reviewed in Ref. 10. Attempts [11][12][13][14][15][16] to distinguish between these different possibilities using numerical simulations are hampered by the fact that in the most interesting regimes -weak magnetic field or weak interlayer coupling -the localisation length is never short, making asymptotic behaviour hard to reach.…”

Symmetry, Universality and Critical Transitions in the Network Models

“…This doubling of the critical exponent was discussed in detail and calculated numerically in Ref. [27].…”

“…It was later generalized 27 to study the levitation scenario as p − q model. The levitation scenario, proposed by Khmelnitskii in 1984, 29 was introduced to reconcile the result of the scaling theory for 2d systems, that there are no extended states, and the necessity of a delocalized state for a quantum Hall transition.…”

“…Later, in Ref. [27], similarly to the CC model, the critical exponent was obtained from the scaling analysis of the smallest positive Lyapunov exponents. A high quality scaling was achieved for = 2 36, which is in good agreement with simulations of the CC model.…”

“…[28], and described in detail and generalized in Ref. [27]. In a chiral triangular model it is assumed that smooth potential has 120 rotational symmetry.…”