Spin quantum Hall effect and plateau transitions in multilayer network models

Abstract

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 localization, 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 localization 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. Focusing 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.

Phase diagram of the weak-magnetic-field quantum Hall transition quantified from classical percolation

“…As in Ref. 62, for a given realization, the two-terminal conductance between the opposite open faces was identified with the number of such spanning trajectories. Different disorder realizations generate the conductivity distribution with average, σ(q, p, L).…”

“…For this reason there is a mapping between the simulation in Ref. 62 and treatment of the model A in the present paper. Namely, p 1 in Ref.…”

“…Namely, p 1 in Ref. 62 should be identified with the backscattering probability p in the present paper, while the probability that the given bond is present in Ref. 62, p, should be identified with our parameter q.…”

“…Relation to Ref. 62 In Ref. 62, spin quantum Hall effect in bilayer and trilayer systems was studied numerically, in order to trace the emergence of macroscopic metallic phase upon adding the third dimension 65 .…”

“…62 In Ref. 62, spin quantum Hall effect in bilayer and trilayer systems was studied numerically, in order to trace the emergence of macroscopic metallic phase upon adding the third dimension 65 . The authors made use of the fact that in a strictly 2D system there is a mapping between the spin quantum Hall transition and classical bond percolation 66,67 .…”