Spin-orbit interaction and time reversal symmetry are at the heart of the physics of topological insulators and this SFB. 2D (3D) topological insulators are characterized by gapless edge states (gapless surface states) where the direction of motion follows the spin. The usual focus on the properties of topological insulators (TIs) is directed at strictly 2D or 3D materials while almost nothing is known about the transition of the surface states of 3D TIs into one-dimensional edge states of 2D TIs. Therefore, the first goal of this proposal is to develop band structure models to understand how an overlap (hybridization) of surface states leads to the formation of edge states, to investigate if the generated one-dimensional edge states have the same properties as expected for 2D TIs. In particular, we intend to study this transition in the presence of magnetic fields.
Further, magnetic disorder should play a central role in the modification of the properties of TIs. Although one could expect that magnetic impurities would destroy the topological order preserved by time reversal symmetry, there are at least a few examples where the magnetic impurities lead to new states of matter: (1) the quantum anomalous Hall effect (QAHE) (2) the formation of a Weyl semimetal and (3) topological Kondo insulators. In the QAHE, magnetic impurities create an effective Zeeman field which allows for the inversion of bands for only one spin direction, and formation of a single spin-polarized edge state at the sample boundary. A Weyl semimetal is a three-dimensional topological phase characterized by pairs of nondegenerate, linear touchings of the bulk bands, called Weyl nodes, and can be generated when magnetic impurities are added to 3D TIs. The topological Kondo insulator (TKI) is the topologically nontrivial insulating state produced by the spin-orbit coupling associated with the hybridization between itinerant and localized f-electrons. Central to our proposal is the understanding of the formation of the QAHE, i.e. the competition between scattering from magnetic impurities and the creation of the coherent topological insulator state, as well as a direct study of the transition between the QAHE and the quantum Hall effect (QHE). Moreover, we will be concerned with spin pumping from a spin-polarized edge state into a semiconductor, i.e. spintronic applications of the QAHE. As a related topic we will investigate the transition of a 3D topological insulator into a Weyl semimetal in the presence of magnetic impurities for realistic systems. Finally, in the long run, we would like to go from non-interacting models with magnetic impurities to interacting models, i.e. to topological Kondo insulators.
[A06.4] G. Tkachov and E. M. Hankiewicz, Ballistic quantum spin Hall state and enhanced edge backscattering in strong magnetic fields, Phys. Rev. Lett. 104, 166803 (2010).