Topological states offer many intriguing possibilities, both for fundamental research as well as for potential applications [1,2]. In this talk, we present properties of two different such systems in reduced dimensionalities: topological superconductivity in quantum dots and quantum rings as well as magneto-optical properties of HgTe/CdTe quantum wells.

Engineering topological superconductivity in semiconductor structures provides fascinating ways to obtain and study Majorana modes in a condensed matter context. Here, we theoretically investigate topological superconductivity in quantum dots and quantum rings [3]. By applying a magnetic field which is expelled from a quantum ring, but which creates a flux that is an odd integer multiple of , Majorana modes, that is, (approximately) degenerate edge modes with zero energy and zero charge density, become possible in the topological regime. Even if a magnetic field penetrates into the superconducting region, the system can under certain circumstances still support such edge modes with approximately zero energy and charge.

The second part of the talk is about magneto-optical properties of HgTe/CdTe quantum wells. In two-dimensional topological insulators, such as inverted HgTe/CdTe quantum wells, helical quantum spin Hall (QSH) states persist even at finite magnetic fields below a critical magnetic field , above which only quantum Hall (QH) states can be found [4,5]. Using linear response theory, we theoretically investigate the magneto-optical properties of inverted HgTe/CdTe quantum wells, both for infinite two-dimensional and finite-strip geometries, and possible signatures of the transition between the QSH and QH regimes. In the QSH regime, we find an additional absorption peak at low energies for the finite-strip geometry. This peak arises due to the presence of edge states in this geometry and persists for any Fermi level in the QSH regime, while in the QH regime the peak vanishes if the Fermi level is situated in the bulk gap. Thus, by sweeping the gate voltage, it is potentially possible to distinguish between the QSH and QH regimes due to this signature. Moreover, we investigate the effect of spin-orbit coupling and finite temperature on this measurement scheme [6].

[1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).

[2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).

[3] B. Scharf and I. Zutic, Phys. Rev. B 91, 144505 (2015).

[4] G. Tkachov and E. M. Hankiewicz, Phys. Rev. Lett. 104, 166803 (2010).

[5] B. Scharf, A. Matos-Abiague, and J. Fabian, Phys. Rev. B 86, 075418 (2012).

[6] B. Scharf, A. Matos-Abiague, I. Zutic, and J. Fabian, Phys. Rev. B 91, 235433 (2015).