## Doctoral Thesis

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Matrix-product-state based methods, in particular the density-matrix renormalization group, are used to numerically investigate several one-dimensional systems, focusing on models with symmetry-protected topological phases that generalize the spin-1 Haldane chain. In the first part, ground state properties such as topological order parameters and the criticality at quantum phase transitions are studied.
The second part deals with dynamic properties of spin chains. Using time-dependent matrix-product-state calculations, the dynamic structure factor, and the transport properties of contacted spin chains are analyzed.

In this thesis, the transport properties of topological insulators are investigated. In contrast to trivial insulators, topological insulators possess conducting boundary states which cross the bulk energy gap that separates the highest occupied electronic band from the lowest unoccupied band. The materials used in this thesis are three-dimensional topological insulators with time-reversal symmetry. Their associated helical surface states are protected against elastic backscattering by Kramers degeneracy. The unique properties of the helical surface states can be utilized to generate spin-polarized currents at the surface of topological insulators and to control their propagation direction. This makes them a promising material class for the field of spintronics.
Here, we perform photocurrent scans of topological insulator Hall bar and nanowire devices. From these measurements, we obtained two-dimensional maps of the polarization-independent and helicity-dependent components of the photocurrents.
We find that the polarization-independent component is dominated by the Seebeck effect and thus driven by thermoelectric currents. On the other hand, the helicity-dependent component is driven by the spin-polarized currents that emerge from the topologically non-trivial helical surface states via the circular photogalvanic effect.
First and foremost, our experiments demonstrate that topological insulator nanowires provide a promising platform for the generation of spin-polarized currents, whose direction can be controlled via the helicity of the excitation light. They also highlight the importance of analysing the spatial distribution of the photocurrent, as we observe a strong enhancement of the spin-polarized current and the thermoelectric current at the interface between the nanowire and the metallic contacts. As our analysis shows, the thermoelectric current is enhanced by the Schottky effect and the spin-polarized current is amplified by the spin Nernst effect. In addition, the spin Nernst effect is also present in Hall bar devices and manifest as an enhancement of the spin-polarized current along the Hall bar sides.

This thesis contains studies on a special class of topological insulators, so called anomalous Floquet topological insulators, which exclusively occur in periodically driven systems. At the boundary of an anomalous Floquet topological insulator, topologically protected transport occurs even though all of the Floquet bands are topologically trivial. This is in stark contrast to ordinary topological insulators of both static and Floquet type, where the topological invariants of the bulk bands completely determine the chiral boundary states via the bulk-boundary correspondence. In anomalous Floquet topological insulators, the boundary states are instead characterized by bulk invariants that account for the full dynamical evolution of the Floquet system.
Here, we explore the interplay between topology, symmetry, and non-Hermiticity in two-dimensional anomalous Floquet topological insulators. The central results of this exploration are (i) new expressions for the topological invariants of symmetry-protected anomalous Floquet topological phases which can be efficiently computed numerically, (ii) the construction of a universal driving protocol for symmetry-protected anomalous Floquet topological phases and its experimental implementation in photonic waveguide lattices, (iii) the discovery of non-Hermitian boundary state engineering which provides unprecedented possibilities to control and manipulate the topological transport of anomalous Floquet topological insulators.