The interest in the properties of edge states in Chern insulators and in Z2 topological insulators has increased rapidly in recent years. We present calculations on how to influence the transport properties of chiral and helical edge states by modifying the edges in the Haldane and in the KaneMele model. The Fermi velocity of the chiral edge states becomes direction dependent as does the spin-dependent Fermi velocity of the helical edge states. Additionally, we explicitly investigate the robustness of edge states against local disorder. The edge states can be reconstructed in the Brillouin zone in the presence of disorder. The influence of the width and of the length of the system is studied as well as the dependence of the edge states on the strength of the disorder.

We study topological properties of one-triplon bands in an extended Shastry-Sutherland model relevant for the frustrated quantum magnet SrCu2(BO3)2. To this end perturbative continuous unitary transformations are applied about the isolated dimer limit allowing to calculate the one-triplon dispersion up to high order in various couplings including intra and inter Dzyaloshinskii-Moriya interactions and a general uniform magnetic field. We determine the Berry curvature and the Chern number of the different one-triplon bands. We demonstrate the occurance of Chern numbers ±1 and ±2 for the case that two components of the magnetic field are finite. Finally, we also calculate the triplon Hall effect arising at finite temperatures.

Topological excitations in magnetically ordered systems have attracted much attention lately. We report on topological magnon bands in ferromagnetic Shastry-Sutherland lattices whose edge modes can be put to use in magnonic devices. The synergy of Dzyaloshinskii-Moriya interactions and geometrical frustration are responsible for the topologically nontrivial character. Using exact spin-wave theory, we determine the finite Chern numbers of the magnon bands which give rise to chiral edge states. The quadratic band crossing point vanishes due the present anisotropies, and the system enters a topological phase. We calculate the thermal Hall conductivity as an experimental signature of the topological phase. Different promising compounds are discussed as possible physical realizations of ferromagnetic Shastry-Sutherland lattices hosting the required antisymmetric Dzyaloshinskii-Moriya interactions. Routes to applications in magnonics are pointed out. PACS numbers:Topological phases 1,2 exist in both fermionic and bosonic systems and constitute a fast developing research area. Although the theoretical understanding of fermionic topological systems has made impressive progress, topological bosonic excitations have gained considerable attention only in the past few years. Despite the increasing conceptual knowledge of topological matter, only very few materials have been identified with topological properties compared to the large number of potential topological materials 3 . Even less is known about potential applications. This is, in particular, true for topological bosonic signatures 4 . Thus, it is a major challenge to theoretically predict and experimentally verify topological bosonic fingerprints in order to move towards useful applications.In the research of topological properties in condensed matter, the magnetic degrees of freedom have increased in importance. Magnetic data storage is already a ubiquitous everyday technology 5 . Recently, magnetic spin waves, so-called magnons, themselves are used to carry and to process information which is called " magnonics " 6-8 . Adding topological aspects the field of magnonics 9 considerably enhances the possibilities to build efficient devices for which we will make a proposal in this paper.The challenge in finding topological signatures in magnetically ordered spin systems are the small Dzyaloshinskii-Moriya (DM) interactions 10,11 which induce only small Berry curvatures. The size of the DM terms relative to the isotropic coupling is roughly as large as |g − 2|/2, i.e., the deviation of the g factor from 2, because both result from spin-orbit coupling. Thus, the DM terms are generically too small to induce detectable topological effects. In strongly frustrated systems, however, the relative size of the DM terms can indeed be comparable to the isotropic couplings 12 .Another issue is the localization of edge modes. Employing the wording of semiconductor physics, one must distinguish direct (at fixed wave vector) and indirect gaps (allowing for changes in the wave vector)...

Topological properties play an increasingly important role in future research and technology. This also applies to the field of topological magnetic excitations which has recently become a very active and broad field. In this Perspective article, we give an insight into the current theoretical and experimental investigations and try an outlook on future lines of research.

Topological aspects represent currently a boosting area in condensed matter physics. Yet there are very few suggestions for technical applications of topological phenomena. Still, the most important is the calibration of resistance standards by means of the integer quantum Hall effect. We propose modifications of samples displaying the integer quantum Hall effect which render the tunability of the Fermi velocity possible by external control parameters such as gate voltages. In this way, so far unexplored possibilities arise to realize devices such as tunable delay lines and interferometers.