Graphene and topological insulators are novel materials which have recently attracted a lot of attention. Due to their peculiar fundamental properties a number of new and yet unknown effects arise in these materials.
One of such examples are triplet excitations, magnons, which may be observed in graphene. During the last decade there has been a discussion in the literature regarding their existence. Since no established viewpoint was stated, in our work we reexamine this problem. Furthermore, we study the properties of magnons not only in graphene but also in carbon nanotubes. We calculate the spectrum of these exotic “spin-1” excitations and confirm that they indeed can exist in graphene-based materials in the presence of the Coulomb interactions.
In the second part of our work 2D topological insulators are examined from the perspective of the semiclassical theory. In spite of quantum mechanical study, this approach can give simple and pictorial explanation of the topological edge states. In our work we find the semiclassical orbits for the samples of different geometries and also discuss the influence of the quantum effects, the Berry phase, on the semiclassical electron dynamics. Finally, we try to find the semiclassical mechanism responsible for topological protection of the edge states.