The thesis "Anlre resolved dielectric response in carbon nanotubes" is dedicated to expounding the the anisotropy in the fundamental dielectric response of carbon nanotubes. While nanotubes are along their axis essentially planar graphene, the rolled up topology gives rise to entirely new features for perpendicular polarizations.

Topological insulators (TIs) are a new state of quantum matter that has fundamentally challenged our knowledge of insulator and metals. They are insulators in the bulk, but metallic on the edge. A TI is characterized by a so-called topological invariant. This characteristic integer number is associated to every mapping between two topological spaces and can be defined for an electronic system on the lattice. Due to the bulk-edge correspondence a non-trivial value leads to topologically protected edge states. To get insight into the electronic characteristics of these edge/surface states, however, an effective continuum theory is needed. Continuum models are analytical and are also able to model transport. In this thesis we will address the suitability of continuum low-energy theories to describe the topological characteristics of TIs. The models which are topologically well-defined are called topological k.p Hamiltonians. After introducing a necessary background in chapter 1 and 2, we will discuss in the methodological chapter 3 the strategies that have to be taken into account to allow for studying topological surface states. In chapter 4 we will study two different model classes associated to a spherical basis manifold. Both have an integer topological invariant, but one shows a marginal bulk-edge correspondence. In chapter 5 we will study a different continuum theory where the basis manifold corresponds to a hemisphere. We then apply all these ideas to a time-reversal invariant TI -- uniaxially strained Mercury Telluride (HgTe). We determine the spin textures of the topological surface states of strained HgTe using their close relations with the mirror Chern numbers of the system and the orbital composition of the surface states. We show that at the side surfaces with $C_{2v}$ point group symmetry an increase in the strain magnitude triggers a topological phase transition where the winding number of the surface state spin texture is flipped while the four topological invariants characterizing the bulk band structure are unchanged. In the last chapter we will give a summary.

The concept of topology in condensed matter physics has led to the discovery of rich and exotic physics in recent years. Especially when strong correlations are included, phenomenons such as fractionalization and anyonic particle statistics can arise. In this thesis, we study several systems hosting topological phases of interacting fermions. In the first part, we consider one-dimensional systems of parafermions, which are generalizations of Majorana fermions, in the presence of a Z_N charge symmetry. We classify the symmetry-protected topological (SPT) phases that can occur in these systems using the projective representations of the symmetries and find a finite number of distinct phases depending on the prime factorization of N. The different phases exhibit characteristic degeneracies in their entanglement spectrum (ES). Apart from these SPT phases, we report the occurrence of parafermion condensate phases for certain values of N. When including an additional Z_N symmetry, we find a non-Abelian group structure under the addition of phases. In the second part of the thesis, we focus on two-dimensional lattice models of spinless fermions. First, we demonstrate the detection of a fractional Chern insulator (FCI) phase in the Haldane honeycomb model on an infinite cylinder by means of the density-matrix renormalization group (DMRG). We report the calculation of several quantities characterizing the topological order of the state, i.e., (i)~the Hall conductivity, (ii)~the spectral flow and level counting in the ES, (iii)~the topological entanglement entropy, and (iv)~the charge and topological spin of the quasiparticles. Since we have access to sufficiently large system sizes without band projection with DMRG, we are in addition able to investigate the transition from a metal to the FCI at small interactions which we find to be of first order. In a further study, we consider a time-reversal symmetric model on the honeycomb lattice where a Chern insulator (CI) induced by next-nearest neighbor interactions has been predicted by mean field theory. However, various subsequent studies challenged this picture and it was still unclear whether the CI would survive quantum fluctuations. We therefore map out the phase diagram of the model as a function of the interactions on an infinite cylinder with DMRG and find evidence for the absence of the CI phase. However, we report the detection of two novel charge-ordered phases and corroborate the existence of the remaining phases that had been predicted in mean field theory. Furthermore, we characterize the transitions between the various phases by studying the behavior of correlation length and entanglement entropy at the phase boundaries. Finally, we develop an improvement to the DMRG algorithm for fermionic lattice models on cylinders. By using a real space representation in the direction along the cylinder and a real space representation in the perpendicular direction, we are able to use the momentum around the cylinder as conserved quantity to reduce computational costs. We benchmark the method by studying the interacting Hofstadter model and report a considerable speedup in computation time and a severely reduced memory usage.