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.

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.