Topological phases in condensed matter systems: A study of symmetries, quasiparticles and phase transitions

The research described in this thesis focuses on topological phases in condensed matter systems. It can be roughly divided into two parts. In the first part noninteracting systems are studied. The symmetry algebra of a charged spin-1/2 particle coupled to a non-Abelian magnetic field is determined, which explains the finite and infinite degeneracy of the energy. This system is a candidate for a continuum model of a three-dimensional topological insulator.
Next, a two-dimensional version is considered on a sphere, where its spectrum is solved. The planar version of the sam model is probed by the insertion of a non-Abelian flux. Starting from a spin-polarized state, the adiabatic insertion of the flux results in a state with nontrivial spin-texture which is recognized as a quantum Hall skyrmion.
The second part covers topological phases which stem from an underlying interacting model and that carry quasiparticles with fractional statistics. By applying a technique called topological symmetry breaking transitions between different phases can be induced. A careful treatment shows that different domains may appear in the broken phase separated by domain walls and it leads to a clear interpretation of confined particles.
Moreover, phase transitions induced by multilayered condensates are considered. Non-Abelian phases as well as an entire hierarchy of Abelian fractional quantum Hall states are treated. A special focus is given to the study of the one-dimensional boundary between the two phases.