Quasiparticles in topological quantum liquids such as the fractional
Quantum Hall states and certain two-dimensional frustrated magnets
display unconventional quantum statistics. The conserved topological
charge of these non-Abelian anyons is protected and has spawned
interest for such systems in the context of quantum computation.
In this project we plan to study the properties of interacting many-anyon systems whose construction is based on the mathematical structures describing the fundamental operations of fusion and braiding. Upon fine-tuning of the interactions these models can be embedded into a family of commuting operators. We shall develop functional methods to exploit local identities present in these integrable models for the solution of their spectral problem. Our investigation of integrable anyon chains will be complemented by studies of non-integrable deformations thereof to gain understanding into the emergence of unconventional boundary degrees of freedom and their realization as topological quantum impurities in electronic systems.