Interactions and Geometry in Topological Systems

Topological materials are one of the lead candidates for developing viable noise resilient quantum computers. The properties that make these materials so suited to the task include their degenerate ground states and anyonic excitation statistics. However, it is often the case that the more exotic the statistics are the more complex the under- lying Hamiltonian is. This can make them challenging to work with. Alternate representations of these Hamiltonians can prove useful in solving the systems and investigating the behaviour of their physical observables.
This thesis explores the construction and advantages of alternate rep- resentations of certain topological quantum systems. Initially, unitary transformations are presented, which map the Z2 surface code and toric code to free fermions and fermions coupled to global symmetry operators, respectively. The methods presented in this thesis could be employed to find possible free fermion solvable descriptions of other more complex interacting topological systems. It also is found that the Kitaev honeycomb model has an effective geometric description in terms of massless Majorana spinors obeying the Dirac equation em- bedded in a Riemann-Cartan spacetime. This description is shown numerically to be faithful for the low energy limit of the model, pre- dicting the response of two-point correlations to variations of the cou- pling parameters of the model. These results suggest that geometric descriptions of topological materials could provide useful insights into the behaviour of their physical observables that make them so useful for quantum computation.