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

Free-fermion descriptions of parafermion chains and string-net models

Topological phases of matter remain a focus of interest due to their unique properties: fractionalization, ground-state degeneracy, and exotic excitations. While some of these properties can occur in systems of free fermions, their emergence is generally associated with interactions between particles. Here, we quantify the role of interactions in general classes of topological states of matter in one and two spatial dimensions, including parafermion chains and string-net models. Surprisingly, we find that certain topological states can be exactly described by free fermions, while others saturate the maximum possible distance from their optimal free-fermion description [C. J. Turner et al., Nat. Commun. 8, 14926 (2017)]. Our work opens the door to understanding the complexity of topological models by establishing new types of fermionization procedures to describe their low-energy physics, thus making them amenable to experimental realizations