  The content of this thesis can be broadly summarised into two categories:
first, I constructed modified numerical algorithms based on tensor networks to
simulate systems of anyons in low dimensions, and second, I used those methods
to study the topological phases the anyons form when they braid around one
another. In the first phase of my thesis, I extended the anyonic tensor network
algorithms, by incorporating U(1) symmetry to give a modified ansatz,
Anyon-U(1) tensor networks, which are capable of simulating anyonic systems at
any rational filling fraction. In the second phase, I used the numerical
methods to study some models of non-Abelian anyons that naturally allows for
exchange of anyons. I proposed a lattice model of anyons, which I dubbed
anyonic Hubbard model, which is a pair of coupled chains of anyons (or simply
called anyonic ladder). Each site of the ladder can either host a single
anyonic charge, or it can be empty. The anyons are able to move around,
interact with one another, and exchange positions with other anyons, when
vacancies exist. Exchange of anyons is a non-trivial process which may
influence the formation of different kinds of new phases of matter. I studied
this model using the two prominent species of anyons: Fibonacci and Ising
anyons, and made a number of interesting discoveries about their phase
diagrams. I identified new phases of matter arising from both the interaction
between these anyons and their exchange braid statistics.
