Studies of braided non-Abelian anyons using anyonic tensor networks

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. Anyons are point-like particles which are neither bosons nor fermions. All point-like particles in our three dimensional world are either bosons or fermions and have an exchange factor of +1 and -1 respectively, when a pair of those particles exchange positions. Anyons on the other hand represent new possibilities that are capable of existing only in two dimensions, and have non-trivial exchange factors, which can either be a complex number or even a matrix acting on a space of degenerate states. These unusual particles have some surprising properties. For example, states with anyonic excitations are known be resilient against local perturbations, which simply means that, you may perturb their local environments, and they would still retain their quantum properties - a property exclusive only to anyons. This property has motivated scientists to sometimes call them topological charges. Anyons have motivated many recent developments in science and technology. I mention two of them. Firstly, just as bosons give rise to unique phases of matter such as Bose-Einstein condensates, and fermions allow for degenerate Fermi gases, interacting anyons can form new phases of matter, and it is still a subject of intense research, as we have as yet little understanding of these new forms of matter. Secondly, the topological nature of anyons, i.e. their fault tolerance, make them to be among the leading prospective candidate particles to realise a quantum computer. This inspired a different field of research called topological quantum computation. As important as anyons tend to be, in reality, collective systems of anyons are very challenging to study analytically and numerically, due to the exponential growth of the state space of such systems with the number of particles. It is in fact this exponential growth that makes anyons to be usable for quantum computation. In the past, common numerical techniques used to study quantum many body systems included exact diagonalisation (ED) and Monte Carlo (MC) methods. Tensor network techniques are a more modern approach which have lots of advantages. I will list only two of them: 1) they are able to simulate very large system sizes and even infinite-sized systems which are not possible with ED, and 2) they can treat systems of fermions and anyons which are problematic for MC methods. In the first phase of this thesis, I extended the anyonic tensor network algorithms, by incorporating U(1) symmetry to give a modified ansatz AnyonxU(1) tensor networks, which are capable of simulating anyonic systems at any rational filling fraction. By testing the ansatz with the time evolving block decimation algorithm, I benchmarked these methods by using it to confirm previously known results for simple models, including a one-dimensional chain of itinerant and localised anyons, and a two-leg ladder where anyons hop between sites and use available vacancies to exchange positions with other anyons. I performed test simulations with four different particle types, namely, Fibonacci and Ising anyons, which are non-Abelian anyons, and spinless fermions and hardcore bosons, which can be treated as "simple types" of anyons. I compared the results with known exact results obtained using other methods,and got very good performances with relative error of around e < 10-4 in the ground state energy. After being satisfied with the performance of the numerical algorithms, I then proceeded to the second phase where 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, where 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 the interaction of these anyons and their exchange "braid" statistics.