Anyons in infinite quantum systems : QFT in d=2+1 and the Toric Code

Anyons, comprising a special type of particles that exhibit braid statistics, have received considerable attention in the past decade, from physicists, mathematicians, and computer scientists. It turns out that so-called modular tensor categories naturally appear in the description of the key properties of anyons, which partly explains the interest of mathematicians in the subject. In this thesis, we study the way such modular tensor categories emerge in two classes of models, viz. quantum field theory in low-dimensional Minkowski space-time, and quantum spin systems on an infinite spatial lattice. Although the nature of these two classes of models is quite different, we show in this thesis that they may both be discussed in a similar (mathematically rigorous) framework. This approach is inspired by the work of Dpplicher, Haag and Roberts (DHR) in algebraic quantum field theory. The DHR approach leads to braided tensor categories, which describe the charges in the theory (in our case, these are anyons). These categories need not be modular. In this thesis we present a method to remove an obstruction for modularity, in the case of a space-time of dimension three. We then study (a variant of) Kitaev's toric code, regarded as a quantum spin model on an infinite lattice. Using techniques inspired by the DHR programme, we show that one can obtain a modular tensor category describing all relevant properties of the anyons in this system. Finally, some generalisations to non-abelian models are discussed