Abstract

When studying gapped topological phases, there are two broad approaches that exist in the literature. One may either study macroscopic anyon data by looking at their Braided Tensor Category (BTC). Alternatively, one may investigate microscopic models like the Toric Code that demonstrate topological phenomena. A natural question is to ask if it is possible to extract anyon data from the microscopic lattice models and organise them into the nice form of a BTC . Kitaev’s Quantum Double Models (QDM) are a celebrated class of lattice models that demonstrate important topological phenomena like anyons. We answer the above question in the context of QDM , proving a longstanding conjecture in our setting and concluding that Kitaev was right (yet again…)

Our approach is the following. We start with Kitaev’s Quantum Double Model (QDM) in the infinite volume. Using a selection criterion, we pick out ‘anyon sectors’ that roughly describe global anyon types. We work out that there are finitely many global anyon types, each uniquely labelled by the irreducible representations of $D(G)$, the Quantum Double algebra. We then show these anyon sectors form a Braided Tensor Category. Moreover, this category is braided monoidal equivalent to $Rep(D(G))$, the category of representations of $D(G)$. Thus it is possible to extract all the macroscopic anyon data from the QDM , including F/R symbols, by studying the much simpler category, $Rep(D(G))$, whose anyon data is already known in the literature.

This talk is based on a joint work with Alex Bols (arXiv:2310.19661) and its sequel with Alex Bols, Mahdie Hamdan, Pieter Naaijkens (work in progress).