Abstract

We will consider recent developments in topological characterisations of free fermion systems. Starting from some simple models to recapitulate some of the basics concepts we will then move on to recently developed classification schemes. This will involve an analysis that profits from an interplay of symmetry and topology, taking into account the role of crystalline symmetries. In particular, we will outline on a qualitative level how this connects to a deeper framework known as equivariant K-theory. Using these concepts we will address recent developments known as topological quantum chemistry and symmetry indicators. Finally, we will turn our attention to new kinds of fragile topologies and multi-gap phases that are emerging at the moment. The overarching idea is to give an intuitive understanding of these new concepts, using explicit examples, that will make it easier to read this rapidly evolving branch of literature.

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