Manifestations of topological effects in graphene

Graphene is a monoatomic layer of graphite with carbon atoms arranged in a two-dimensional honeycomb lattice configuration. It has been known for more than 60 years that the electronic structure of graphene can be modelled by two-dimensional massless relativistic fermions. This property gives rise to numerous applications, both in applied sciences and in theoretical physics. Electronic circuits made out of graphene could take advantage of its high electron mobility that is witnessed even at room temperature. In the theoretical domain the Dirac-like behaviour of graphene can simulate high energy effects, such as the relativistic Klein paradox. Even more surprisingly, topological effects can be encoded in graphene such as the generation of vortices, charge fractionalisation and the emergence of anyons. The impact of the topological effects on graphene's electronic properties can be elegantly described by the Atiyah-Singer index theorem. Here we present a pedagogical encounter of this theorem and review its various applications to graphene. A direct consequence of the index theorem is charge fractionalisation that is usually known from the fractional quantum Hall effect. The charge fractionalisation gives rise to the exciting possibility of realising graphene based anyons that unlike bosons or fermions exhibit fractional statistics. Besides being of theoretical interest, anyons are a strong candidate for performing error free quantum information processing.