The lack of electron localization in graphene is conveniently reformulated in terms of the asymptotic decay of the Wannier functions corresponding to the valence and the conduction band. To quantify the decay of the Wannier functions,we introduce a topological invariant for the family of Bloch eigenspaces, baptized eigenspace vorticity. This invariant characterizes the behavior of such eigenspaces around an eigenvalue intersection. If time permits, a comparison with the pseudospin winding number of the physics literature will be also outlined. For each value $n\in\mathbb Z$ of the eigenspace vorticity, a canonical model for the local topology of the eigenspaces is exhibited, and a suitable universality theorem for these models is stated. This allows us to extract the asymptotic decrease of the Wannier functions for the valence and conduction band of graphene, both in the monolayer and the bilayer case. We show that that the single band Wannier function satisfies, in a suitable weak sense, $w(x)\sim |x|^{-3/2}$ as $|x|\to \infty$. In particular, the expectation value of the modulus of the position operator is infinite, yielding the expected delocalization of the electrons.
The talk is based on a joint work with D. Monaco.
