## Topological invariants of eigenvalue intersections and decrease of Wannier functions in graphene    [PDF]

Domenico Monaco, Gianluca Panati
We investigate the asymptotic decrease of the Wannier functions for the valence and conduction band of graphene, both in the monolayer and the multilayer case. Since the decrease of the Wannier functions is characterised by the structure of the Bloch eigenspaces around the Dirac points, we introduce a topological invariant of the family of eigenspaces, baptised eigenspace vorticity. We compare it with the pseudospin winding number. For every value n in Z of the eigenspace vorticity, we exhibit a canonical model for the local topology of the eigenspaces. The set of the canonical models is proved to be universal, in a suitable sense. With the help of this universality theorem, we show that the single band Wannier function w satisfies \$w(x) \asymp |x|^{- 3/2}\$ as |x| tends to infinity, both in monolayer and bilayer graphene.
View original: http://arxiv.org/abs/1306.3904