Topological invariants of eigenvalue intersections and decrease of Wannier functions in graphene

TL;DR

This paper studies how the decay of Wannier functions in graphene relates to topological invariants of eigenspaces, introducing a new invariant called eigenspace vorticity and analyzing its effects on function decay.

Contribution

It introduces the eigenspace vorticity as a new topological invariant and links it to the decay properties of Wannier functions in graphene.

Findings

Wannier functions decay as |x|^{-2} at infinity in graphene.

Eigenspace vorticity characterizes local topology of eigenspaces.

Canonical models are constructed for each vorticity value.

Abstract

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 geometric 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. With the help of these canonical models, we show that the single band Wannier function $w$ satisfies $|w(x)| \leq \mathrm{const} |x|^{- 2}$ as $|x| \rightarrow \infty$, both in monolayer and bilayer graphene.