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

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 $|w(x)|\le \mathrm{const}|x{|}^{-2}$ as $|x|\to \infty$, both in monolayer and bilayer graphene.

54 pages, 4 figures. Version 2: Section 1.0 added; improved results on the decay rate of Wannier functions in graphene (Th. 4.3 and Prop. 4.6). Version 3: final version, to appear in JSP. New in V3: previous Sections 3.1 and 3.2 are now Section 2.2; Lemma 2.4 modified (previous statement was not correct); major modifications to Section 2.3; Assumption 4.1(v) on the Hamiltonian changed

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• Localised Wannier functions in metallic systems
• Spectral analysis near a Dirac type crossing in a weak non-constant magnetic field