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Journal of Statistical Physics

, Volume 155, Issue 6, pp 1027–1071 | Cite as

Topological Invariants of Eigenvalue Intersections and Decrease of Wannier Functions in Graphene

  • Domenico Monaco
  • Gianluca PanatiEmail author
Article

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 \(n \in \mathbb {Z}\) 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 \(|w(x)| \le {\mathrm {const}} \cdot |x|^{-2}\) as \(|x| \rightarrow \infty \), both in monolayer and bilayer graphene.

Keywords

Wannier functions Bloch bundles Conical intersections Eigenspace vorticity Pseudospin winding number Graphene 

Notes

Acknowledgments

We are indebted with D. Fiorenza and A. Pisante for many inspiring discussions, and with R. Bianco, R. Resta and A. Trombettoni for interesting comments and remarks. We are also grateful to the anonymous reviewers for their useful observations and suggestions. Financial support from the INdAM-GNFM project “Giovane Ricercatore 2011”, and from the AST Project 2009 “Wannier functions” is gratefully acknowledged

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.SISSATriesteItaly
  2. 2.Dipartimento di Matematica “G. Castelnuovo”“La Sapienza” Università di RomaRomeItaly

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