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


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.


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



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


  1. 1.
    Agrachev, A.A.: Space of symmetric operators with multiple ground states, (Russian). Funktsional. Anal. i Prilozhen. 45(4), 1–15 (2011). Translation in Funct. Anal. Appl.45 (2011), no. 4, 241–251.CrossRefMathSciNetGoogle Scholar
  2. 2.
    Amrein, W.O.: Hilbert Space Methods in Quantum Mechanics. EPFL Press, Lausanne (2009)zbMATHGoogle Scholar
  3. 3.
    Bellissard, J., Schulz-Baldes, H., van Elst, A.: The non commutative geometry of the quantum Hall effect. J. Math. Phys. 35, 5373–5471 (1994)CrossRefzbMATHMathSciNetADSGoogle Scholar
  4. 4.
    Bena, C., Montambaux, G.: Remarks on the tight-binding model of graphene. New J. Phys. 11, 095003 (2009)CrossRefADSGoogle Scholar
  5. 5.
    Brouder, Ch., Panati, G., Calandra, M., Mourougane, Ch., Marzari, N.: Exponential localization of Wannier functions in insulators. Phys. Rev. Lett. 98, 046402 (2007)CrossRefADSGoogle Scholar
  6. 6.
    Castro Neto, A.H., Guinea, F., Peres, N.M.R., Novoselov, K.S., Geim, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009)CrossRefADSGoogle Scholar
  7. 7.
    des Cloizeaux, J.: Energy bands and projection operators in a crystal: analytic and asymptotic properties. Phys. Rev. 135, A685–A697 (1964)CrossRefADSGoogle Scholar
  8. 8.
    des Cloizeaux, J.: Analytical properties of \(n\)-dimensional energy bands and Wannier functions. Phys. Rev. 135, A698–A707 (1964)CrossRefADSGoogle Scholar
  9. 9.
    Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Modern Geometry—Methods and Applications. Part II: The Geometry and Topology of Manifolds. No. 93 in Graduate Texts in Mathematics. Springer, New York (1985)Google Scholar
  10. 10.
    Fermanian Kammerer, C., Lasser, C.: Wigner measures and codimension two crossings. J. Math. Phys. 44, 507–527 (2003)CrossRefzbMATHMathSciNetADSGoogle Scholar
  11. 11.
    Fefferman, C.L.; Weinstein, M.I.: Waves in Honeycomb Structures, preprint arXiv:1212.6684 (2012)
  12. 12.
    Goerbig, M.O.: Electronic properties of graphene in a strong magnetic field, preprint arXiv:1004.3396v4 (2011)
  13. 13.
    Graf, G.M.: Aspects of the integer quantum Hall effect. Proc. Symp. Pure Math. 76, 429–442 (2007)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Hagedorn, G.A.: Classification and normal forms for quantum mechanical eigenvalue crossings. Astérisque 210, 115–134 (1992)MathSciNetGoogle Scholar
  15. 15.
    Hagedorn, G.A.: Classification and normal forms for avoided crossings of quantum mechanical energy levels. J. Phys. A 31, 369–383 (1998)CrossRefzbMATHMathSciNetADSGoogle Scholar
  16. 16.
    Hainzl, C., Lewin, M., Sparber, C.: Ground state properties of graphene in Hartree–Fock theory. J. Math. Phys. 53, 095220 (2012)CrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Haldane, F.D.M.: Model for a auantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2017 (1988)MathSciNetADSGoogle Scholar
  18. 18.
    Hasan, M.Z., Kane, C.L.: Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010)CrossRefADSGoogle Scholar
  19. 19.
    Kane, C.L., Mele, E.J.: \({\mathbb{Z}}_2\) topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005)CrossRefADSGoogle Scholar
  20. 20.
    Kato, T.: Perturbation theory for linear operators. Springer, Berlin (1966)CrossRefzbMATHGoogle Scholar
  21. 21.
    King-Smith, R.D., Vanderbilt, D.: Theory of polarization of crystalline solids. Phys. Rev. B 47, 1651–1654 (1993)CrossRefADSGoogle Scholar
  22. 22.
    Kohn, W.: Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115, 809 (1959)CrossRefzbMATHMathSciNetADSGoogle Scholar
  23. 23.
    Kuchment, P.: Floquet Theory for Partial Differential Equations. Operator Theory: Advances and Applications. Birkhäuser, Basel (1993)CrossRefGoogle Scholar
  24. 24.
    Lepori, L., Mussardo, G., Trombettoni, A.: \((3+1)\) Massive Dirac Fermions with Ultracold Atoms in Optical Lattices. EPL (Europhysics Letters) 92, 50003 (2010)CrossRefADSGoogle Scholar
  25. 25.
    Luke, Y.L.: Integrals of Bessel Functions. McGraw-Hill, New York (1962)zbMATHGoogle Scholar
  26. 26.
    Mc Cann, E., Falko, V.I.: Landau-level degeneracy and quantum Hall effect in a graphite bilayer. Phys. Rev. Lett. 96, 86805 (2006)CrossRefADSGoogle Scholar
  27. 27.
    Marzari, N., Mostofi, A.A., Yates, J.R., Souza, I., Vanderbilt, D.: Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 84, 1419 (2012)CrossRefADSGoogle Scholar
  28. 28.
    Marzari, N., Vanderbilt, D.: Maximally localized generalized Wannier functions for composite energy bands. Phys. Rev. B 56, 12847–12865 (1997)CrossRefADSGoogle Scholar
  29. 29.
    Milnor, J.W., Stasheff, J.D.: Characteristic Classes. No. 76 in Annals of Mathematical Studies. Princeton Univesity Press, Princeton (1974)Google Scholar
  30. 30.
    Min, H., MacDonald, A.H.: Chiral decomposition in the electronic structure of graphene multilayers. Phys. Rev. B 77, 155416 (2008)CrossRefADSGoogle Scholar
  31. 31.
    Nenciu, G.: Existence of the exponentially localized Wannier functions. Commun. Math. Phys. 91, 81–85 (1983)CrossRefzbMATHMathSciNetADSGoogle Scholar
  32. 32.
    Nenciu, G.: Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians. Rev. Mod. Phys. 63, 91–127 (1991)CrossRefADSGoogle Scholar
  33. 33.
    Novoselov, K.S., McCann, E., Morozov, S.V., Falko, V.I., Katsnelson, M.I., Geim, A.K., Schedin, F., Jiang, D.: Unconventional quantum Hall effect and Berry’s phase of \(2 \pi \) in bilayer graphene. Nat. Phys. 2, 177 (2006)CrossRefGoogle Scholar
  34. 34.
    Panati, G., Pisante, A.: Bloch bundles, Marzari–Vanderbilt functional and maximally localized Wannier functions. Commun. Math. Phys. 322(3), 835–875 (2013)CrossRefzbMATHMathSciNetADSGoogle Scholar
  35. 35.
    Panati, G., Sparber, C., Teufel, S.: Geometric currents in piezoelectricity. Arch. Ration. Mech. Anal. 91, 387–422 (2009)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Panati, G., Spohn, H., Teufel, S.: Effective dynamics for Bloch electrons: Peierls substitution and beyond. Commun. Math. Phys. 242, 547–578 (2003)CrossRefzbMATHMathSciNetADSGoogle Scholar
  37. 37.
    Panati, G.: Triviality of Bloch and Bloch–Dirac bundles. Ann. Henri Poincaré 8, 995–1011 (2007)CrossRefzbMATHMathSciNetADSGoogle Scholar
  38. 38.
    Park, C.-H., Marzari, N.: Berry phase and pseudospin winding number in bilayer graphene. Phys. Rev. B 84, 1–5 (2011)Google Scholar
  39. 39.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. I: Functional Analysis (revised and enlarged edition). Academic Press, New York (1980)Google Scholar
  40. 40.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. IV. Analysis of Operators. Academic Press, New York (1978)Google Scholar
  41. 41.
    Resta, R.: Theory of the electric polarization in crystals. Ferroelectrics 136, 51–75 (1992)CrossRefGoogle Scholar
  42. 42.
    Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. No. 3 in De Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter, Berlin (1996)Google Scholar
  43. 43.
    Soluyanov, A.A., Vanderbilt, D.: Wannier representation of \({\mathbb{Z}}_2\) topological insulators. Phys. Rev. B 85, 115415 (2012)CrossRefADSGoogle Scholar
  44. 44.
    Steenrod, N.: The Topology of Fibre Bundles. Princeton University Press, Princeton (1960)Google Scholar
  45. 45.
    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)zbMATHGoogle Scholar
  46. 46.
    Tarruel, L., Greif, D., Uehlinger, Th, Jotzu, G., Esslinger, T.: Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice. Nature 483, 302–305 (2012)CrossRefADSGoogle Scholar
  47. 47.
    Thouless, D.J., Kohmoto, M., Nightingale, M.P., de Nijs, M.: Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982)CrossRefADSGoogle Scholar
  48. 48.
    Voisin, C.: Hodge Theory and Complex Algebraic Geometry I. No. 76 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  49. 49.
    von Neumann, J., Wigner, E.: On the behaviour of eigenvalues in adiabatic processes, Phys. Z. 30:467 (1929). Republished in: Hettema, H. (ed.) Quantum Chemistry: Classic Scientific Papers. World Scientific Series in 20th Century Chemistry. World Scientific, Singapore (2000)Google Scholar
  50. 50.
    Wallace, P.R.: The band theory of graphite. Phys. Rev. 71, 622–634 (1947)CrossRefzbMATHADSGoogle Scholar
  51. 51.
    Wannier, G.H.: The structure of electronic excitation levels in insulating crystals. Phys. Rev. 52, 191–197 (1937)CrossRefADSGoogle Scholar
  52. 52.
    Xiao, D., Chang, M.-C., Niu, Q.: Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010)CrossRefzbMATHMathSciNetADSGoogle Scholar
  53. 53.
    Zhu, S.L., Wang, B., Duan, L.M.: Simulation and detection of Dirac fermions with cold atoms in an optical lattice. Phys. Rev. Lett. 98, 260402 (2007)CrossRefADSGoogle Scholar

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© 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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