Advertisement

Annales Henri Poincaré

, Volume 11, Issue 8, pp 1409–1452 | Cite as

Anomalous Behavior in an Effective Model of Graphene with Coulomb Interactions

  • Alessandro Giuliani
  • Vieri MastropietroEmail author
  • Marcello Porta
Article

Abstract

We analyze by exact Renormalization Group (RG) methods the infrared properties of an effective model of graphene, in which two-dimensional (2D) massless Dirac fermions propagating with a velocity smaller than the speed of light interact with a 3D quantum electromagnetic field. The fermionic correlation functions are written as series in the running coupling constants, with finite coefficients that admit explicit bounds at all orders. The implementation of Ward Identities in the RG scheme implies that the effective charges tend to a line of fixed points. At small momenta, the quasi-particle weight tends to zero and the effective Fermi velocity tends to a finite value. These limits are approached with a power law behavior characterized by non-universal critical exponents.

Keywords

Beta Function Ward Identity Fermi Velocity Feynman Graph Luttinger Liquid 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adler S. and Bardeen W. (1969). Absence of higher-order corrections in the anomalous axial-vector divergence equation. Phys. Rev. 182: 1517–1536 CrossRefADSGoogle Scholar
  2. 2.
    Benfatto G., Falco P. and Mastropietro V. (2010). Universal relations for nonsolvable statistical models. Phys. Rev. Lett. 104: 075701 CrossRefADSGoogle Scholar
  3. 3.
    Benfatto G. and Gallavotti G. (1990). Perturbation theory of the Fermi surface in a quantum liquid. A general quasiparticle formalism and one-dimensional systems. J. Stat. Phys. 59: 541–664 zbMATHCrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Benfatto G. and Gallavotti G. (1995). Renormalization Group. Princeton University Press, NJ zbMATHGoogle Scholar
  5. 5.
    Benfatto G., Gallavotti G., Procacci A. and Scoppola B. (1994). Beta function and Schwinger functions for a many fermions system in one dimension. Anomaly of the fermi surface. Commun. Math. Phys. 160: 93–171 zbMATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Benfatto G. and Mastropietro V. (2005). Ward identities and chiral anomaly in the Luttinger liquid. Commun. Math. Phys. 258: 609–655 zbMATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Benfatto G., Giuliani A. and Mastropietro V. (2006). Fermi liquid behavior in the 2D Hubbard model at low temperatures. Ann. Henri Poincaré 7: 809–898 zbMATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Bonini M., D’Attanasio M. and Marchesini G. (1994). Ward identities and Wilson renormalization group for QED. Nucl. Phys. B 418: 81–112 CrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Bostwick A., Ohta T., Seyller T., Horn K. and Rotenberg E. (2007). Quasiparticle dynamics in graphene. Nature Phys. 3: 36–40 CrossRefADSGoogle Scholar
  10. 10.
    Castro Neto A.H., Guinea F., Peres N., Novoselov K. and Geim K. (2009). The electronic properties of graphene. Rev. Mod. Phys. 81: 109 CrossRefADSGoogle Scholar
  11. 11.
    Disertori M. and Rivasseau V. (2000). Interacting Fermi liquid in two dimensions at finite temperature. Part I: convergent attributions Commun. Math. Phys. 215: 251–290 zbMATHMathSciNetADSGoogle Scholar
  12. 12.
    Disertori M. and Rivasseau V. (2000). Interacting Fermi liquid in two dimensions at finite temperature. Part II: renormalization 215: 291–341 zbMATHMathSciNetGoogle Scholar
  13. 13.
    Feldman J., Knoerrer H. and Trubowitz E. (2004). Commun. Math. Phys. 247: 1–320 zbMATHCrossRefADSGoogle Scholar
  14. 14.
    Feldman J. and Trubowitz E. (1990). Perturbation theory for many fermion systems. Helvetica Phys. Acta 63: 156–260 zbMATHMathSciNetGoogle Scholar
  15. 15.
    Gallavotti G. (1985). Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods. Rev. Mod. Phys. 57: 471–562 CrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Gentile G. and Mastropietro V. (2001). Renormalization group for one-dimensional fermions. A review on mathematical results. Phys. Rep. 352: 273–437 zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Giuliani A. and Mastropietro V. (2004). Anomalous critical exponents in the anisotropic Ashkin–Teller model. Phys. Rev. Lett. 93: 190603 CrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Giuliani A. and Mastropietro V. (2010). The two-dimensional Hubbard model on the honeycomb lattice. Commun. Math. Phys. 293: 301–346 CrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Giuliani, A., Mastropietro, V.: Rigorous construction of ground state correlations in graphene: renormalization of the velocities and Ward identities. Phys. Rev. B 79, 201403(R) (2009)Google Scholar
  20. 20.
    González J., Guinea F. and Vozmediano M.A.H. (1994). Non-Fermi liquid behavior of electrons in the half-filled honeycomb lattice (a renormalization group approach). Nucl. Phys. B 424: 595–618 CrossRefADSGoogle Scholar
  21. 21.
    González J., Guinea F. and Vozmediano M.A.H. (1999). Marginal-Fermi-liquid behavior from two-dimensional Coulomb interaction. Phys. Rev. B 59: R2474 CrossRefADSGoogle Scholar
  22. 22.
    González J., Guinea F. and Vozmediano M.A.H. (2001). Electron–electron interactions in graphene sheets. Phys. Rev. B 63: 134421 CrossRefADSGoogle Scholar
  23. 23.
    Herbut I.F. (2006). Interactions and phase transitions on graphene’s honeycomb lattice. Phys. Rev. Lett. 97: 146401 CrossRefADSGoogle Scholar
  24. 24.
    Herbut I.F., Juricic V. and Roy B. (2009). Theory of interacting electrons on the honeycomb lattice. Phys. Rev. B 79: 085116 CrossRefADSGoogle Scholar
  25. 25.
    Jiang Z., Henriksen E.A., Tung L.C., Wang Y.-J., Schwartz M.E., Han M., Kim P. and Stormer H.L. (2007). Infrared spectroscopy of Landau levels of graphene. Phys. Rev. Lett. 98: 197403 CrossRefADSGoogle Scholar
  26. 26.
    Kotov V.N., Uchoa B. and Castro Neto A.H. (2008). Electron–electron interactions in the vacuum polarization of graphene. Phys. Rev. B 78: 035119 CrossRefADSGoogle Scholar
  27. 27.
    Li G., Luican A. and Andrei E. (2009). Scanning tunneling spectroscopy of graphene on graphite, Phys. Rev. Lett. 102: 176804 CrossRefADSGoogle Scholar
  28. 28.
    Mastropietro V. (2008). Non-Perturbative Renormalization. World Scientific, Singapore zbMATHCrossRefGoogle Scholar
  29. 29.
    Mishchenko E.G. (2007). Effect of electron–electron interactions on the conductivity of clean graphene. Phys. Rev. Lett 98: 216801 CrossRefADSGoogle Scholar
  30. 30.
    Novoselov K.S., Geim A.K., Morozov S.V., Jiang D., Katsnelson M.I., Grigorieva I.V., Dubonos S.V. and Firsov A.A. (2005). Two-dimensional gas of massless Dirac fermions in graphene. Nature 438: 197 CrossRefADSGoogle Scholar
  31. 31.
    Novoselov K.S., Geim A.K., Morozov S.V., Jiang D., Zhang Y., Dubonos S.V., Gregorieva I.V. and Firsov A.A. (2004). Electric field effect in atomically thin carbon films. Science 306: 666 CrossRefADSGoogle Scholar
  32. 32.
    Polchinski J. (1984). Renormalization and effective lagrangians. Nucl. Phys. B 231: 269 CrossRefADSGoogle Scholar
  33. 33.
    Rivasseau V. (1991). From Perturbative to Constructive Renormalization. Princeton University Press, NJ Google Scholar
  34. 34.
    Salmhofer M. (1999). Renormalization: An Introduction, Texts and Monographs in Physics. Springer, Berlin Google Scholar
  35. 35.
    Shankar R. (1994). Renormalization-group approach to interacting fermions. Rev. Mod. Phys. 66: 129 CrossRefMathSciNetADSGoogle Scholar
  36. 36.
    Son D.T. (2007). Quantum critical point in graphene approached in the limit of infinitely strong Coulomb interaction. Phys. Rev. B 75: 235423 CrossRefADSGoogle Scholar
  37. 37.
    Zhou S., Siegel D., Fedorov A. and Lanzara A. (2008). Kohn anomaly and interplay of electron– electron and electron–phonon interactions in epitaxial graphene. Phys. Rev. B 78: 193404 CrossRefADSGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Alessandro Giuliani
    • 1
  • Vieri Mastropietro
    • 2
    Email author
  • Marcello Porta
    • 3
  1. 1.Università di Roma TreRomeItaly
  2. 2.Università di Roma Tor VergataRomeItaly
  3. 3.Università di Roma La SapienzaRomeItaly

Personalised recommendations