A Gentle Introduction to the Functional Renormalization Group: The Kondo Effect in Quantum Dots

  • Sabine Andergassen
  • Tilman Enss
  • Christoph Karrasch
  • Volker Meden
Part of the NATO Science for Peace and Security Series book series (NAPSB)


The functional renormalization group provides an efficient description of the interplay and competition of correlations on different energy scales in interacting Fermi systems. An exact hierarchy of flow equations yields the gradual evolution from a microscopic model Hamiltonian to the effective action as a function of a continuously decreasing energy cutoff. Practical implementations rely on suitable truncations of the hierarchy, which capture nonuniversal properties at higher energy scales in addition to the universal low-energy asymptotics. As a specific example we study transport properties through a single-level quantum dot coupled to Fermi liquid leads. In particular, we focus on the temperature T = 0 gate voltage dependence of the linear conductance. A comparison with exact results shows that the functional renormalization group approach captures the broad resonance plateau as well as the emergence of the Kondo scale. It can be easily extended to more complex setups of quantum dots.


Vertex Function Luttinger Liquid Kondo Effect Matsubara Frequency Gate Voltage Versus 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A.C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cam-bridge, UK, 1993)CrossRefGoogle Scholar
  2. 2.
    A.M. Tsvelik, P.B. Wiegmann, Adv. Phys. 32, 453 (1983)CrossRefADSGoogle Scholar
  3. 3.
    L. Glazman, M. Raikh, JETP Lett. 47, 452 (1988)ADSGoogle Scholar
  4. 4.
    T.K. Ng, P.A. Lee, Phys. Rev. Lett. 61, 1768 (1988)CrossRefADSGoogle Scholar
  5. 5.
    T.A. Costi, A.C. Hewson, V. Zlati ć , J. Phys.: Condens. Matter 6, 2519 (1994)CrossRefADSGoogle Scholar
  6. 6.
    U. Gerland, J. von Delft, T.A. Costi, Y. Oreg, Phys. Rev. Lett. 84, 3710 (2000)CrossRefADSGoogle Scholar
  7. 7.
    Y. Meir, N.S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992)CrossRefADSGoogle Scholar
  8. 8.
    D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, M.A. Kastner, Nature 391, 156 (1998)CrossRefADSGoogle Scholar
  9. 9.
    W.G. van der Wiel, S.D. Franceschi, T. Fujisawa, J.M. Elzerman, S. Tarucha, L.P. Kouwenhoven, Science 289, 2105 (2000)CrossRefADSGoogle Scholar
  10. 10.
    K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975)CrossRefADSGoogle Scholar
  11. 11.
    H.R. Krishna-murthy, J.W. Wilkins, K.G. Wilson, Phys. Rev. B 21, 1044 (1980)CrossRefADSGoogle Scholar
  12. 12.
    W. Hofstetter, J. K önig, H. Schoeller, Phys. Rev. Lett. 87, 156803 (2001)Google Scholar
  13. 13.
    M. Salmhofer, C. Honerkamp, Prog. Theor. Phys. 105, 1 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    J.W. Negele, H. Orland, Quantum Many-Particle Systems (Addison-Wesley, Reading, MA, 1987)Google Scholar
  15. 15. T. Enss, Renormalization, conservation laws and transport in correlated electron systems, Ph.D. thesis, University of Stuttgart, Germany (2005), arXiv:cond-mat/0504703, URL
  16. 16.
    W. Metzner, AIP Conf. Proc. 846, 130 (2006)Google Scholar
  17. 17.
    F.J. Wegner, A. Houghton, Phys. Rev. A 8, 401 (1973)CrossRefADSGoogle Scholar
  18. 18.
    J. Polchinski, Nucl. Phys. B 231, 269 (1984)CrossRefADSGoogle Scholar
  19. 19.
    C. Wetterich, Phys. Lett. B 301, 90 (1993)CrossRefADSGoogle Scholar
  20. 20.
    R. Hedden, V. Meden, Th. Pruschke, K. Sch önhammer, J. Phys.: Condens. Matter 16, 5279 (2004)CrossRefADSGoogle Scholar
  21. 21.
    T.R. Morris, Int. J. Mod. Phys. A 9, 2411 (1994)zbMATHCrossRefADSGoogle Scholar
  22. 22.
    C. Karrasch, T. Enss, V. Meden, Phys. Rev. B 73, 235337 (2006)Google Scholar
  23. 23.
    T.A. Costi, Phys. Rev. B 64, 241310(R) (2001)CrossRefADSGoogle Scholar
  24. 24.
    V. Meden, F. Marquardt, Phys. Rev. Lett. 96, 146801 (2006)Google Scholar
  25. 25.
    C. Karrasch, T. Hecht, Y. Oreg, J. von Delft, V. Meden, Phys. Rev. Lett. 98, 186802 (2007)Google Scholar
  26. 26.
    V. Meden, W. Metzner, U. Schollw öck, K. Sch önhammer, Phys. Rev. B 65, 045318 (2002)Google Scholar
  27. 27.
    S. Andergassen, T. Enss, V. Meden, W. Metzner, U. Schollw öck, K. Sch önhammer, Phys. Rev. B 70, 075102 (2004)Google Scholar
  28. 28.
    T. Enss, V. Meden, S. Andergassen, X. Barnab é -Th ériault, W. Metzner, K. Schönhammer, Phys. Rev. B 71, 155401 (2005)Google Scholar
  29. 29.
    S. Andergassen, T. Enss, V. Meden, W. Metzner, U. Schollw öck, K. Sch önhammer, Phys. Rev. B 73, 045125 (2006)Google Scholar
  30. 30.
    X. Barnab é -Th ériault, A. Sedeki, V. Meden, K. Sch önhammer, Phys. Rev. Lett. 94, 136405 (2005)Google Scholar
  31. 31.
    S. Andergassen, T. Enss, V. Meden, Phys. Rev. B 73, 153308 (2006)Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  • Sabine Andergassen
    • 1
    • 2
  • Tilman Enss
    • 3
  • Christoph Karrasch
    • 4
  • Volker Meden
    • 4
  1. 1.Max-Planck-Institut für FestkörperforschungStuttgartGermany
  2. 2.Institut Néel / CNRS - UJFGrenobleFrance
  3. 3.INFM-SMC-CNR and Dipt. di FisicaUniversitá di Roma “La Sapienza”RomaItaly
  4. 4.Institut für Theoretische PhysikUniversität GöttingenGöttingenGermany

Personalised recommendations