Skip to main content
SpringerLink
Log in

Wilson’s numerical renormalization group

  • Introductory Lectures
  • Conference paper
  • First Online:
Density-Matrix Renormalization

Part of the book series: Lecture Notes in Physics ((LNP,volume 528))

  • 856 Accesses

Abstract

The NRG transformation for the Kondo model has developed into a powerful tool for the study of quantum impurity models. It gives information on the many-body eigenvalues and eigenstates of such models on all energy scales and thereby allows the direct calculation of their thermodynamic, dynamic and transport properties.

There is room for further improvement and extensions of the method. The use of a logarithmic discretization of the conduction band, for example, gives rise to insufficient resolution at higher energies. This limits the accuracy with which certain models can be investigated, for examples those arising in the dynamical mean field theory of correlated lattice systems [53]. New approaches [56] for overcoming these difficulties are therefore of interest.

The NRG also has potential to give information on the non-equilibrium transport through correlated impurity systems such as quantum dots [51]. However, away from equilibrium, the absence of a groundstate requires new criteria other than energy for eliminating unimportant states. Ideas based on the DMRG method described in this book are being explored and may prove useful in this context.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. K.G. Wilson Rev. Mod. Phys. 47, 773 (1975)

    Article  ADS  Google Scholar 

  2. H.B. Kirshnamurthy, J.W. Wilkins and K.G. Wilson, Phys. Rev. B 21, 1044 (1980)

    Article  ADS  Google Scholar 

  3. S.-T. Chui and J.W. Bray, Phys. Rev. 18, 2426 (1978)

    Article  ADS  Google Scholar 

  4. J.W. Bray and S.-T. Chui, Phys. Rev. 19, 4876 (1979)

    Article  ADS  Google Scholar 

  5. S.R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev. B 48, 10345 (1993); Phys. Rep. 301, 187 (1998)

    Article  ADS  Google Scholar 

  6. S.R. White and R.M. Noack, Phys. Rev. Lett. 68, 3487 (1992)

    Article  ADS  Google Scholar 

  7. A.C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press (1993) gives an extensive discussion of the Kondo model and related quantum impurity models.

    Google Scholar 

  8. A.J. Leggett, in Percolation, Localization and Superconductivity, NATO ASI Series: Physics Vol. 109, edited by M. Goldman and S. A. Wolf

    Google Scholar 

  9. F. Lesage, H. Saleur and S. Skorik, Phys. Rev. Lett. 76, 3388 (1996)

    Article  ADS  Google Scholar 

  10. R. Konik and A. LeClair, Phys. Rev. B 58, 1872 (1998)

    Article  ADS  Google Scholar 

  11. K.S. Ralls and R.A. Buhrman, Phys. Rev. Lett. 60, 998 (1988)

    Article  Google Scholar 

  12. B. Golding, N. Zimmermann and S. Coppersmith, Phys. Rev. Lett. 68, 998 (1992)

    Article  ADS  Google Scholar 

  13. A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987); 67, 725 (1995)

    Article  ADS  Google Scholar 

  14. U. Weiss, Series in Modern Condensed Matter Physics, Vol. 2, 2nd editon, World Scientific, Singapore (1999)

    Google Scholar 

  15. F. Guinea, V. Hakim and A. Muramatsu, Phys. Rev. B 32 4410, (1985)

    Article  ADS  Google Scholar 

  16. T.A. Costi and G. Zaránd, to appear in Phys. Rev. B 59 (1999)

    Google Scholar 

  17. A.M. Tsvelick and P.B. Wiegmann, Adv. Phys. 32, 453 (1983)

    Article  ADS  Google Scholar 

  18. P.W. Anderson, Phys. Rev. 124, 41 (1961)

    Article  ADS  MathSciNet  Google Scholar 

  19. J.R. Schrieffer and P.A. Wolff, Phys. Rev. 149, 491 (1967)

    Article  ADS  Google Scholar 

  20. P.W. Anderson and G. Yuval, Phys. Rev. Lett. 23, 89 (1969); Phys. Rev. B 1, 4464 (1970)

    Article  ADS  Google Scholar 

  21. P.W. Anderson, J. Phys. C 3, 2439 (1970)

    ADS  Google Scholar 

  22. L.N. Oliveira and J.W. Wilkins, Phys. Rev. 23, 1553 (1981)

    Article  Google Scholar 

  23. B.A. Jones, in Field Theories in Condensed Matter, Ed. Z. Tesanovic, Addison-Wesley, p 87 (1990)

    Google Scholar 

  24. O. Sakai and Y. Shimizu, J. Phys. Soc. Japan, 61, 2333, (1991)

    ADS  Google Scholar 

  25. ibid 61, 2348 (1991)

    ADS  Google Scholar 

  26. D.M. Cragg, P. Lloyd and P. Nozieres, J. Phys. C 13, 803 (1980)

    Article  ADS  Google Scholar 

  27. H.B. Pang and D.L. Cox, Phys. Rev. B 44, 9454 (1991)

    Article  ADS  Google Scholar 

  28. H.B. Pang, Phys. Rev. Lett. 73, 2736 (1994)

    Article  ADS  Google Scholar 

  29. T.A. Costi and A.C. Hewson, Physica C. 185–189, 2649 (1991)

    Article  Google Scholar 

  30. T.A. Costi and A.C. Hewson, J. Magn. Magn. Mat. 108, 129 (1992)

    Article  ADS  Google Scholar 

  31. I.E. Perakis, C.M. Varma and A.E. Ruckenstein, Phys. Rev. Lett. 70, 3467 (1993)

    Article  ADS  Google Scholar 

  32. I.E. Perakis and C.M. Varma, Phys. Rev. 49, 9041 (1994)

    Article  ADS  Google Scholar 

  33. K. Ingersent, B.A. Jones and J.W. Wilkins, Phys. Rev. Lett. 69, 2594 (1992)

    Article  ADS  Google Scholar 

  34. H. Kusunose, K. Miyake, Y. Shimizu and O. Sakai, Phys. Rev. Lett. 76, 271 (1996)

    Article  ADS  Google Scholar 

  35. K. Satori, H. Shiba, O. Sakai and Y. Shimizu, J. Phys. Soc. Japan, 61, 3239 (1992)

    Article  ADS  Google Scholar 

  36. ibid, 62, 3181 (1993)

    Article  ADS  Google Scholar 

  37. K. Chen and C. Jayaprakash, J. Phys. Cond. Matt. 7, 491 (1995)

    Article  ADS  Google Scholar 

  38. K. Ingersent, Phys. Rev. B 54, 11936 (1996); C. Gonzalez-Buxton and K. Ingersent, preprint cond-mat/9803256, (1998)

    Article  ADS  Google Scholar 

  39. R. Bulla, A.C. Hewson and G.M. Zhang, Phys. Rev. B 56, 1171 (1997)

    Article  Google Scholar 

  40. R. Bulla, Th. Pruschke and A.C. Hewson, J. Phys. Cond. Matt. 9, 10463 (1997)

    Article  ADS  Google Scholar 

  41. L.N. Oliveira and J.W. Wilkins, Phys. Rev. B 24, 4863 (1981)

    Article  ADS  Google Scholar 

  42. ibid 32, 696 (1985)

    Article  ADS  Google Scholar 

  43. D.L. Cox, H.O. Frota, L.N. Oliveira and J.W. Wilkins, Phys. Rev. B 32, 555 (1985)

    Article  ADS  Google Scholar 

  44. H.O. Frota and L.N. Oliveira, Phys. Rev. B 43, 7871 (1986)

    Article  ADS  Google Scholar 

  45. O. Sakai, Y. Shimizu and T. Kasuya, J. Phys. Soc. Jap. 58 3666 (1989)

    Article  ADS  Google Scholar 

  46. O. Sakai, Y. Shimizu and T. Kasuya, Physica B 163, (1990)

    Google Scholar 

  47. T.A. Costi and A.C. Hewson, Physica B 163, 179–181 (1990)

    Article  ADS  Google Scholar 

  48. J.J.S. Brito and H.O. Frota, Phys. Rev. B 42, 6378 (1990)

    Article  ADS  Google Scholar 

  49. T.A. Costi and A.C. Hewson, Phil. Mag. B 65, 1165 (1992)

    Article  Google Scholar 

  50. T.A. Costi and C. Kieffer, Phys. Rev. Lett. 76, 1683 (1996)

    Article  ADS  Google Scholar 

  51. T.A. Costi, Phys. Rev. Lett. 80, 1038 (1998)

    Article  ADS  Google Scholar 

  52. T.A. Costi, P. Schmitteckert, J. Kroha and P. Wölfle, Phys. Rev. Lett. 73, 1275 (1994)

    Article  ADS  Google Scholar 

  53. T.A. Costi, P. Schmitteckert, J. Kroha and P. Wölfle, Physica C 235–240, 2287 (1994)

    Article  Google Scholar 

  54. T.A. Costi, J. Kroha and P. Wölfle, Phys. Rev. B 53, 1850 (1996)

    Article  ADS  Google Scholar 

  55. O. Sakai, Y. Shimizu and N. Kaneko, Physica B 186–188, 323, (1993)

    Article  Google Scholar 

  56. Y. Shimizu and O. Sakai, Physica B 186–188, 891 (1993)

    Article  Google Scholar 

  57. T.A. Costi and A.C. Hewson, J. Phys. Cond. Matt. 30, 361 (1993)

    Article  Google Scholar 

  58. T.A. Costi, A.C. Hewson and V. Zlatić J. Phys. Cond. Matt. 6, 2251 (1994)

    Google Scholar 

  59. S. Suzuki, O. Sakai and Y. Shimizu, J. Phys. Soc. Japan 65, 4034 (1996)

    Article  ADS  Google Scholar 

  60. D. Goldhaber-Gordon, J. Göres, M.A. Kastner, H. Shtrikman, D. Mahalu and U. Meirav, Phys. Rev. Lett. 81, 5225 (1998)

    Article  ADS  Google Scholar 

  61. H.O. Frota and K. Flensberg, Phys. Rev. B 46, 15207 (1992)

    Article  ADS  Google Scholar 

  62. W. Izumida, O. Sakai and Y. Shimizu, J. Phys. Soc. Japan 66, 717 (1997); ibid 67, (1998)

    Article  ADS  Google Scholar 

  63. T.A. Costi, Phys. Rev. B55, 3003 (1997)

    Article  ADS  Google Scholar 

  64. R. Bulla, Th. Pruschke, J. Low Temp. Phys. 99, 619 (1995)

    Google Scholar 

  65. W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989)

    Article  ADS  Google Scholar 

  66. A. Georges, G. Kotliar, W. Krauth and M.J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  67. O. Sakai and Y. Shimizu, Sol. St. Commun. 89, 307 (1994)

    Article  ADS  Google Scholar 

  68. R. Bulla and Th. Pruschke and A.C. Hewson, J. Phys. Cond. Matt. 10, 8365 (1998)

    Article  ADS  Google Scholar 

  69. M. Yoshida, M.A. Whitaker and L.N. Oliveira, Phys. Rev. B 41, 9403 (1990)

    Article  ADS  Google Scholar 

  70. W.C. Oliveira and L.N. Oliveira, Phys. Rev. B 49, 11992 (1994)

    ADS  Google Scholar 

  71. S.C. Costa, C.A. Paula, V.L. Libero and L.N. Oliveira, Phys. Rev. B 55, 30 (1997)

    Article  ADS  Google Scholar 

  72. D.C. Langreth, Phys. Rev. 150, 516 (1966)

    Article  ADS  Google Scholar 

  73. K. Völker, Phys. Rev. B 58, 1862 (1998)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretische Physik III, Universität Augsburg, D-86135, Augsburg, Germany

    Theo Costi

Authors
  1. Theo Costi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Ingo Peschel Matthias Kaulke Xiaoqun Wang Karen Hallberg

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag

About this paper

Cite this paper

Costi, T. (1999). Wilson’s numerical renormalization group. In: Peschel, I., Kaulke, M., Wang, X., Hallberg, K. (eds) Density-Matrix Renormalization. Lecture Notes in Physics, vol 528. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0106063

Download citation

  • DOI: https://doi.org/10.1007/BFb0106063

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66129-0

  • Online ISBN: 978-3-540-48750-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation