The European Physical Journal B

, Volume 67, Issue 1, pp 77–82 | Cite as

Optimisation of multifractal analysis at the 3D Anderson transition using box-size scaling

Computational Methods

Abstract

We study various box-size scaling techniques to obtain the multifractal properties, in terms of the singularity spectrum f(α), of the critical eigenstates at the metal-insulator transition within the 3-D Anderson model of localisation. The typical and ensemble averaged scaling laws of the generalised inverse participation ratios are considered. In pursuit of a numerical optimisation of the box-scaling technique we discuss different box-partitioning schemes including cubic and non-cubic boxes, use of periodic boundary conditions to enlarge the system and single and multiple origins for the partitioning grid are also implemented. We show that the numerically most reliable method is to divide a system of linear size L equally into cubic boxes of size l for which L/l is an integer. This method is the least numerically expensive while having a good reliability.

PACS

71.30.+h Metal-insulator transitions and other electronic transitions 72.15.Rn Localization effects 05.45.Df Fractals 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Janssen, Int. J. Mod. Phys. B 8, 943 (1994)Google Scholar
  2. A.B. Chabra, R.V. Jensen, Phys. Rev. Lett. 62, 1327 (1989)Google Scholar
  3. F. Milde, R.A. Römer, M. Schreiber, Phys. Rev. B 55, 9463 (1997)Google Scholar
  4. P.W. Anderson, Phys. Rev. 109, 1492 (1958)Google Scholar
  5. P.A. Lee, T.V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985)Google Scholar
  6. E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979)Google Scholar
  7. R.A. Römer, M. Schreiber, in The Anderson Transition and its Ramifications – Localisation, Quantum Interference, and Interactions, Lecture Notes in Physics, Chap. Numerical investigations of scaling at the Anderson transition, edited by T. Brandes, S. Kettemann (Springer, Berlin, 2003), Vol. 630, pp. 3–19Google Scholar
  8. M. Schreiber, H. Grussbach, Phys. Rev. Lett. 67, 607 (1991)Google Scholar
  9. H. Grussbach, M. Schreiber, Phys. Rev. B 57, 663 (1995)Google Scholar
  10. A.B. Chhabra, K.R. Sreenivasan, Phys. Rev. A 43, 1114(R) (1991)Google Scholar
  11. M. Yamaguti, C.P.C. Prado, Phys. Rev. E 55, 7726 (1997)Google Scholar
  12. P. Kestener, A. Arneodo, Phys. Rev. Lett. 91, 194501 (2003)Google Scholar
  13. E. Cuevas, Phys. Rev. B 68, 024206 (2003)Google Scholar
  14. L.J. Vasquez, A. Rodriguez, R.A. Römer, Phys. Rev. B 78, 195106 (2008)Google Scholar
  15. A. Rodriguez, L.J. Vasquez, R.A. Römer, Phys. Rev. B 78, 195107 (2008)Google Scholar
  16. F. Evers, A. Mildenberg, A.D. Mirlin, Phys. Stat. Sol. b 245, 284 (2008)Google Scholar
  17. M.A. Lebyodkin, T.A. Lebedkina, Phys. Rev. E 77, 026111 (2008)Google Scholar
  18. M. Morgenstern et al., Phys. Rev. Lett. 89, 136806 (2002), e-print arXiv:cond-mat/0202239 Google Scholar
  19. K. Hashimoto et al., Phys. Rev. Lett. 101, 256802 (2008)Google Scholar
  20. K. Slevin, P. Markos, T. Ohtsuki, Phys. Rev. B 67, 155106 (2003)Google Scholar
  21. K. Slevin, P. Markos, T. Ohtsuki, Phys. Rev. Lett. 86, 3594 (2001)Google Scholar
  22. T. Ohtsuki, K. Slevin, T. Kawarabayashi, Ann. Phys. (Leipzig) 8, 655 (1999), e-print arXiv:cond-mat/9911213 Google Scholar
  23. F. Milde, R.A. Römer, M. Schreiber, V. Uski, Eur. Phys. J. B 15, 685 (2000), e-print arXiv:cond-mat/9911029 Google Scholar
  24. M. Bollhöfer, Y. Notay, Comp. Phys. Comm. 177, 951 (2007)Google Scholar
  25. A.D. Mirlin, F. Evers, Phys. Rev. B 62, 7920 (2000)Google Scholar
  26. A.D. Mirlin, Y.V. Fyodorov, A. Mildenberg, F. Evers, Phys. Rev. Lett. 97, 046803 (2006)Google Scholar
  27. H. Obuse et al., Phys. Rev. Lett. 98, 156802 (2007); H. Obuse, Physica E 40, 1404 (2008)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of Physics and Centre for Scientific ComputingUniversity of WarwickCoventryUK
  2. 2.Departamento de Fisica FundamentalUniversidad de SalamancaSalamancaSpain

Personalised recommendations