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Percolation, Renormalization and Quantum Hall Transition

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Summary

In this article, I give a pedagogical introduction and overview of percolation theory. Special emphasis will be put on the review of some of the most prominent algorithms devised to study percolation numerically. The real-space renormalization group treatment of the percolation problem is then discussed. As a rather novel application of this approach to percolation, I will review recent results using similar real-space renormalization ideas that have been applied to the quantum Hall transition.

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References

  1. We note that the WEH-Ferienkurs, following which this article was prepared, took place during early fall.

    Google Scholar 

  2. S.R. Broadbent, J.M. Hammersley: Proc. Camb Philos. Soc. 53, 629 (1957)

    MathSciNet  MATH  Google Scholar 

  3. J.M. Hammersley: Proc. Camb. Philos. Soc. 53, 642 (1957)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. J.M. Hammersley: Ann. Math. Statist. 28, 790 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Stauffer, A. Aharony: Perkolationstheorie (VCH, Weinheim 1995 )

    Google Scholar 

  6. K. Binder: Rep. Prog. Phys. 60, 487 (1997)

    Article  ADS  Google Scholar 

  7. G. Grimmett: Percolation ( Springer, Berlin 1989 )

    MATH  Google Scholar 

  8. A. Bunde, S. Havlin (Eds.): Percolation and Disordered Systems: Theory and Applications ( North-Holland, Amsterdam 1999 )

    Google Scholar 

  9. J. Voit: The Statistical Mechanics of Capital Markets ( Springer, Heidelberg 2001 )

    Google Scholar 

  10. J. Goldenberg, B. Libai, S. Solomon, N. Jan, D. Stauffer: Marketing Percolation (2000). Cond-mat/9905426

    Google Scholar 

  11. http://de.arXiv.org/, 1993–2000

  12. J. Hoshen, R. Kopelman: Phys. Rev. B 14, 3438 (1976)

    Article  ADS  Google Scholar 

  13. C. Adami: (1997), http://www.krl.caltech.edu/“adami/CD1/Percolation/percolation.html, likely to change without prior notice

  14. P. Leath: Phys. Rev. B 14, 5056 (1976)

    Article  ADS  Google Scholar 

  15. W. Kinzel, G. Reents: Physics by Computer ( Springer, Berlin 1998 )

    MATH  Google Scholar 

  16. W. Kinzel, G. Reents: (1999), http://wptxl5.physik.uni-wuerzburg.de/TP3/applet_java/percgr.html, likely to change without prior notice

    Google Scholar 

  17. R.F. Voss: J. Phys. A: Math. Gen. 17 (7), L373 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  18. R.M. Ziff, P.T. Cummings, G. Stell: J. Phys. A Math. Gen. 17, 3009 (1984)

    Article  ADS  Google Scholar 

  19. M. Rosso, J.F. Gouyet, B. Sapoval: Phys. Rev. B 32, 6053 (1985)

    Article  ADS  Google Scholar 

  20. R.M. Ziff, B. Sapoval: J. Phys. A Math. Gen. 19, L1169 (1986)

    Article  ADS  Google Scholar 

  21. P.N. Suding, R.M. Ziff: Phys. Rev. E 60, 275 (1999)

    Article  ADS  Google Scholar 

  22. M.F. Sykes, J.W. Essam: Phys. Rev. Lett. 10, 3 (1963)

    Article  ADS  Google Scholar 

  23. R.M. Ziff, P.N. Suding: J. Phys. A Math. Gen. 30, 5351 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. C.D. Lorenz, R.M. Ziff: Phys. Rev. B 57, 230 (1998)

    ADS  Google Scholar 

  25. P. Kleban, R.M. Ziff: Phys. Rev. B 57, R8075 (1998)

    Article  ADS  Google Scholar 

  26. M.E.J. Newman, R.M. Ziff: Phys. Rev. Lett. 85, 4104 (2000). Cond-mat/0005264

    Google Scholar 

  27. A.B. Harris, T.C. Lubensky, W.K. Holcomb, C. Dasgupta: Phys. Rev. Lett. 35, 327 (1975)

    Article  ADS  Google Scholar 

  28. P.J. Reynolds, W. Klein, H.E. Stanley: J. Phys. C Solid State Phys. 10, L167 (1977)

    Article  ADS  Google Scholar 

  29. J. Bernasconi: Phys. Rev. B 18, 2185 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  30. P.J. Reynolds, H.E. Stanley, W. Klein: Phys. Rev. B 21, 1223 (1980)

    Article  ADS  Google Scholar 

  31. P.D. Eschbach, D. Stauffer, H. Herrmann: Phys. Rev. B 23, 422 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  32. K. von Klitzing, G. Dorda, M. Pepper: Phys. Rev. Lett. 45, 494 (1980)

    Article  ADS  Google Scholar 

  33. http://www.tu-chemnitz.de/physik/HERAEUS/2000/Springer.html

  34. M. Janssen, O. Viehweger, U. Fastenrath, J. Hajdu: Introduction to the Theory of the Integer Quantum Hall effect ( VCH, Weinheim 1994 )

    Google Scholar 

  35. T. Chakraborty, P. Pietilänen: The Quantum Hall Effects ( Springer, Berlin 1995 )

    Book  Google Scholar 

  36. R.B. Laughlin: Phys. Rev. B 23, 5632 (1981)

    Article  ADS  Google Scholar 

  37. D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs: Phys. Rev. Lett. 49, 405 (1982)

    Article  ADS  Google Scholar 

  38. R.E. Prange: Phys. Rev. B 23, 4802 (1981)

    Article  ADS  Google Scholar 

  39. A.M.M. Pruisken: Nucl. Phys. B 235, 277 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  40. D.L. Landau: Z. Phys. 64, 629 (1930)

    Article  ADS  MATH  Google Scholar 

  41. S.V. Iordanskii: Solid State Commun. 43, 1 (1982)

    Article  ADS  Google Scholar 

  42. J.T. Chalker, P.D. Coddington: J. Phys. Condens. Matter 21, 2665 (1988)

    Google Scholar 

  43. D.H. Lee, Z. Wang, S. Kivelson: Phys. Rev. Lett. 70, 4130 (1993)

    Article  ADS  Google Scholar 

  44. S. Koch, R.J. Haug, K. von Klitzing, K. Ploog: Phys. Rev. B 43, 6828 (1991)

    Article  ADS  Google Scholar 

  45. R.T.F. van Schaijk, A. de Visser, S.M. Olsthoorn, H.P. Wei, A.M.M. Pruisken: Phys. Rev. Lett. 84, 1567 (2000)

    Article  ADS  Google Scholar 

  46. A.G. Galstyan, M.E. Raikh: Phys. Rev. B 56, 1422 (1997)

    Article  ADS  Google Scholar 

  47. M. Büttiker, Y. Imry, R. Landauer, S. Pinhas• Phys. Rev. B 31, 6207 (1985)

    Article  Google Scholar 

  48. W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling: Numerical Recipes in FORTRAN, 2nd edn. ( Cambridge University Press, Cambridge 1992 )

    MATH  Google Scholar 

  49. Z. Wang, B. Jovanovic, D.H. Lee: Phys. Rev. Lett. 77, 4426 (1996)

    Article  ADS  Google Scholar 

  50. A. Weymer, M. Janssen: Ann. Phys. (Leipzig) 7, 159 (1998). Cond-mat/9805063

    Google Scholar 

  51. Y. Avishai, Y. Band, D. Brown: Phys. Rev. B 60, 8992 (1999)

    Article  ADS  Google Scholar 

  52. D.H. Cobden, E. Kogan: Phys. Rev. B 54, R17 316 (1996)

    Google Scholar 

  53. P. Cain, R.A. Römer, M.E. Raikh, M. Schreiber: Integer Quantum Hall Transition in the Presence of a Quenched Disorder,submitted to Phys. Rev. B (2001). Condmat/0104045

    Google Scholar 

  54. A. Aho, J.E. Hoperoft, J.D. Ullman: Data Structures and Algorithms ( Addison-Wesley, New York 1983 )

    MATH  Google Scholar 

Download references

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Römer, R.A. (2002). Percolation, Renormalization and Quantum Hall Transition. In: Computational Statistical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04804-7_17

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  • DOI: https://doi.org/10.1007/978-3-662-04804-7_17

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-07571-1

  • Online ISBN: 978-3-662-04804-7

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