Skip to main content
SpringerLink
Log in

General form of DMPK Equation

  • Order, Disorder, And Phase Transition In Condensed System
  • Published:
Journal of Experimental and Theoretical Physics Aims and scope Submit manuscript

Abstract

The Dorokhov–Mello–Pereyra–Kumar (DMPK) equation, using in the analysis of quasi-onedimensional systems and describing evolution of diagonal elements of the many-channel transfer-matrix, is derived under minimal assumptions on the properties of channels. The general equation is of the diffusion type with a tensor character of the diffusion coefficient and finite values of non-diagonal components. We suggest three different forms of the diagonal approximation, one of which reproduces the usual DMPK equation and its generalization suggested by Muttalib and co-workers. Two other variants lead to equations of the same structure, but with different definitions of their parameters. They contain additional terms, which are absent in the first variant. The coefficients of additional terms are shown to be finite beyond the metallic phase by calculation of the Lyapunov exponents and their comparison with numerical experiments. Consequences of the obtained equations for the problem of the conductance distribution and the status of the nonlinear sigma-models are discussed.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. O. N. Dorokhov, JETP Lett. 36, 318 (1982).

    ADS  Google Scholar 

  2. P. A. Mello, P. Pereyra, and N. Kumar, Ann. Phys. (N.Y) 181, 290 (1988).

    Article  ADS  Google Scholar 

  3. P. A. Mello and A. D. Stone, Phys. Rev. 44, 3559 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  4. A. M. S. Macêdo and J. T. Chalker, Phys. Rev. 46, 14985 (1992).

    Article  Google Scholar 

  5. C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).

    Article  ADS  Google Scholar 

  6. M. L. Mehta, Random Matrices (Academic, New York, 1991).

    MATH  Google Scholar 

  7. P. W. Brower and K. Frahm, Phys. Rev. 53, 1490 (1996).

    Article  ADS  Google Scholar 

  8. K. B. Efetov, Adv. Phys. 32, 53 (1983).

    Article  ADS  MathSciNet  Google Scholar 

  9. S. Iida, H. A. Weidenmüller, and M. R. Zirnbauer, Ann. Phys. (N.Y) 200, 219 (1990).

    Article  ADS  Google Scholar 

  10. B. L. Al’tshuler, JETP Lett. 41, 648 (1985).

    ADS  Google Scholar 

  11. P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).

    Article  ADS  Google Scholar 

  12. J. L. Pichard and G. Sarma, J. Phys. C 14, L127 (1981)

    Article  ADS  Google Scholar 

  13. A. MacKinnon and B. Kramer, Phys. Rev. Lett. 47, 1546 (1981).

    Article  ADS  Google Scholar 

  14. J. T. Chalker and M. Bernhardt, Phys. Rev. Lett. 70, 982 (1993).

    Article  ADS  Google Scholar 

  15. I. M. Suslov, J. Exp. Theor. Phys. 124, 763 (2017).

    Article  ADS  Google Scholar 

  16. K. A. Muttalib and J. R. Klauder, Phys. Rev. Lett. 82, 4272 (1999).

    Article  ADS  Google Scholar 

  17. K. A. Muttalib and V. A. Gopar, Phys. Rev. 66, 11538 (2002).

    Article  Google Scholar 

  18. A. Douglas, P. Markoš, and K. A. Muttalib, J. Phys. A 47, 125103 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  19. P. Markoš, Acta Phys. Slov. 56, 561 (2006).

    ADS  Google Scholar 

  20. P. A. Mello and J. L. Pichard, J. Phys. I 1, 493 (1991).

    Google Scholar 

  21. E. N. Economou and C. M. Soukoulis, Phys. Rev. Lett. 46, 618 (1981).

    Article  ADS  Google Scholar 

  22. D. S. Fisher and P. A. Lee, Phys. Rev. 23, 6851 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  23. C. W. J. Beenakker and B. Rejaei, Phys. Rev. Lett. 71, 3689 (1993).

    Article  ADS  Google Scholar 

  24. R. Landauer, IBM J. Res. Dev. 1, 223 (1957); Philos. Mag. 21, 863 (1970).

    Article  MathSciNet  Google Scholar 

  25. V. I. Mel’nikov, Sov. Phys. Solid Stat. 23, 444 (1981).

    Google Scholar 

  26. A. A. Abrikosov, Solid State Commun. 37, 997 (1981).

    Article  ADS  Google Scholar 

  27. N. Kumar, Phys. Rev. 31, 5513 (1985).

    Article  ADS  Google Scholar 

  28. B. Shapiro, Phys. Rev. 34, 4394 (1986).

    Article  ADS  Google Scholar 

  29. P. Mello, Phys. Rev. 35, 1082 (1987).

    Article  ADS  Google Scholar 

  30. B. Shapiro, Philos. Mag. 56, 1031 (1987).

    Article  ADS  Google Scholar 

  31. K. A. Muttalib, P. Markoš, and P. Wölfle, Phys. Rev. 72, 125317 (2005).

    Article  Google Scholar 

  32. K. G. Wilson and J. Kogut, The Renormalisation Group and E-Expansion (Mir, Moscow, 1975); Phys. Rep. 12, 75 (1974).

    Google Scholar 

  33. Sh. Ma, Modern Theory of Critical Phenomena (Mir, Moscow, 1980; Benjamin, Reading, MA, 1976).

    Google Scholar 

  34. I. M. Suslov, J. Exp. Theor. Phys. 115, 897 (2012).

    Article  ADS  Google Scholar 

  35. A. D. Stone and A. Szafer, IBM J. Res. Dev. 32, 384 (1988).

    Article  Google Scholar 

  36. P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, Phys. Rev. 22, 3519 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  37. D. C. Langreth and E. Abrahams, Phys. Rev. 24, 2978 (1981).

    Article  ADS  Google Scholar 

  38. M. Ya. Azbel, J. Phys. C 14, L225 (1981).

    Article  Google Scholar 

  39. M. Buttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. 31, 6207 (1985).

    Article  ADS  Google Scholar 

  40. M. Buttiker, Phys. Rev. Lett. 57, 1761 (1986).

    Article  ADS  Google Scholar 

  41. B. I. Shklovskii, B. Shapiro, B. R. Sears, et al., Phys. Rev. 47, 11487 (1993).

    Article  Google Scholar 

  42. V. I. Oseledets, Tr. Mosk. Mat. Ob-v. 19, 197 (1968).

    Google Scholar 

  43. I. M. Suslov, J. Exp. Theor. Phys. 101, 661 (2005).

    Article  ADS  Google Scholar 

  44. I. M. Suslov, J. Exp. Theor. Phys. 114, 107 (2012).

    Article  ADS  Google Scholar 

  45. J. L. Pichard and G. Andre, Europhys. Lett. 2, 477 (1986).

    Article  ADS  Google Scholar 

  46. P. Markoš, J. Phys.: Condens. Matter 7, 8361 (1995).

    ADS  Google Scholar 

  47. K. Efetov, Supersymmetry in Disorder and Haos (Cambridge Univ. Press, Cambridge, 1995).

    MATH  Google Scholar 

  48. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishman, Phys. Rev. Lett. 42, 673 (1979).

    Article  ADS  Google Scholar 

  49. K. B. Efetov, Sov. Phys. JET. 56, 467 (1982).

    Google Scholar 

  50. V. E. Kravtsov, I. V. Lerner, and V. I. Yudson, Sov. Phys. JET. 67, 1441 (1988).

    Google Scholar 

  51. F. Wegner, Z. Phys. B 35, 207 (1979)

    Article  ADS  Google Scholar 

  52. L. Schäfer and F. Wegner, Z. Phys. B 38, 113 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  53. S. Hikami, Phys. Rev. 24, 2671 (1981).

    Article  ADS  Google Scholar 

  54. K. B. Efetov, A. I. Larkin, and D. E. Khmel’nitskii, Sov. Phys. JET. 52, 568 (1980).

    ADS  Google Scholar 

  55. K. B. Efetov, Sov. Phys. JET. 61, 606 (1985).

    Google Scholar 

  56. B. Mühlschlegel, D. J. Scalapino, and R. Denton, Phys. Rev. 6, 1767 (1972)

    Article  Google Scholar 

  57. G. Deutscher, Y. Imry, and L. Gunter, Phys. Rev. 10, 4598 (1974).

    Article  ADS  Google Scholar 

  58. K. B. Efetov, Sov. Phys. JET. 65, 360 (1987); Sov. Phys. JETP 66, 634 (1987)

    MathSciNet  Google Scholar 

  59. M. R. Zirnbauer, Phys. Rev. 34, 6394 (1986); Nucl. Phys. B 265, 375 (1986).

    Article  ADS  MathSciNet  Google Scholar 

  60. I. M. Suslov, J. Exp. Theor. Phys. 119, 1115 (2014).

    Article  ADS  Google Scholar 

  61. N. N. Bogolyubov and D. V. Shirkov, Introduction to the Theory of Quantized Fields (Nauka, Moscow, 1984; Wiley, New York, 1980)

    MATH  Google Scholar 

  62. E. Brezin, J. C. Le Guillou, and J. Zinn-Justin, in Phase Transitions and Critical Phenomena, Ed. by C. Domb and M. S. Green (Academic, New York, 1976), Vol. 6.

  63. A. Nitzan, K. F. Freed, and M. N. Cohen, Phys. Rev. 15, 4476 (1977)

    Article  ADS  Google Scholar 

  64. M. V. Sadovskii, Sov. Phys. Usp. 24, 96 (1981).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Kapitza Institute for Physical Problems, Moscow, 119334, Russia

    I. M. Suslov

Authors
  1. I. M. Suslov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. M. Suslov.

Additional information

Original Russian Text © I.M. Suslov, 2018, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2018, Vol. 154, No. 1, pp. 152–165.

The article was translated by the authors.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Suslov, I.M. General form of DMPK Equation. J. Exp. Theor. Phys. 127, 131–142 (2018). https://doi.org/10.1134/S1063776118070129

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063776118070129

Search

Navigation