Journal of Statistical Physics

, Volume 62, Issue 1–2, pp 373–387

Statistics of transfer matrices for disordered quantum thin metallic slabs

  • Pierre Devillard


In the quantum transport problem of a tight-binding Anderson model, the statistics of eigenvalues for the transfer matrices of thin disordered slabs is studied. Numerical simulations indicate that the probability distribution of nearest neighbor eigenvalue spacing and theΔ3 statistics have already become close to that of the Gaussian orthogonal ensemble for sample lengths of the order of the mean free path, provided that transverse localization effects are not important. An intuitive argument is given why this should occur independently of the size of the matrix. Therefore, good mixing of the channels is not essential for obtaining Gaussian orthogonal ensemble type statistics and universal conductance fluctuations.

Key words

Localization random matrices conductance fluctuations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Washburn and R. A. Webb,Adv. Phys. 35:375 (1986), and references therein.Google Scholar
  2. 2.
    A. D. Stone,Phys. Rev. Lett. 54:2692 (1985).Google Scholar
  3. 3.
    P. A. Lee and A. D. Stone,Phys. Rev. Lett. 55:1622 (1985); P. A. Lee, A. D. Stone, and H. Fukuyama,Phys. Rev. B 35:1039 (1987).Google Scholar
  4. 4.
    B. L. Al'tshuler,Pis'ma Zh. Eksp. Teor. Fiz. 41:530 (1985) [JETP Lett. 41:648 (1985)]; B. L. Alt'shuler and D. E. Khmel'nitskii,Pis'ma Zh. Eksp. Teor. Fiz. 42:291 (1985) [JETP Lett. 42:359 (1985)]; B. L. Alt'sshuler and B. I. Shklovskii,Zh. Eksp. Teor. Fiz. 91:220 (1986) [Sov. Phys. JETP 64:127 (1986)].Google Scholar
  5. 5.
    Y. Imry,Europhys. Lett. 1:249 (1986).Google Scholar
  6. 6.
    P. A. Mello,Phys. Rev. Lett. 60:1089 (1988); P. A. Mello, P. Pereyra, and N. Kumar,Ann. Phys. 181:290 (1988).Google Scholar
  7. 7.
    K. A. Muttalib, J.-L. Pichard, and A. Douglas Stone,Phys. Rev. Lett. 59:2475 (1987).Google Scholar
  8. 8.
    P. A. Mello, E. Akkermans, and B. Shapiro,Phys. Rev. Lett. 61:459 (1988).Google Scholar
  9. 9.
    N. Giordano,Phys. Rev. B 38:4746 (1988);36:4190 (1987).Google Scholar
  10. 10.
    P. W. Anderson,Phys. Rev. B 23:4828 (1981).Google Scholar
  11. 11.
    D. S. Fisher and P. A. Lee,Phys. Rev. B 23:6851 (1981); P. A. Lee and D. S. Fisher,Phys. Rev. Lett. 47:882 (1981).Google Scholar
  12. 12.
    M. Büttiker, Y. Imry, R. Landauer, and S. Pinhas,Phys. Rev. Lett. 61:459 (1988).Google Scholar
  13. 13.
    F. J. Dyson,J. Math. Phys. 3:140 (1962).Google Scholar
  14. 14.
    M. L. Mehta,Random Matrices and the Statistical Theory of Energy Levels (Academic Press, New York, 1967).Google Scholar
  15. 15.
    R. Balian,Nuovo Cimento 57:183 (1968).Google Scholar
  16. 16.
    A. Benoit, C. P. Umbach, R. B. Laibowitz, and R. A. Webb,Phys. Rev. Lett. 58:2343 (1987).Google Scholar
  17. 17.
    W. J. Skocpol, P. M. Mankiewich, R. E. Howard, L. D. Jackel, D. M. Tennant, and A. Douglas Stone,Phys. Rev. Lett. 58:2347 (1987).Google Scholar
  18. 18.
    J.-L. Pichard and G. Sarma,J. Phys. C 14:L127, L617 (1981).Google Scholar
  19. 19.
    J.-L. Pichard and G. André,Europhys. Lett. 2:477 (1986).Google Scholar
  20. 20.
    S. Ida, H. A. Weidenmüller, and J. A. Zuk,Phys. Rev. Lett. 64:583 (1990).Google Scholar
  21. 21.
    C. E. Román, T. H. Seligman, and J. J. M. Verbaarschot, T. H. Seligman, and J. J. M. Verbaarschot, inProceedings of the 4th International Conference on Quantum Chaos and the 2nd Colloquium on Statistical Nuclear Physics, T. H. Seligman and H. Nishioka, eds. (Springer, Berlin, 1986), pp. 131 and 256.Google Scholar
  22. 22.
    O. Bohigas and M.-J. Giannoni, inMathematical and Computational Methods in Nuclear Physics, J. Dehesa, J. Gomez, and A. Polls, eds. (Springer-Verlag, Berlin, 1984), p. 1.Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • Pierre Devillard
    • 1
  1. 1.HLRZc/o KFA JülichJülich 1Germany

Personalised recommendations