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Universality in chaotic quantum transport: The concordance between random-matrix and semiclassical theories

Gregory Berkolaiko1 and Jack Kuipers2

  • 1Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368, USA
  • 2Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany

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Phys. Rev. E 85, 045201(R) – Published 26 April, 2012

DOI: https://doi.org/10.1103/PhysRevE.85.045201

Abstract

Electronic transport through chaotic quantum dots exhibits universal, system-independent properties, consistent with random-matrix theory. The quantum transport can also be rooted, via the semiclassical approximation, in sums over the classical scattering trajectories. Correlations between such trajectories can be organized diagrammatically and have been shown to yield universal answers for some observables. Here, we develop the general combinatorial treatment of the semiclassical diagrams, through a connection to factorizations of permutations. We show agreement between the semiclassical and random matrix approaches to the moments of the transmission eigenvalues. The result is valid for all moments to all orders of the expansion in inverse channel number for all three main symmetry classes (with and without time-reversal symmetry and spin-orbit interaction) and extends to nonlinear statistics. This finally explains the applicability of random-matrix theory to chaotic quantum transport in terms of the underlying dynamics as well as providing semiclassical access to the probability density of the transmission eigenvalues.

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References (49)

  1. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).
  2. F. Haake, Quantum Signatures of Chaos, 3rd ed. (Springer, Berlin, 2010).
  3. M. C. Gutzwiller, J. Math. Phys. 12, 343 (1971).
  4. R. Balian and C. Bloch, Ann. Phys. (NY) 85, 514 (1974).
  5. J. H. Hannay and A. M. Ozorio de Almeida, J. Phys. A 17, 3429 (1984).
  6. M. V. Berry, Proc. R. Soc. A 400, 229 (1985).
  7. O. Bohigas, M. J. Giannoni, and C. Schmit, Phys. Rev. Lett. 52, 1 (1984).
  8. M. L. Mehta, Random Matrices, 3rd ed. (Elsevier, Amsterdam, 2004).
  9. M. Sieber and K. Richter, Phys. Scr. T90, 128 (2001).
  10. S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, Phys. Rev. Lett. 93, 014103 (2004); Phys. Rev. E 72, 046207 (2005).
  11. S. Heusler, S. Müller, A. Altland, P. Braun, and F. Haake, Phys. Rev. Lett. 98, 044103 (2007).
  12. J. P. Keating and S. Müller, Proc. R. Soc. A 463, 3241 (2007).
  13. S. Müller, S. Heusler, A. Altland, P. Braun, and F. Haake, New J. Phys. 11, 103025 (2009).
  14. A. M. Chang, H. U. Baranger, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 73, 2111 (1994).
  15. R. Landauer, IBM J. Res. Dev. 1, 223 (1957); 33, 306 (1988).
  16. M. Büttiker, Phys. Rev. Lett. 57, 1761 (1986).
  17. R. Blümel and U. Smilansky, Phys. Rev. Lett. 60, 477 (1988); 64, 241 (1990).
  18. K. Richter and M. Sieber, Phys. Rev. Lett. 89, 206801 (2002).
  19. S. Heusler, S. Müller, P. Braun, and F. Haake, Phys. Rev. Lett. 96, 066804 (2006).
  20. S. Müller, S. Heusler, P. Braun, and F. Haake, New J. Phys. 9, 12 (2007).
  21. G. Berkolaiko, J. M. Harrison, and M. Novaes, J. Phys. A 41, 365102 (2008).
  22. G. Berkolaiko and J. Kuipers, New J. Phys. 13, 063020 (2011).
  23. R. S. Whitney and Ph. Jacquod, Phys. Rev. Lett. 96, 206804 (2006); Ph. Jacquod and R. S. Whitney, Phys. Rev. B 73, 195115 (2006).
  24. P. W. Brouwer and S. Rahav, Phys. Rev. B 74, 075322 (2006); 74, 085313 (2006).
  25. D. Waltner and J. Kuipers, Phys. Rev. E 82, 066205 (2010).
  26. D. Waltner, J. Kuipers, and K. Richter, Phys. Rev. B 83, 195315 (2011).
  27. C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).
  28. H. U. Baranger and P. A. Mello, Phys. Rev. Lett. 73, 142 (1994).
  29. R. A. Jalabert, J. Pichard, and C. W. J. Beenakker, Europhys. Lett. 27, 255 (1994).
  30. P. W. Brouwer and C. W. J. Beenakker, J. Math. Phys. 37, 4904 (1996).
  31. D. V. Savin and H.-J. Sommers, Phys. Rev. B 73, 081307(R) (2006).
  32. D. V. Savin, H.-J. Sommers, and W. Wieczorek, Phys. Rev. B 77, 125332 (2008).
  33. P. Vivo and E. Vivo, J. Phys. A 41, 122004 (2008).
  34. M. Novaes, Phys. Rev. B 78, 035337 (2008).
  35. G. Livan and P. Vivo, Acta Phys. Pol. B 42, 1081 (2011).
  36. F. Mezzadri and N. J. Simm, J. Math. Phys. 52, 103511 (2011); e-print arXiv:1108.2859.
  37. W. H. Miller, Adv. Chem. Phys. 30, 77 (1975).
  38. K. Richter, Semiclassical Theory of Mesoscopic Quantum Systems (Springer, Berlin, 2000).
  39. M. Novaes, e-print arXiv:1111.5179.
  40. S. Samuel, J. Math. Phys. 21, 2695 (1980).
  41. P. A. Mello, J. Phys. A 23, 4061 (1990).
  42. S. Matsumoto and J. Novak, Discrete Math. Theor. Comput. Sci. (FPSAC 2010), 403 (2010).
  43. O. Zaitsev, D. Frustaglia, and K. Richter, Phys. Rev. Lett. 94, 026809 (2005); Phys. Rev. B 72, 155325 (2005).
  44. J. Bolte and D. Waltner, Phys. Rev. B 76, 075330 (2007); D. Waltner (private communication).
  45. V. A. Osipov and E. Kanzieper, Phys. Rev. Lett. 101, 176804 (2008); J. Phys. A 42, 475101 (2009).
  46. B. A. Khoruzhenko, D. V. Savin, and H.-J. Sommers, Phys. Rev. B 80, 125301 (2009).
  47. J. Kuipers, D. Waltner, C. Petitjean, G. Berkolaiko, and K. Richter, Phys. Rev. Lett. 104, 027001 (2010); J. Kuipers, T. Engl, G. Berkolaiko, C. Petitjean, D. Waltner, and K. Richter, Phys. Rev. B 83, 195316 (2011).
  48. J. A. Melsen, P. W. Brouwer, K. M. Frahm, and C. W. J. Beenakker, Europhys. Lett. 35, 7 (1996); Phys. Scr. T69, 223 (1997).
  49. G. Berkolaiko and J. Kuipers, J. Phys. A 43, 035101 (2010).

Outline

Information

Phys. Rev. E 85, 045201(R)– Published 26 April, 2012
  • Received 21 November 2011
Article part of CHORUS

©2012 American Physical Society

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