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Semiclassical theory of nonlocal statistical measures: Residual Coulomb interactions

Denis Ullmo1, Steven Tomsovic1,2,3, and Arnd Bäcker4

  • 1CNRS, Université Paris-Sud, LPTMS UMR 8626, 91405 Orsay Cedex, France
  • 2Max-Planck-Institut für Physik komplexer Systeme, D-01187 Dresden, Germany
  • 3Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA*
  • 4Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany

  • *Permanent address.
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Phys. Rev. E 79, 056217 – Published 15 May, 2009

DOI: https://doi.org/10.1103/PhysRevE.79.056217

Abstract

In a recent paper [Phys. Rev. Lett. 100, 164101 (2008)] and within the context of quantized chaotic billiards, random plane-wave and semiclassical theoretical approaches were applied to an example of a relatively new class of statistical measures, i.e., measures involving both complete spatial integration and energy summation as essential ingredients. A quintessential example comes from the desire to understand the short-range approximation to the first-order ground-state contribution of the residual Coulomb interaction. Billiards, fully chaotic or otherwise, provide an ideal class of systems on which to focus as they have proven to be successful in modeling the single-particle properties of a Landau-Fermi liquid in typical mesoscopic systems, i.e., closed or nearly closed quantum dots. It happens that both theoretical approaches give fully consistent results for measure averages, but that somewhat surprisingly for fully chaotic systems the semiclassical theory gives a much improved approximation for the fluctuations. Comparison of the theories highlights a couple of key shortcomings inherent in the random plane-wave approach. This paper contains a complete account of the theoretical approaches, elucidates the two shortcomings of the oft-relied-upon random plane-wave approach, and treats non-fully-chaotic systems as well.

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References (57)

  1. O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223, 43 (1993).
  2. I. L. Aleiner and A. I. Larkin, Phys. Rev. E 55, R1243 (1997).
  3. A. D. Mirlin, Phys. Rep. 326, 259 (2000).
  4. H. Schomerus and P. Jacquod, J. Phys. A 38, 10663 (2005).
  5. O. Bohigas, M. J. Giannoni, and C. Schmit, Phys. Rev. Lett. 52, 1 (1984).
  6. O. Bohigas, in Chaos and Quantum Physics, edited by M. J. Giannoni, A. Voros, and J. Jinn-Justin (North-Holland, Amsterdam, 1991), pp. 87–199.
  7. M. L. Mehta, Random Matrices, 3rd ed. (Elsevier, Amsterdam, 2004).
  8. M. V. Berry, J. Phys. A 10, 2083 (1977).
  9. A. Voros, in Stochastic Behaviour in Classical and Quantum Hamiltonian Systems, edited by G. Casati and G. Ford (Springer-Verlag, Berlin, 1979), p. 334.
  10. D. Ullmo, T. Nagano, and S. Tomsovic, Phys. Rev. Lett. 90, 176801 (2003).
  11. M. Miller, D. Ullmo, and H. U. Baranger, Phys. Rev. B 72, 045305 (2005).
  12. A. M. Garcia-Garcia, J. D. Urbina, E. A. Yuzbashyan, K. Richter, and B. L. Altshuler, Phys. Rev. Lett. 100, 187001 (2008).
  13. U. Sivan, R. Berkovits, Y. Aloni, O. Prus, A. Auerbach, and G. Ben-Yoseph, Phys. Rev. Lett. 77, 1123 (1996).
  14. F. Simmel, T. Heinzel, and D. A. Wharam, Europhys. Lett. 38, 123 (1997).
  15. S. R. Patel, S. M. Cronenwett, D. R. Stewart, A. G. Huibers, C. M. Marcus, C. I. Duruöz, J. S. Harris, K. Campman, and A. C. Gossard, Phys. Rev. Lett. 80, 4522 (1998).
  16. F. Simmel, D. Abusch-Magder, D. A. Wharam, M. A. Kastner, and J. P. Kotthaus, Phys. Rev. B 59, R10441 (1999).
  17. S. Lüscher, T. Heinzel, K. Ensslin, W. Wegscheider, and M. Bichler, Phys. Rev. Lett. 86, 2118 (2001).
  18. S. Tarucha, D. G. Austing, T. Honda, R. J. van der Hage, and L. P. Kouwenhoven, Phys. Rev. Lett. 77, 3613 (1996).
  19. Y. M. Blanter, A. D. Mirlin, and B. A. Muzykantskii, Phys. Rev. Lett. 78, 2449 (1997).
  20. D. Ullmo and H. U. Baranger, Phys. Rev. B 64, 245324 (2001).
  21. G. Usaj and H. U. Baranger, Phys. Rev. B 64, 201319(R) (2001).
  22. G. Usaj and H. U. Baranger, Phys. Rev. B 66, 155333 (2002).
  23. I. L. Aleiner, P. W. Brouwer, and L. I. Glazman, Phys. Rep. 358, 309 (2002).
  24. D. Ullmo, H. Jiang, W. Yang, and H. U. Baranger, Phys. Rev. B 70, 205309 (2004).
  25. D. Pines and P. Nozières, Theory of Quantum Liquids (Benjamin, New York, 1966), Vol. I.
  26. S. Tomsovic, D. Ullmo, and A. Bäcker, Phys. Rev. Lett. 100, 164101 (2008).
  27. M. C. Gutzwiller, J. Math. Phys. 12, 343 (1971) and references therein.
  28. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990).
  29. A. Bäcker, R. Schubert, and P. Stifter, Phys. Rev. E 57, 5425 (1998); 58, 5192(E) (1998).
  30. D. Ullmo, Rep. Prog. Phys. 71, 026001 (2008).
  31. L. Hörnander, The Analysis of Linear Partial Differential Operators III (Springer, Berlin, 1985), see theorem 17.5.10.
  32. M. S. Longuet-Higgins, J. Mar. Res. 11, 1245 (1952).
  33. E. B. Bogomolny, Physica D 31, 169 (1988).
  34. M. V. Berry, J. Phys. A 35, 3025 (2002).
  35. M. Robnik, J. Phys. A 16, 3971 (1983).
  36. M. Robnik, J. Phys. A 17, 1049 (1984).
  37. A. Bäcker, F. Steiner, and P. Stifter, Phys. Rev. E 52, 2463 (1995).
  38. L. A. Bunimovich, Funct. Anal. Appl. 8, 254 (1974).
  39. L. A. Bunimovich, Commun. Math. Phys. 65, 295 (1979).
  40. S. Tomsovic and E. J. Heller, Phys. Rev. E 47, 282 (1993).
  41. A. Bäcker and R. Schubert, J. Phys. A 35, 539 (2002).
  42. A. M. Ozorio de Almeida, Hamiltonian Systems: Chaos and Quantization (Cambridge University Press, Cambridge, 1988).
  43. J. D. Urbina and K. Richter, Eur. Phys. J. Spec. Top. 145, 255 (2007).
  44. L. Kaplan and Y. Alhassid, Phys. Rev. B 78, 085305 (2008).
  45. N. Argaman, Phys. Rev. B 53, 7035 (1996).
  46. M. V. Berry and M. Tabor, Proc. R. Soc. London, Ser. A 349, 101 (1976).
  47. M. V. Berry and M. Tabor, J. Phys. A 10, 371 (1977).
  48. S. Tomsovic, M. Grinberg, and D. Ullmo, Phys. Rev. Lett. 75, 4346 (1995).
  49. D. Ullmo, M. Grinberg, and S. Tomsovic, Phys. Rev. E 54, 136 (1996).
  50. A. Einstein, Verh. Dtsch. Phys. Ges. 19, 82 (1917); english translation by C. Jaffe, JILA Report No. 116, 1980 (unpublished).
  51. L. Brillouin, C. R. Acad. Sci. 183, 24 (1926).
  52. J. B. Keller, Ann. Phys. (N.Y.) 4, 180 (1958).
  53. M. Sieber, U. Smilansky, S. Creagh, and R. G. Littlejohn, J. Phys. A 26, 6217 (1993).
  54. G. Tanner, J. Phys. A 30, 2863 (1997).
  55. A. Bäcker, R. Schubert, and P. Stifter, J. Phys. A 30, 6783 (1997).
  56. H. Olofsson, S. Äberg, and P. Leboeuf, Phys. Rev. Lett. 100, 037005 (2008).
  57. N. Chernov, J. Stat. Phys. 88, 1 (1997).

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Phys. Rev. E 79, 056217– Published 15 May, 2009
  • Received 30 December 2008

© 2009 American Physical Society

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