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2007 | 112 | 4 | 635-653
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Anderson Localization in Quantum Chaos: Scaling and Universality

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EN
The one-parameter scaling theory is a powerful tool to investigate Anderson localization effects in disordered systems. In this paper we show that this theory can be adapted to the context of quantum chaos provided that the classical phase space is homogeneous, not mixed. The localization problem in this case is defined in momentum, not in real space. We then employ the one-parameter scaling theory to: (a) propose a precise characterization of the type of classical dynamics related to the Wigner-Dyson and Poisson statistics which also predicts in which situations Anderson localization corrections invalidate the relation between classical chaos and random matrix theory encoded in the Bohigas-Giannoni-Schmit conjecture, (b) to identify the universality class associated with the metal-insulator transition in quantum chaos. In low dimensions it is characterized by classical superdiffusion, in higher dimensions it has in general a quantum origin as in the case of disordered systems. We illustrate these two cases by studying 1d kicked rotors with non-analytical potentials and a 3d kicked rotor with a smooth potential.
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Contributors
  • Physics Department, Princeton University, Princeton, New Jersey 08544, USA
  • The Abdus Salam International Centre for Theoretical Physics, P.O.B. 586, 34100 Trieste, Italy
author
  • Temasek Laboratories, National University of Singapore, 117542 Singapore
  • Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems, National University of Singapore, 117542 Singapore
References
  • 1. O. Bohigas, M.J. Gianonni, C. Schmit, Phys. Rev. Lett., 52, 1, 1984
  • 2. S.W. McDonald, A.N. Kaufman, Phys. Rev. Lett., 42, 1189, 1979
  • 3. G. Casati, F. Valz-Gris, I. Guarneri, Lett. Nuovo Cimento, 28, 279, 1980
  • 4. Chaos in Classical and Quantum Mechanics, Ed. M.C. Gutzwiller, Springer-Verlag, New York 1990; M.V. Berry, M. Tabor, Proc. R. Soc. Lond. A, 356, 375, 1977
  • 5. E.B. Bogomolny, B. Georgeot, M.-J. Giannoni, C. Schmit, Phys. Rev. Lett., 69, 1477, 1992
  • 6. S. Aubry, G. Andr, Ann. Israel Phys. Soc., 3, 133, 1980; M.Ya. Azbel, Zh. Eksp. Teor. Fiz., 46, 929, 1964; D.R. Hofstadter, Phys. Rev. B, 14, 2239, 1976; D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett., 49, 405, 1982
  • 7. G. Casati, B.V. Chirikov, F.M. Izrailev, J. Ford, in: Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Eds. G. Casati, J. Ford, Lecture Notes in Physics, Vol. 93, Springer, Berlin 1979, p. 334; B.V. Chirikov, F.M. Izrailev, D.L. Shepelyansky, Sov. Sci. Rev., 2C, 209, 1981; B.V. Chirikov, Phys. Rep., 52, 265, 1979
  • 8. S. Fishman, D.R. Grempel, R.E. Prange, Phys. Rev. Lett., 49, 509, 1982
  • 9. I.L. Aleiner, A.I. Larkin, Phys. Rev. B, 54, 14 423, 1996; Phys. Rev. E, 55, R1243, 1997
  • 10. A.V. Andreev, B.L. Altshuler, Phys. Rev. Lett., 75, 902, 1995
  • 11. A.V. Andreev, O. Agam, B.D. Simons, B.L. Altshuler, Phys. Rev. Lett., 76, 3947, 1996
  • 12. P.W. Anderson, Phys. Rev., 109, 1492, 1958
  • 13. E. Abrahams, P.W. Anderon, D.C. Licciardello, T.V. Ramakrishnan, Phys. Rev. Lett., 42, 673, 1979
  • 14. M.L. Mehta, Random Matrices, 2nd ed., Academic Press, San Diego 1991
  • 15. J.T. Edwards, D.J. Thouless, J. Phys. C, 5, 807, 1972
  • 16. V.E. Kravtsov, K.A. Muttalib, Phys. Rev. Lett., 79, 1913, 1997; S. Nishigaki, Phys. Rev. E, 59, 2853, 1999
  • 17. B.I. Shklovskii, B. Shapiro, B.R. Sears, P. Lambrianides, H.B. Shore, Phys. Rev. B, 47, 11487, 1993
  • 18. B.L. Altshuler, I.K. Zharekeshev, S.A. Kotochigova, B.I. Shklovskii, JETP, 67, 62, 1988
  • 19. M. Schreiber, H. Grussbach, Phys. Rev. Lett., 67, 607, 1991; H. Aoki, J. Phys. C, 16, L205, 1983
  • 20. F. Evers, A.D. Mirlin, Phys. Rev. Lett., 84, 3690, 2000; E. Cuevas, M. Ortuno, V. Gasparian, A. Pérez-Garrido, Phys. Rev. Lett., 88, 016401, 2002; A.D. Mirlin, Y.V. Fyodorov, F.-M. Dittes, J. Quezada, T.H. Seligman, Phys. Rev. E, 54, 3221, 1996; Y.V. Fyodorov, A.D. Mirlin, Phys. Rev. Lett., 67, 2405, 1991
  • 21. M. Janssen, Int. J. Mod. Phys. B, 8, 943, 1994; B. Huckestein, Rev. Mod. Phys., 67, 357, 1995; E. Cuevas, Phys. Rev. B, 68, 184206, 2003
  • 22. A.D. Mirlin, Phys. Rep., 326, 259, 2000
  • 23. F. Wegner, Z. Phys. B, 36, 209, 1980; F. Wegner, Z. Phys. B, 25, 327, 1976
  • 24. D. Vollhardt, P. Wolfle, Phys. Rev. Lett., 48, 699, 1982; ibid., 45, 842, 1980
  • 25. R. Abou-Chacra, P. Anderson, D.J. Thouless, J. Phys. C, Solid State Phys., 6, 1734, 1973
  • 26. A.M. Garcia-Garcia, J. Wang, Phys. Rev. Lett., 94, 244102, 2005
  • 27. B. Hu, B. Li, J. Liu, Y. Gu, Phys. Rev. Lett., 82, 4224, 1999
  • 28. J. Liu, W.T. Cheng, C.G. Cheng, Comm. Theor. Phys., 33, 15, 2000
  • 29. A.M. Garcia-Garcia, J. Wang, Phys. Rev. E, 73, 036210, 2006
  • 30. A.M. Garcia-Garcia, J. Wang, arXiv:0707.3964
  • 31. E.B. Bogomolny, C. Schmit, Phys. Rev. Lett., 93, 254102, 2004; O. Giraud, J. Marklof, S. O'Keefe, J. Phys. A, 37, L303, 2004
  • 32. B.L. Altshuler, L.S. Levitov, Phys. Rep., 288, 487, 1997
  • 33. D. Wintgen, H. Marxer, Phys. Rev. Lett., 60, 971, 1988
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bwmeta1.element.bwnjournal-article-appv112n406kz
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