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Onset of chaos and relaxation in isolated systems of interacting spins: Energy shell approach

L. F. Santos*

F. Borgonovi†

F. M. Izrailev‡

  • Department of Physics, Yeshiva University, 245 Lexington Avenue, New York, New York 10016, USA

  • Dipartimento di Matematica e Fisica and Interdisciplinary Laboratories for Advanced Materials Physics, Universitá Cattolica, via Musei 41, 25121 Brescia, Italy and Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Italy

  • Instituto de Física, Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla, Puebla 72570, Mexico and National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824-1321, USA

  • *lsantos2@yu.edu
  • †fausto.borgonovi@unicatt.it
  • ‡felix.izrailev@gmail.com
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Phys. Rev. E 85, 036209 – Published 20 March, 2012

DOI: https://doi.org/10.1103/PhysRevE.85.036209

Abstract

We study the onset of chaos and statistical relaxation in two isolated dynamical quantum systems of interacting spins 1/2, one of which is integrable and the other chaotic. Our approach to identifying the emergence of chaos is based on the level of delocalization of the eigenstates with respect to the energy shell, the latter being determined by the interaction strength between particles or quasiparticles. We also discuss how the onset of chaos may be anticipated by a careful analysis of the Hamiltonian matrices, even before diagonalization. We find that despite differences between the two models, their relaxation processes following a quench are very similar and can be described analytically with a theory previously developed for systems with two-body random interactions. Our results imply that global features of statistical relaxation depend on the degree of spread of the eigenstates within the energy shell and may happen to both integrable and nonintegrable systems.

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References (72)

  1. J. M. Deutsch, Phys. Rev. A 43, 2046 (1991); M. Srednicki, Phys. Rev. E 50, 888 (1994); M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) 452, 854 (2008).
  2. M. Horoi, V. Zelevinsky, and B. A. Brown, Phys. Rev. Lett. 74, 5194 (1995); N. Frazier, B. A. Brown, and V. Zelevinsky, Phys. Rev. C 54, 1665 (1996).
  3. V. Zelevinsky, B. A. Brown, N. Frazier, and M. Horoi, Phys. Rep. 276, 85 (1996).
  4. V. V. Flambaum, F. M. Izrailev, and G. Casati, Phys. Rev. E 54, 2136 (1996); V. V. Flambaum and F. M. Izrailev, ibid. 55, R13 (1997).
  5. V. V. Flambaum and F. M. Izrailev, Phys. Rev. E 56, 5144 (1997).
  6. F. Borgonovi, I. Guarneri, F. M. Izrailev, and G. Casati, Phys. Lett. A 247, 140 (1998); F. Borgonovi and F. M. Izrailev, Phys. Rev. E 62, 6475 (2000).
  7. F. M. Izrailev, Phys. Scr. T90, 95 (2001).
  8. M. Rigol, Phys. Rev. Lett. 103, 100403 (2009); Phys. Rev. A 80, 053607 (2009).
  9. M. Rigol and M. Fitzpatrick, Phys. Rev. A 84, 033640 (2011).
  10. L. F. Santos and M. Rigol, Phys. Rev. E 81, 036206 (2010).
  11. L. F. Santos and M. Rigol, Phys. Rev. E 82, 031130 (2010); M. Rigol and L. F. Santos, Phys. Rev. A 82, 011604(R) (2010).
  12. C. Kollath, A. M. Läuchli, and E. Altman, Phys. Rev. Lett. 98, 180601 (2007); S. R. Manmana, S. Wessel, R. M. Noack, and A. Muramatsu, ibid. 98, 210405 (2007); G. Roux, Phys. Rev. A 79, 021608R (2009); C. Neuenhahn and F. Marquardt, e-print arXiv:1007.5306; A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011); M. C. Bañuls, J. I. Cirac, and M. B. Hastings, Phys. Rev. Lett. 106, 050405 (2011); L. F. Santos, A. Polkovnikov, and M. Rigol, ibid. 107, 040601 (2011); V. K. B. Kota, A. Relao, J. Retamosa, and M. Vyas, J. Stat. Mech. (2011) P10028. E. Canovi, D. Rossini, R. Fazio, G. E. Santoro, and A. Silva, Phys. Rev. B 83, 094431 (2011); G. P. Brandino, A. De Luca, R. M. Konik, G. Mussardo, e-print arXiv:1111.6119.
  13. M. Greiner, O. Mandel, T. W. Hansch, and I. Bloch, Nature (London) 419, 51 (2002); T. Kinoshita, T. Wenger, and D. S. Weiss, ibid. 440, 900 (2006); S. Hofferberth et al., ibid. 449, 324 (2007); I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008); S. Trotzky, Y. A. Chen, A. Flesch, I. P. McCulloch, U. Schollwöck, J. Eisert, and I. Bloch, Nature Physics (2012), doi: .10.1038/nphys2232
  14. B. V. Chirikov, Open Sys. Information Dyn. 4, 241 (1997).
  15. L. Casetti, M. Cerruti-Sola, M. Pettini, and E. G. D. Cohen, Phys. Rev. E 55, 6566 (1997).
  16. B. V. Chirikov, Phys. Rep. 52, 263 (1979).
  17. A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983).
  18. G. Casati and B.V. Chirikov, Quantum Chaos: Between Order and Disorder (Cambridge University Press, Cambridge, 1995).
  19. L. E. Reichl, The Transition to Chaos: Conservative Classical Systems and Quantum Manifestations (Springer, New York, 2004).
  20. H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, 1999).
  21. G. Casati, B. V. Chirikov, I. Guarneri, and F. M. Izrailev, Phys. Rev. E 48, R1613 (1993).
  22. G. Casati, B. V. Chirikov, I. Guarneri, and F. M. Izrailev, Phys. Lett. A 223, 430 (1996).
  23. F. Borgonovi, I. Guarneri, and F. M. Izrailev, Phys. Rev. E 57, 5291 (1998); G. A. Luna-Acosta, J. A. Méndes-Bermúdez, and F. M. Izrailev, Phys. Lett. A 274, 192 (2000); L. Benet et al., ibid. 277, 87 (2000); , Phys. Rev. E 64, 036206 (2001); , Physica E 12, 267 (2002); , J. Phys. A 36, 1289 (2003).
  24. B. V. Chirikov, Phys. Lett. A 108, 68 (1985).
  25. V. V. Flambaum and F. M. Izrailev, Phys. Rev. E 61, 2539 (2000); 64, 026124 (2001); F. M. Izrailev and A. Castañeda-Mendoza, Phys. Lett. A 350, 355 (2006).
  26. V. K. B. Kota and R. Sahu, Phys. Lett. B 429, 1 (1998).
  27. V. K. B. Kota, Phys. Rep. 347, 223 (2001); D. Angom and V. K. B. Kota, Phys. Rev. A 71, 042504 (2005); Phys. Rep. 499, 103 (2011).
  28. V. V. Flambaum and F. M. Izrailev, Phys. Rev. E 64, 036220 (2001).
  29. N. Gershenfeld and I. L. Chuang, Science 275, 350 (1997).
  30. D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 (1998).
  31. B. Kane, Nature (London) 393, 133 (1998).
  32. R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Science 327, 177 (2010).
  33. X. Zotos, F. Naef, and P. Prelovsek, Phys. Rev. B 55, 11029 (1997).
  34. A. V. Sologubenko, K. Giannò, H. R. Ott, U. Ammerahl, and A. Revcolevschi, Phys. Rev. Lett. 84, 2714 (2000).
  35. F. Heidrich-Meisner, A. Honecker, D. C. Cabra, and W. Brenig, Phys. Rev. Lett. 92, 069703 (2004).
  36. X. Zotos, J. Phys. Soc. Jpn. Suppl. 74, 173 (2005).
  37. N. Hlubek, P. Ribeiro, R. Saint-Martin, A. Revcolevschi, G. Roth, G. Behr, B. Büchner, and C. Hess, Phys. Rev. B 81, 020405(R) (2010).
  38. J. Sirker, R. G. Pereira, and I. Affleck, Phys. Rev. B 83, 035115 (2011).
  39. L. F. Santos and A. Mitra, Phys. Rev. E 84, 016206 (2011).
  40. L. M. Duan, E. Demler, and M. D. Lukin, Phys. Rev. Lett. 91, 090402 (2003).
  41. S. Trotzky, P. Cheinet, S. Fölling, M. Feld, U. Schnorrberger, A. M. Rey, A. Polkovnikov, E. A. Demler, M. D. Lukin, and I. Bloch, Science 319, 295 (2008).
  42. J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner, Nature (London) 472, 307 (2011).
  43. Y.-A. Chen, S. Nascimbène, M. Aidelsburger, M. Atala, S. Trotzky, and I. Bloch, Phys. Rev. Lett. 107, 210405 (2011).
  44. T. C. Hsu and J. C. Anglès d’Auriac, Phys. Rev. B 47, 14291 (1993).
  45. Y. Avishai, J. Richert, and R. Berkovits, Phys. Rev. B 66, 052416 (2002).
  46. L. F. Santos, J. Phys. A 37, 4723 (2004).
  47. D. A. Rabson, B. N. Narozhny, and A. J. Millis, Phys. Rev. B 69, 054403 (2004).
  48. K. Kudo and T. Deguchi, J. Phys. Soc. Jpn. 74, 1992 (2005).
  49. F. Dukesz, M. Zilbergerts, and L. F. Santos, New J. Phys. 11, 043026 (2009).
  50. P. Jordan and E. Wigner, Z. Phys. 47, 631 (1928).
  51. T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940).
  52. H. A. Bethe, Z. Phys. 71, 205 (1931).
  53. F. C. Alcaraz, M. N. Barber, M. T. Batchelor, R. J. Baxter, and G. R. W. Quispel, J. Phys. A 20, 6397 (1987).
  54. M. Karbach and G. Müller, Comput. Phys. 11, 36 (1997).
  55. W. G. Brown, L. F. Santos, D. J. Starling, and L. Viola, Phys. Rev. E 77, 021106 (2008).
  56. G. P. Berman and F. M. Izrailev, Chaos 15, 015104 (2005).
  57. M. L. Mehta, Random Matrices (Academic, Boston, 1991).
  58. F. Haake, Quantum Signatures of Chaos (Springer-Verlag, Berlin, 1991).
  59. T. Guhr, A. Mueller-Gröeling, and H. A. Weidenmüller, Phys. Rep. 299, 189 (1998).
  60. J. B. French and S. S. M. Wong, Phys. Lett. B 33, 449 (1970).
  61. T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, Rev. Mod. Phys. 53, 385 (1981).
  62. There are in fact two transitions. The system is integrable in both extremes, when the NNN couplings are nonexistent (λ=0) and also when the NN couplings become negligible (λ→∞), but this second limit is unrealistic.
  63. V. V. Flambaum and F. M. Izrailev, Phys. Rev. E 61, 2539 (2000).
  64. V. V. Flambaum and F. M. Izrailev, Phys. Rev. E 64, 026124 (2001).
  65. F. M. Izrailev, Phys. Rep. 196, 299 (1990).
  66. Formulas for delocalization measures for Gaussian ensembles with few-body interactions were obtained in Ref. [26].
  67. J. M. Deutsch, Phys. Rev. A 43, 2046 (1991).
  68. M. Srednicki, Phys. Rev. E 50, 888 (1994).
  69. V. V. Flambaum, Aust. J. Phys. 53, 489 (2000).
  70. B. L. Altshuler, Y. Gefen, A. Kamenev, and L. S. Levitov, Phys. Rev. Lett. 78, 2803 (1997).
  71. G. P. Berman, F. Borgonovi, F. M. Izrailev, and A. Smerzi, Phys. Rev. Lett. 92, 030404 (2004).
  72. V. Balachandran, G. Benenti, G. Casati, and J. Gong, Phys. Rev. E 82, 046216 (2010).

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Phys. Rev. E 85, 036209– Published 20 March, 2012
  • Received 30 December 2011
Article part of CHORUS

©2012 American Physical Society

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