The European Physical Journal Special Topics

, Volume 180, Issue 1, pp 217–235 | Cite as

New theoretical approaches for correlated systems in nonequilibrium

  • M. Eckstein
  • A. Hackl
  • S. Kehrein
  • M. Kollar
  • M. Moeckel
  • P. Werner
  • F.A. Wolf
Review

Abstract.

We review recent developments in the theory of interacting quantum many-particle systems that are not in equilibrium. We focus mainly on the nonequilibrium generalizations of the flow equation approach and of dynamical mean-field theory (DMFT). In the nonequilibrium flow equation approach one first diagonalizes the Hamiltonian iteratively, performs the time evolution in this diagonal basis, and then transforms back to the original basis, thereby avoiding a direct perturbation expansion with errors that grow linearly in time. In nonequilibrium DMFT, on the other hand, the Hubbard model can be mapped onto a time-dependent self-consistent single-site problem. We discuss results from the flow equation approach for nonlinear transport in the Kondo model, and further applications of this method to the relaxation behavior in the ferromagnetic Kondo model and the Hubbard model after an interaction quench. For the interaction quench in the Hubbard model, we have also obtained numerical DMFT results using quantum Monte Carlo simulations. In agreement with the flow equation approach they show that for weak coupling the system relaxes to a “prethermalized” intermediate state instead of rapid thermalization. We discuss the description of nonthermal steady states with generalized Gibbs ensembles.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975)CrossRefADSGoogle Scholar
  2. 2.
    W. Metzner, D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989)CrossRefADSGoogle Scholar
  3. 3.
    A. Georges, G. Kotliar, W. Krauth, M.J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996)CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    M. Greiner, O. Mandel, T.W. Hänsch, I. Bloch, Nature 419, 51 (2002)CrossRefADSGoogle Scholar
  5. 5.
    S. Iwai, M. Ono, A. Maeda, H. Matsuzaki, H. Kishida, H. Okamoto, Y. Tokura, Phys. Rev. Lett. 91, 057401 (2003)CrossRefADSGoogle Scholar
  6. 6.
    L. Perfetti, P.A. Loukakos, M. Lisowski, U. Bovensiepen, H. Berger, S. Biermann, P.S. Cornaglia, A. Georges, M. Wolf, Phys. Rev. Lett. 97, 067402 (2006)CrossRefADSGoogle Scholar
  7. 7.
    L. Perfetti, P.A. Loukakos, M. Lisowski, U. Bovensiepen, M. Wolf, H. Berger, S. Biermann, A. Georges, New J. Phys. 10, 053019 (2008)CrossRefADSGoogle Scholar
  8. 8.
    Y. Kawakami, S. Iwai, T. Fukatsu, M. Miura, N. Yoneyama, T. Sasaki, N. Kobayashi, Phys. Rev. Lett. 103, 066403 (2009)CrossRefADSGoogle Scholar
  9. 9.
    S. Wall, D. Brida, S.R. Clark, H.P. Ehrke, D. Jaksch, A. Ardavan, S. Bonora, H. Uemura, Y. Takahashi, T. Hasegawa, H. Okamoto, G. Cerullo, A. Cavalleri [arXiv:0910.3808]Google Scholar
  10. 10.
    D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, M.A. Kastner, Nature 391, 156 (1998)CrossRefADSGoogle Scholar
  11. 11.
    S.M. Cronenwett, T.H. Oosterkamp, L.P. Kouwenhoven, Science 281, 540 (1998)CrossRefADSGoogle Scholar
  12. 12.
    J. Schmid, J. Weis, K. Eberl, K.V. Klitzing, Physica B: Cond. Matt. 256-258, 182(1998)CrossRefADSGoogle Scholar
  13. 13.
    W.G. van der Wiel, W. De Franceschi, T. Fujisawa, J.M. Elzerman, S. Tarucha, L.P. Kouwenhoven, Science 289, 2105 (2000)CrossRefADSGoogle Scholar
  14. 14.
    F. Wegner, Ann. Phys. (Leipzig) 3, 77 (1994)MATHADSGoogle Scholar
  15. 15.
    S. Kehrein, The Flow Equation Approach to Many-Particle Systems (Springer Verlag, 2006)Google Scholar
  16. 16.
    P. Schmidt, H. Monien [ arXiv:cond-mat/0202046]Google Scholar
  17. 17.
    J.K. Freericks, V.M. Turkowski, V. Zlatić, Phys. Rev. Lett. 97, 266408 (2006)CrossRefADSGoogle Scholar
  18. 18.
    A. Hackl, D. Roosen, S. Kehrein, W. Hofstetter, Phys. Rev. Lett. 102, 196601 (2009)CrossRefADSGoogle Scholar
  19. 19.
    S.D. Głazek, K.G. Wilson, Phys. Rev. D 48, 5863 (1993)CrossRefADSGoogle Scholar
  20. 20.
    S.D. Głazek, K.G. Wilson, Phys. Rev. D 49, 4214 (1994)CrossRefADSGoogle Scholar
  21. 21.
    A. Hackl, S. Kehrein, Phys. Rev. B 78, 092303 (2008)CrossRefADSGoogle Scholar
  22. 22.
    H. Goldstein, Ch. P. Poole, J.L. Safko, Classical Mechanics, 3rd edn. (Addison-Wesley, 2002), p. 526Google Scholar
  23. 23.
    A. Hackl, S. Kehrein, J. Phys. C 21, 015601 (2009)Google Scholar
  24. 24.
    P. Anderson, J. Phys. C 3, 2436 (1970)CrossRefADSGoogle Scholar
  25. 25.
    A. Abrikosov, A.A. Migdal, J. Low Temp. Phys. 3, 519 (1970)CrossRefADSGoogle Scholar
  26. 26.
    A. Hackl, S. Kehrein, M. Vojta, Phys. Rev. B 80, 195117 (2009)CrossRefADSGoogle Scholar
  27. 27.
    M. Moeckel, S. Kehrein, Ann. Phys. 324, 2146 (2009)MATHCrossRefADSGoogle Scholar
  28. 28.
    J. Sabio, S. Kehrein, in New J. Phys. (to appear) [arXiv:0911.1302]Google Scholar
  29. 29.
    J. Berges, Sz. Borsanyi, C. Wetterich, Phys. Rev. Lett. 93, 142002 (2004)CrossRefADSGoogle Scholar
  30. 30.
    M. Moeckel, S. Kehrein, Phys. Rev. Lett. 100, 175702 (2008)CrossRefADSGoogle Scholar
  31. 31.
    M. Moeckel, Real-time Evolution of Quenched Quantum Systems, PhD thesis (LMU München, 2009)Google Scholar
  32. 32.
    G.S. Uhrig, Phys. Rev. A 80, 061602(R) (2009)CrossRefADSGoogle Scholar
  33. 33.
    M. Moeckel, S. Kehrein, New J. Phys. (to appear) [arXiv:0911.0875]Google Scholar
  34. 34.
    A. Rosch, J. Kroha, P. Wölfle, Phys. Rev. Lett. 87, 156802 (2001)CrossRefADSGoogle Scholar
  35. 35.
    A. Rosch, J. Paaske, P. Wölfle, Phys. Rev. Lett. 90, 076804 (2003)CrossRefADSGoogle Scholar
  36. 36.
    A. Rosch, J. Paaske, J. Kroha, P. Wölfle, J. Phys. Soc. Jpn. 74, 118 (2005)MATHCrossRefADSGoogle Scholar
  37. 37.
    H. Schoeller, Lect. Notes Phys. 544, 137 (2000)CrossRefADSGoogle Scholar
  38. 38.
    H. Schoeller, F. Reininghaus, Phys. Rev. B 80, 045117 (2009)CrossRefADSGoogle Scholar
  39. 39.
    H. Schoeller, F. Reininghaus, Phys. Rev. B 80, 209901(E) (2009)ADSGoogle Scholar
  40. 40.
    S. Kehrein, Phys. Rev. Lett. 95, 056602 (2005)CrossRefADSGoogle Scholar
  41. 41.
    P. Fritsch, S. Kehrein, Ann. Phys. 324, 1105 (2009)MATHCrossRefADSGoogle Scholar
  42. 42.
    P. Fritsch, S. Kehrein, Phys. Rev. B 81, 035113 (2010)CrossRefADSGoogle Scholar
  43. 43.
    J.K. Freericks, Phys. Rev. B 77, 075109 (2008)CrossRefADSGoogle Scholar
  44. 44.
    A.V. Joura, J.K. Freericks, T. Pruschke, Phys. Rev. Lett. 101, 196401 (2008)CrossRefADSGoogle Scholar
  45. 45.
    N. Tsuji, T. Oka, H. Aoki, Phys. Rev. B 78, 235124 (2008)CrossRefADSGoogle Scholar
  46. 46.
    N. Tsuji, T. Oka, H. Aoki, Phys. Rev. Lett. 103, 047403 (2009)CrossRefADSGoogle Scholar
  47. 47.
    M. Eckstein, M. Kollar, Phys. Rev. Lett. 100, 120404 (2008)CrossRefADSGoogle Scholar
  48. 48.
    M. Eckstein, M. Kollar, in New J. Phys. (to appear) [arXiv:0911.1282]Google Scholar
  49. 49.
    M. Eckstein, M. Kollar, P. Werner, Phys. Rev. Lett. 103, 056403 (2009)CrossRefADSGoogle Scholar
  50. 50.
    J.K. Freericks, H. R. Krishnamurthy, T. Pruschke, Phys. Rev. Lett. 102, 136401 (2009)CrossRefADSGoogle Scholar
  51. 51.
    M. Eckstein, M. Kollar, Phys. Rev. B 78, 245113 (2008)CrossRefADSGoogle Scholar
  52. 52.
    M. Eckstein, M. Kollar, Phys. Rev. B 78, 205119 (2008)CrossRefADSGoogle Scholar
  53. 53.
    M. Eckstein, M. Kollar, P. Werner, Phys. Rev. B 81, 115131 (2010)CrossRefADSGoogle Scholar
  54. 54.
    For an introduction into the Keldysh formalism, seeN. E. van Leeuwen, R. Dahlen, G. Stefanucci, C.-O. Almbladh, U. von Barth, in Time-Dependent Density Functional Theory, edited byM.A.L.Marques, C.A.Ullrich, F.Nogueira, A.Rubio, K.Burke, E.K.U.Gross, Lecture Notes in Physics706 (Springer, Berlin, 2006) [arXiv:cond-mat/0506130]Google Scholar
  55. 55.
    P. Werner, T. Oka, A.J. Millis, Phys. Rev. B 79, 035320 (2009)CrossRefADSGoogle Scholar
  56. 56.
    U. Brandt, C. Mielsch, Z. Phys. B: Condens. Matter 75, 365 (1989)CrossRefADSGoogle Scholar
  57. 57.
    E.N. Economou, Green’s Functions in Quantum Physics (Springer-Verlag, Berlin, 1979)Google Scholar
  58. 58.
    G.D. Mahan, Phys. Rev. B 63, 155110 (2001)CrossRefADSGoogle Scholar
  59. 59.
    M. Eckstein, M. Kollar, K. Byczuk, D. Vollhardt, Phys. Rev. B 71, 235119 (2005)CrossRefADSGoogle Scholar
  60. 60.
    M. Kollar, M. Eckstein, K. Byczuk, N. Blümer, P. van Dongen, M.H.R. de Cuba, W. Metzner, D. Tanasković, V. Dobrosavljević, G. Kotliar, D. Vollhardt, Ann. Phys. 14 (2005)Google Scholar
  61. 61.
    N. Blümer, P.G.J. van Dongen, Concepts in Electron Correlation, edited byA.C.Hewson, V.Zlatić (NATO Science Series, Kluwer, 2003)Google Scholar
  62. 62.
    A. Weisse, G. Wellein, A. Alvermann, H. Fehske, Rev. Mod. Phys. 78, 275 (2006)CrossRefMathSciNetADSGoogle Scholar
  63. 63.
    S.P. Heims, Am. J. Phys. 33, 722 (1965)CrossRefADSGoogle Scholar
  64. 64.
    M.D. Girardeau, Phys. Lett. A 30, 442 (1969)CrossRefADSGoogle Scholar
  65. 65.
    K. Sengupta, S. Powell, S. Sachdev, Phys. Rev. A 69, 053616 (2004)CrossRefADSGoogle Scholar
  66. 66.
    M. Rigol, A. Muramatsu, M. Olshanii, Phys. Rev. A 74, 053616 (2006)CrossRefADSGoogle Scholar
  67. 67.
    M.A. Cazalilla, Phys. Rev. Lett. 97, 156403 (2006)CrossRefADSGoogle Scholar
  68. 68.
    M. Rigol, V. Dunjko, V. Yurovsky, M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007)CrossRefADSGoogle Scholar
  69. 69.
    C. Kollath, A. Läuchli, E. Altman, Phys. Rev. Lett. 98, 180601 (2007)CrossRefADSGoogle Scholar
  70. 70.
    S.R. Manmana, S. Wessel, R.M. Noack, A. Muramatsu, Phys. Rev. Lett. 98, 210405 (2007)CrossRefADSGoogle Scholar
  71. 71.
    M. Kollar, M. Eckstein, Phys. Rev. A 78, 013626 (2008)CrossRefADSGoogle Scholar
  72. 72.
    M. Rigol, V. Dunjko, M. Olshanii, Nature 452, 854 (2008)CrossRefADSGoogle Scholar
  73. 73.
    D. Rossini, A. Silva, G. Mussardo, G.E. Santoro, Phys. Rev. Lett. 102, 127204 (2009)CrossRefADSGoogle Scholar
  74. 74.
    P. Barmettler, M. Punk, V. Gritsev, E. Demler, E. Altman, Phys. Rev. Lett. 102, 130603 (2009)CrossRefADSGoogle Scholar
  75. 75.
    J. Dziarmaga [arXiv:0912.4034]Google Scholar
  76. 76.
    F. Gebhard, A.E. Ruckenstein, Phys. Rev. Lett. 68, 244 (1992)MATHCrossRefMathSciNetADSGoogle Scholar
  77. 77.
    F. Gebhard, A. Girndt, A.E. Ruckenstein, Phys. Rev. B 49, 10926 (1994)CrossRefADSGoogle Scholar
  78. 78.
    T. Kinoshita, T. Wenger, D.S. Weiss, Nature 440, 900 (2006)CrossRefADSGoogle Scholar
  79. 79.
    R. Balian, From Microphysics to Macrophysics: Methods and Applications of Statistical Physics, Vol. 1 (Springer, Berlin, 1991)Google Scholar
  80. 80.
    M. Kollar, F.A. Wolf, M. Eckstein (to be published)Google Scholar

Copyright information

© EDP Sciences and Springer 2010

Authors and Affiliations

  • M. Eckstein
    • 1
  • A. Hackl
    • 2
  • S. Kehrein
    • 3
  • M. Kollar
    • 4
  • M. Moeckel
    • 5
  • P. Werner
    • 1
  • F.A. Wolf
    • 4
  1. 1.Institute of Theoretical Physics, ETH ZürichZürichSwitzerland
  2. 2.Institute of Theoretical Physics, Universität zu KölnKölnGermany
  3. 3.Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Physics DepartmentLudwig-Maximilians-Universität MünchenMünchenGermany
  4. 4.Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of PhysicsUniversität AugsburgAugsburgGermany
  5. 5.Max Planck Institute of Quantum OpticsGarchingGermany

Personalised recommendations