Current Density Functional Theory and Orbital Magnetism

Dedicated to Mark Rasolt
  • G. Vignale
Part of the NATO ASI Series book series (NSSB, volume 337)


The basic motivations and ideas underlying the formulation of the current and spin density functional theory for electronic systems in a magnetic field are reviewed. We emphasize the advantages of using the canonical current as a basic variable, the fundamental constraints imposed by gauge symmetry on the structure of the exchange-correlation energy functional, and the role played by this symmetry in enforcing the conservation laws for the particle density and spin density. Next, we focus on the local density approximation to the exact theory. A generalized Thomas-Fermi theory for density and current is derived. Using this theory, a universal relation between density and current distributions of electronic systems, valid in the limit of large magnetic field, is obtained. The usefulness of this relation is demonstrated in some exactly solvable examples of electrons in parabolic confining potentials. Finally, we present a selection of results from some recent applications of the current-density functional theory: the surface structure of an electron-hole droplet, the current and density distributions of harmonically confined systems, and the ground-state energy of the two-dimensional Wigner crystal.


Magnetic Field Random Phase Approximation Lower Landau Level Magnetic Length Large Magnetic Field 


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  1. 1.
    L. D. Landau and E. M. Lifshitz, “Quantum Mechanics,” Chapter XV.Google Scholar
  2. 2.
    L. D. Landau and E. M. Lifshitz, “Statistical Physics II,” Chapt. 63 (Pergamon Press, 1980); L. D. Landau and E. M. Lifshitz, “Physical Kinetics,” Chapt. 90 (Pergamon Press 1981).Google Scholar
  3. 3.
    B. L. Altshuler and A. G. Aronov, “Electron-electron interactions in disordered systems,” A. L. Efros and M. Pollak, ed., Chapter 1 (North Holland, 1985).Google Scholar
  4. 4.
    V. Celli and N. D. Mermin, Phys. Rev. 140:A839 (1965).ADSCrossRefGoogle Scholar
  5. 5.
    E. P. Wigner, Phys. Rev. 46, 1002 (1934).ADSCrossRefGoogle Scholar
  6. 6.
    Z. Tesanovic and B.I. Halperin, Phys. Rev. B 36:4888 (1987).ADSCrossRefGoogle Scholar
  7. 7.
    B.I. Halperin, Jap. Journal of Applied Physics 26 Suppl. 26-3:1913 (1987).MathSciNetGoogle Scholar
  8. 8.
    “The Quantum Hall Effect,” R. E. Prange and S. M. Girvin, ed., (Springer Verlag, 1987).Google Scholar
  9. 9.
    H. Hasegawa, M. Robnik and G. Wunner, Prog. Theor. Phys. Suppl 98:98 (1989).MathSciNetCrossRefGoogle Scholar
  10. 10.
    E. H. Lieb, J. P. Solovej, and J. Yngvason, Phys. Rev. Lett. 69:749 (1992).ADSCrossRefGoogle Scholar
  11. 11.
    M. Rasolt and F. Perrot, Phys. Rev. Lett. 69:2563 (1992).ADSCrossRefGoogle Scholar
  12. 12.
    G. Vignale and M. Rasolt, Phys. Rev. Lett 59:2360 (1987).ADSCrossRefGoogle Scholar
  13. G. Vignale and M. Rasolt, Phys. Rev. B 37:10685 (1988).ADSCrossRefGoogle Scholar
  14. G. Vignale, M. Rasolt, and D. J. W. Geldart, Adv. Quantum Physics 21:235 (1990).Google Scholar
  15. M. Rasolt and G. Vignale, Phys. Rev. Lett. 65:1498 (1990).ADSCrossRefGoogle Scholar
  16. 13.
    P. Hohenberg and W. Kohn, Phys. Rev. 136:B846 (1964).Google Scholar
  17. W. Kohn and L. J. Sham, Phys. Rev. 144:A1133 (1965).MathSciNetCrossRefGoogle Scholar
  18. 14.
    O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13:4274 (1976).ADSCrossRefGoogle Scholar
  19. For a review, see R. M. Dreizler and E. K. U. Gross, “Density Functional Theory”, Springer-Verlag, Berlin (1990).MATHCrossRefGoogle Scholar
  20. 15.
    G. Diener, J. Phys. C 3:9417 (1991).Google Scholar
  21. 16.
    For a pedagogical review, see L. Spruch, Rev. Mod. Phys. 63:151 (1991).ADSCrossRefGoogle Scholar
  22. 17.
    G. Vignale and P. Skudlarski, Phys. Rev. B 46:10232 (1992).ADSCrossRefGoogle Scholar
  23. 18.
    A. H. MacDonald and S. M. Girvin, Phys. Rev. B 38:6295 (1988).ADSCrossRefGoogle Scholar
  24. 19.
    K. Maki and X. Zotos, Phys. Rev. B 28:4349 (1983).ADSCrossRefGoogle Scholar
  25. 20.
    M. Shayegan et al., Appl. Phys. Lett. 53:791 (1988).ADSCrossRefGoogle Scholar
  26. E. G. Gwinn et al., Phys. Rev. B 39:6260 (1989).ADSCrossRefGoogle Scholar
  27. A. J. Rimberg and R. M. Westervelt, Phys. Rev. B 40:3970 (1989).ADSCrossRefGoogle Scholar
  28. 21.
    P. Streda, Journal of Physics C: Solid State Physics 15:L717 (1982).ADSCrossRefGoogle Scholar
  29. P. Streda, Journal of Physics C: Solid State Physics 15:L1299 (1982).ADSCrossRefGoogle Scholar
  30. 22.
    We take this opportunity to point out that the results for γxc were incorrectly plotted in Ref. 15. The present Fig. 1 (lower panel) corrects that error.Google Scholar
  31. 23.
    U. Merkt, Adv. Solid State Physics 30:77 (1990).CrossRefGoogle Scholar
  32. M. A. Kastner, Physics Today 46:24 (1993).ADSCrossRefGoogle Scholar
  33. 24.
    G. Vignale, M. Rasolt, and D. J. W. Geldart, Phys. Rev. B 37:2502 (1988).ADSCrossRefGoogle Scholar
  34. 25.
    L. V. Keldysh and T. A. Onishchenko, JETP Lett 24:59 (1976).ADSGoogle Scholar
  35. N. J. Morgenstern Horing, R. W. Danz and M. L. Glasser, Phys. Rev. A 6:2391 (1972).ADSCrossRefGoogle Scholar
  36. 26.
    P. Skudlarski and G. Vignale, Phys. Rev., B 48:8547 (1993).ADSCrossRefGoogle Scholar
  37. 27.
    D. Levesque, J. J. Weis, and A. H. MacDonald, Phys. Rev. B 30:1056 (1984).ADSCrossRefGoogle Scholar
  38. 28.
    G. Fano and F. Ortolani, Phys. Rev. B 37:8179 (1988).ADSCrossRefGoogle Scholar
  39. 29.
    G. Vignale, unpublished.Google Scholar
  40. 30.
    B. Tanatar and D. M. Ceperley, Phys. Rev. B 39:5005 (1989).ADSCrossRefGoogle Scholar
  41. 31.
    G. Vignale, P. Skudlarski, and M. Rasolt, Phys. Rev. B 45:8494 (1992).ADSCrossRefGoogle Scholar
  42. P. Skudlarski and G. Vignale, Phys. Rev. B 47, 16647 (1993).ADSCrossRefGoogle Scholar
  43. 32.
    T. M. Rice, “Solid State Physics,” F. Seitz, D. Turnbull, and H. Ehrenreich, ed., Academic, New York, (1977), Vol. 32, p. 1.Google Scholar
  44. 33.
    For a review, see Ref. 12.Google Scholar
  45. 34.
    H. W. Jiang et al., Phys. Rev. B 44:8107 (1991).ADSCrossRefGoogle Scholar
  46. V. J. Goldman et al. Phys. Rev. Lett. 65:2189 (1990).ADSCrossRefGoogle Scholar
  47. F. I. B. Williams et al., Phys. Rev. Lett. 66:3285 (1991).ADSCrossRefGoogle Scholar
  48. 35.
    Pui K. Lam and S. M. Girvin, Phys. Rev. B 30:473 (1984).ADSCrossRefGoogle Scholar
  49. K. Esfarjani and S. T. Chui, Phys. Rev. B 42:10758 (1990).ADSCrossRefGoogle Scholar
  50. 36.
    G. Vignale, Phys. Rev. B 47:10105 (1993).ADSCrossRefGoogle Scholar
  51. 37.
    C.S. Wang, D.R. Grempel, and R.E. Prange, Phys. Rev. B 28:4284 (1983).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • G. Vignale
    • 1
  1. 1.University of Missouri-ColumbiaUSA

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