Metal to Insulator Transition in the 2-D Hubbard Model: A Slave-Boson Approach

  • Raymond Frésard
  • Klaus Doll
Part of the NATO ASI Series book series (NSSB, volume 343)


Since the discovery of the High-T c superconductors, [1], the Hubbard model has been the subject of intense investigations following Anderson’s proposal [2] that the model should capture the essential physics of the cuprate superconductors. From the earlier attempts to obtain the magnetic phase diagram on the square lattice (for an overview see the book by Mattis [3]) one can deduce that antiferromagnetic order exists in the vicinity of the half-filled band whereas ferromagnetic ordering might take place in the phase diagram for strong repulsive interaction strength and moderate hole doping of the half-filled band. Obviously antiferromagnetic and ferromagnetic orders compete in this part of the phase diagram. More recent calculations [4] established that the ground state of the Hubbard model on the square lattice shows long-ranged antiferromagnetic ordering with a charge transfer gap. However, the problem of mobile holes in an antiferromagnetic background remains mostly unsolved. Suggestions for a very wide ferromagnetic domain in the phase diagram based on the restricted Hartree-Fock Approximation have been made by several authors [5] on the cubic lattice, and on the square lattice [6–8]. This domain appears for large interaction and moderate hole doping in which case the Hartree-Fock Approximation ceases to be controlled. Within this framework one expects to obtain reliable results for moderate U where the paramagnetic phase is indeed unstable towards an incommensurate spin structure at a critical density n c (U) [9]. The Gutzwiller Approximation (GA) [10–12] has been applied [13], even for large U, yielding results similar to the Hartree-Fock Approximation. However, for large U, a ferromagnetic domain appears only if the density is larger than some critical value. In the Kotliar and Ruckenstein slave boson technique [14] the GA appears as a saddle-point approximation of this field theoretical representation of the Hubbard model. In the latter a metal-insulator transition occurs at half-filling as recently discussed by Lavagna [15]. The contribution of the thermal fluctuations has been calculated [16] and turned out to be incomplete as this representation, even though exact, is not manifestly spin-rotation invariant. Spin-rotation invariant [17] and spin and charge-rotation invariant [18] formulations have been proposed and the first one was used to calculate correlation functions [19] and spin fluctuation contributions to the specific heat [20]. Comparisons of ground state energy with Quantum Monte-Carlo simulations, including antiferromagnetic ordering [21] and spiral states [22], or with exact diagonalisation data [23] have been done and yield excellent agreement, and a magnetic phase diagram has been proposed [24].


Hubbard Model Paramagnetic State Honeycomb Lattice Spin Susceptibility Magnetic Phase Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    See, for example, Proceedings of the International Conference on Materials and Mechanisms of Superconductivity-High Temperature Superconductors II, Physica C 162-164, (1989).Google Scholar
  2. [2]
    Anderson P.W., in “Frontiers and Borderlines in Many-Particle Physics”, Proceedings of the International School of Physics, Enrico Fermi, Varenna 1987, eds. Broglia R.A. and Schrieffer, J.R. (North Holland, Amsterdam 1988), p. 1.Google Scholar
  3. [3]
    Mattis D.C., in “The theory of Magnetism I (Springer Series in Solid State Sciences 17)”, ed. Fulde P. (Springer, Berlin 1981).Google Scholar
  4. [4]
    Liang S., Douçot B. and Anderson P.W., Phys. Rev. Lett. 61 365 (1988)ADSCrossRefGoogle Scholar
  5. Trivedi N. and Ceperley D., Phys. Rev. B 40 2737 (1989).ADSCrossRefGoogle Scholar
  6. [5]
    Penn D.R., Phys. Rev. 142 350 (1966)ADSCrossRefGoogle Scholar
  7. Cyrot M.J., J. Physique 33 125 (1972).CrossRefGoogle Scholar
  8. [6]
    Dzierzawa M., Z. Phys. B-Condensed Matter 86 49 (1992).ADSCrossRefGoogle Scholar
  9. [7]
    John S., Voruganti P. and Goff W., Phys. Rev. B 43 13365 (1991).ADSCrossRefGoogle Scholar
  10. [8]
    Yoshioka D.J., Phys. Soc. Jpn. 58, 1516 (1989)ADSCrossRefGoogle Scholar
  11. Jayaprakash G., Krishnamurthy H.R., and Sarker S., Phys. Rev. B 40, 2610 (1989)ADSCrossRefGoogle Scholar
  12. Kane C.L., Lee P.A., Ng T.K., Chakraborty B. and Read N., Phys. Rev. B 41, 2653 (1990).ADSCrossRefGoogle Scholar
  13. [9]
    Schulz H., Phys. Rev. Lett. 64 1445 (1990).ADSCrossRefGoogle Scholar
  14. [10]
    Gutzwiller M., Phys. Rev. Lett. 10, 159 (1963). Phys. Rev. 134 A, 923 (1964); 137 A, 1726 (1965).ADSCrossRefGoogle Scholar
  15. [11]
    Brinkman W.F. and Rice T.M., Phys. Rev. B 2, 4302 (1970).ADSCrossRefGoogle Scholar
  16. [12]
    Vollhardt D., Rev. Mod. Phys. 56, 99 (1984).ADSCrossRefGoogle Scholar
  17. [13]
    Metzner W. and Vollhardt D., Phys. Rev. Lett. 62, 324 (1989)ADSCrossRefGoogle Scholar
  18. Metzner W., Z. Phys. B-Condensed Matter 77 253 (1989).ADSCrossRefGoogle Scholar
  19. [14]
    Kotliar G. and Ruckenstein A.E., Phys. Rev. Lett. 57, 1362 (1986).MathSciNetADSCrossRefGoogle Scholar
  20. [15]
    Lavagna M., Phys. Rev. B 41, 142 (1990); Helvetica Physica Acta 63, 310 (1990); Int. J. Mod. Phys. B 5, 885 (1991).ADSCrossRefGoogle Scholar
  21. [16]
    Rasul J.W., Li T., J. Phys. C 21, 5119 (1988)ADSCrossRefGoogle Scholar
  22. Li T.C., Rasul J.W., Phys. Rev. B 39, 4630 (1989)ADSCrossRefGoogle Scholar
  23. Rasul J.W., Li T., Beck H., Phys. Rev. B 39, 4191 (1989).ADSCrossRefGoogle Scholar
  24. [17]
    Li T., Wölfle P., and Hirschfeld P.J., Phys. Rev. B 40, 6817 (1989).ADSCrossRefGoogle Scholar
  25. [18]
    Frésard R. and Wölfle P., Int. J. of Mod. Phys. B 6, 685 (1992), Proceedings of the Adriatico Research Conference and Miniworkshop “Strongly Correlated Electrons Systems III”, eds. Baskaran G., Ruckenstein A.E., Tossati E., Yu Lu; Frésard R. and Wölfle P., Erratum of ibid, Int. J. of Mod. Phys. B 6, 3087 (1992).ADSCrossRefGoogle Scholar
  26. [19]
    Li T., Sun Y.S. and Wölfle P., Z. Phys. B-Condensed Matter 82, 369 (1991).ADSCrossRefGoogle Scholar
  27. [20]
    Wölfle P. and Li T., Z. Phys. B-Condensed Matter 78, 45 (1990).ADSCrossRefGoogle Scholar
  28. [21]
    Lilly L., Muramatsu A. and Hanke W., Phys. Rev. Lett. 65, 1379 (1990).ADSCrossRefGoogle Scholar
  29. [22]
    Frésard R., Dzierzawa M. and Wölfle P., Europhys. Lett. 15, 325 (1991).ADSCrossRefGoogle Scholar
  30. [23]
    Frésard R. and Wölfle P., J. Phys. Condensed Matter 4 3625 (1992).ADSCrossRefGoogle Scholar
  31. [24]
    Möller B., Doll K. and Frésard R., J. Phys. Condensed Matter 5 4847 (1993).ADSCrossRefGoogle Scholar
  32. [25]
    Sorella S. and Tossati E., Europhys. Lett. 19, 699 (1992).ADSCrossRefGoogle Scholar
  33. [26]
    Doll K., Dzierzawa M., Frésard R. and Wölfle P., Z. Phys. B-Condensed Matter 90, 297 (1993).ADSCrossRefGoogle Scholar
  34. [27]
    Raimondi R. and Castellani C, Preprint.Google Scholar
  35. [28]
    Zimmermann W., Frésard R. and Wölfle P., Preprint.Google Scholar
  36. [29]
    Moreo A., Phys. Rev. B 48, 3380 (1993).ADSCrossRefGoogle Scholar
  37. [30]
    Tohyama T., Okuda H. and Maekawa S., Physica C 215, 382 (1993).ADSCrossRefGoogle Scholar
  38. [31]
    Chen L. and Tremblay A.-M. S., Preprint.Google Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Raymond Frésard
    • 1
  • Klaus Doll
    • 1
  1. 1.Institut für Theorie der Kondensierten MaterieUniversität KarlsruheKarlsruheDeutschland

Personalised recommendations