Ferromagnetism in Correlated Electron Systems: A New Class of Rigorous Criteria

  • Rainer Strack
  • Dieter Vollhardt
Part of the NATO ASI Series book series (NSSB, volume 343)


Even after several decades of theoretical work the conditions for the occurence of ferromagnetism in itinerant electron systems, e. g. in the transition metals, are still not well understood. In particular, the simplest lattice model for interacting electrons, the Hubbard model (Gutzwiller, 1963; Hubbard, 1963; Kanamori, 1963), which was originally introduced to clarify precisely this problem, did not provide the hoped-for answer. We now know that the single-band Hubbard model is a generic model for the description of a correlation-induced metal-insulator transition, as well as for the formation of antiferromagnetic order, but not for ferromagnetism. Apparently the on-site interaction, which is totally independent of any lattice properties, does not easily provide a mechanism for the generation of ferromagnetism. In the Hubbard model the lattice structure enters only via the kinetic energy due to nearest-neighbor hopping. It is therefore perhaps not surprising that the rigorous proofs of the stability of ferromagnetism in this model by Nagaoka (1966), Lieb (1989), Mielke (1991, 1992) and Tasaki (1992) apply under conditions which are more specific with regard to the lattice structure than the values of the interaction. Indeed, ferromagnetism was proved to be stable either at U = ∞ (in the case of a single hole moving on certain lattices with loops (Nagaoka, 1966)), or else for all U > 0 (namely in the case of asymmetric bipartite lattices in arbitrary dimensions d > 1 at half filling (Lieb, 1989), or for special (“decorated”) lattices where the single-electron ground state has bulk degeneracy, at sufficiently large filling (Mielke, 1991, 1992; Tasaki, 1992)). For details we refer to the recent reviews by Lieb (1993) and Mielke and Tasaki (1993). Investigations of the stability of the Nagaoka state have recently led to increasingly refined bounds for the critical hole density (Hanisch and Müller-Hartmann, 1993). Nevertheless there still does not exist a rigorous proof of the stability of ferromagnetism in the Hubbard model for conventional lattices (e. g. hypercubic, bcc, fcc) and thermodynamically relevant band fillings n ≾ 1.


Hubbard Model Near Neighbor Ferromagnetic State Near Neighbor Interaction Half Filling 
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. Blume, M., Emery, V.J., and Griffiths, R. B., 1971, Phys. Rev. A 4, 1071.ADSGoogle Scholar
  2. Brandt, U., and Giesekus, A., 1992, Phys. Rev. Lett. 68, 2648.ADSCrossRefGoogle Scholar
  3. Campbell, D.K., Gammel, J.T., and Loh, Jr., E.Y., 1988, Phys. Rev. B38, 12043.ADSGoogle Scholar
  4. Campbell, D.K., Gammel, J.T., and Loh, Jr., E.Y.,1990, Phys. Rev. B42, 475.ADSGoogle Scholar
  5. Castellani, C., Di Castro, C., Feinberg, D., and Ranninger, J., 1979, Phys. Rev. Lett. 43, 1959.ADSCrossRefGoogle Scholar
  6. Castellani, C., Di Castro, C., and Grilli, M., 1994, preprint.Google Scholar
  7. van Dongen, P.G.J., and Janiš, V., 1994, preprint RWTH/ITP-C 3/94.Google Scholar
  8. Essler, F.H.L., Korepin, V.E., and Schoutens, K., 1992, Phys. Rev. Lett. 68, 2960.MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. Essler, F.H.L., Korepin, V.E., and Schoutens, K., 1993, Phys. Rev. Lett. 70, 73.ADSCrossRefGoogle Scholar
  10. Gutzwiller, M.C., 1963, Phys. Rev. Lett. 10, 159.ADSCrossRefGoogle Scholar
  11. Hanisch, T., and Müller-Hartmann, E., 1993, Ann. Phys. (Leipzig) 2, 381.ADSGoogle Scholar
  12. Harris, A. B., and Lange, R.V., 1967, Phys. Rev. 157, 295.ADSCrossRefGoogle Scholar
  13. Hirsch, J.E., 1989a, Phys. Rev. B40, 2354.ADSGoogle Scholar
  14. Hirsch, J.E., 1989b, Phys. Rev. B40, 9061.ADSGoogle Scholar
  15. Hirsch, J.E., 1990a, J. Appl. Phys. 67, 4549.ADSCrossRefGoogle Scholar
  16. Hirsch, J.E., 1990b, Physica B163, 291.ADSGoogle Scholar
  17. Hirsch, J.E., 1991, Phys. Rev. B43, 705.ADSGoogle Scholar
  18. Hubbard, J., 1963, Proc. R. Soc. London, Ser. A 276, 238.ADSCrossRefGoogle Scholar
  19. Kanamori, J., 1963, Prog. Theor. Phys. 30, 275.ADSzbMATHCrossRefGoogle Scholar
  20. Lieb, E. H., 1989, Phys. Rev. Lett. 62, 1201.MathSciNetADSCrossRefGoogle Scholar
  21. Lieb, E. H., 1993, in Proc. of the Conference “Advances in Dynamical Systems and Quantum Physics, (World Scientific, Singapore, in press).Google Scholar
  22. Lieb, E.H., and Mattis, D.C., 1962, Phys. Rev. 125, 164.ADSzbMATHCrossRefGoogle Scholar
  23. Mielke, A., 1991, J. Phys. A: Math. Gen. 24, L73.MathSciNetADSCrossRefGoogle Scholar
  24. Mielke, A., 1992, J. Phys. A: Math. Gen. 25, 4335.MathSciNetADSCrossRefGoogle Scholar
  25. Mielke, A., and Tasaki, H., 1993, Commun. Math. Phys. 158, 341.MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. Nagaoka, Y., 1966, Phys. Rev. 147, 392.ADSCrossRefGoogle Scholar
  27. Ovchinnikov, A. A., 1993, Mod. Phys. Lett. B7, 1397.MathSciNetADSGoogle Scholar
  28. Putikka, W. O., Luchini, M. U., and Ogata, M., 1993, Phys. Rev. Lett. 69, 2288.ADSCrossRefGoogle Scholar
  29. Strack, R., 1993, Phys. Rev. Lett. 70, 833.ADSCrossRefGoogle Scholar
  30. Strack, R., and Vollhardt, D., 1993, Phys. Rev. Lett. 70, 2637.ADSCrossRefGoogle Scholar
  31. Strack, R., and Vollhardt, D., 1994a, Physica B, in press.Google Scholar
  32. Strack, R., and Vollhardt, D., 1994b, Phys. Rev. Lett. 72 (May 1994, in press).Google Scholar
  33. Strack, R., and Vollhardt, D., 1994c, in preparation.Google Scholar
  34. Tang, S., and Hirsch, J.E., 1990, Phys. Rev. B42, 771.ADSGoogle Scholar
  35. Tasaki, H., 1992, Phys. Rev. Lett. 69, 1608.MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Rainer Strack
    • 1
  • Dieter Vollhardt
    • 1
  1. 1.Institut für Theoretische Physik CTechnische Hochschule AachenAachenGermany

Personalised recommendations