Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Failure of saturated ferromagnetism for the Hubbard model with two holes


We consider the Hubbard model on a finite set of sites with nonpositive hopping matrix elements and infinitely strong on-site repulsion. Nagaoka's theorem states that in this model the relative ground state in the sector with one unoccupied site is maximally ferromagnetic. We show that this phenomenon is a consequence of a combinatorial coincidence valid in the one-hole regime only. In the case of more than one hole there is no reason to expect maximally ferromagnetic ground states. We prove this claim for the case of two holes for models defined on a class of graphs which contains all tori that are not too small.

This is a preview of subscription content, log in to check access.


  1. 1.

    LiebE. H. and MattisD. C., Phys. Rev. 125, 164–172 (1962).

  2. 2.

    LiebE. H., in Phase Transitions, Proceedings of the Fourteenth Solvay Conference. Wiley Interscience, New York, 1971.

  3. 3.

    LiebE. H., Phys. Rev. Lett. 62, 1201–1204 (1989).

  4. 4.

    NagaokaY., Phys. Rev. 147, 392–405 (1966).

  5. 5.

    TasakiH., Phys. Rev. B 40, 9192–9193 (1989).

  6. 6.

    ThoulessD. J., Proc. Phys. Soc. London. 86, 893–904 (1965).

  7. 7.

    DoucotB. and WenX. G., Phys. Rev. B 40, 2719–2722 (1989).

  8. 8.

    Sütő, A., to be published in Comm. Math. Phys. (1991).

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Tóth, B. Failure of saturated ferromagnetism for the Hubbard model with two holes. Lett Math Phys 22, 321–333 (1991).

Download citation

AMS subject classifications (1991)

  • 81Q99
  • 81V70
  • 82B10