Journal of Statistical Physics

, Volume 116, Issue 1–4, pp 681–697 | Cite as

Segregation in the Asymmetric Hubbard Model

  • Daniel Ueltschi


We study the “asymmetric” Hubbard model, where hoppings of electrons depend on their spin. For strong interactions and sufficiently asymmetric hoppings, it is proved that the ground state displays phase separation away from half-filling. This extends a recent result obtained with Freericks and Lieb for the Falicov–Kimball model. It is based on estimates for the sum of lowest eigenvalues of the discrete Laplacian in arbitrary domains.

Hubbard model Falicov–Kimball model phase separation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    U. Brandt and R. Schmidt, Exact results for the distribution of the f-level ground state occupation in the spinless Falicov-Kimball model, Z. Phys. B 63:45–53 (1986).Google Scholar
  2. 2.
    N. Datta, R. Fernández, and J. Fröhlich, Effective Hamiltonians and phase diagrams for tight-binding models, J. Stat. Phys. 96:545–611 (1999).Google Scholar
  3. 3.
    N. Datta, R. Fernández, J. Fröhlich, and L. Rey-Bellet, Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansion ans stability in systems with infinite degeneracy, Helv. Phys. Acta 69:752–820 (1996).Google Scholar
  4. 4.
    N. Datta, A. Messager, and B. Nachtergaele, Rigidity of interfaces in the Falicov-Kimball model, J. Stat. Phys. 99:461–555 (2000).Google Scholar
  5. 5.
    L. M. Falicov and J. C. Kimball, Simple model for semiconductor-metal transitions: SmB6and transition-metal oxides, Phys. Rev. Lett. 22:997–999 (1969).Google Scholar
  6. 6.
    J. K. Freericks, E. H. Lieb, and D. Ueltschi, Segregation in the Falicov-Kimball model, Comm. Math. Phys. 227:243–279 (2002).Google Scholar
  7. 7.
    J. K. Freericks, E. H. Lieb, and D. Ueltschi, Phase separation due to quantum mechanical correlations, Phys. Rev. Lett. 88:106401 (2002).Google Scholar
  8. 8.
    J. K. Freericks and V. Zlatić, Exact dynamical mean-field theory of the Falicov-Kimball model, Rev. Mod. Phys. 75:1333–1382 (2003).Google Scholar
  9. 9.
    P. Goldbaum, Lower bound for the segregation energy in the Falicov-Kimball model, Physica A 36:2227–2234 (2003).Google Scholar
  10. 10.
    C. Gruber, J. Jędrzejewski, and P. Lemberger, Ground states of the spinless Falicov-Kimball model II, J. Stat. Phys. 66:913–938 (1992).Google Scholar
  11. 11.
    C. Gruber and N. Macris, The Falicov-Kimball model: A review of exact results and extensions, Helv. Phys. Acta 69:850–907 (1996).Google Scholar
  12. 12.
    K. Haller and T. Kennedy, Periodic ground states in the neutral Falicov-Kimball model in two dimensions, J. Stat. Phys. 102:115–134 (2001).Google Scholar
  13. 13.
    J. Hubbard, Electron correlations in narrow energy bands, Proc. Roy. Soc. London A 276:238–257 (1963).Google Scholar
  14. 14.
    J. Jędrzejewski and R. Lemański, Falicov-Kimball models of collective phenomena in solids (a concise guide), Acta Phys. Pol. B 32:3243–3252 (2001).Google Scholar
  15. 15.
    T. Kennedy, Some rigorous results on the ground states of the Falicov-Kimball model, Rev. Math. Phys. 6:901–925 (1994).Google Scholar
  16. 16.
    T. Kennedy, Phase separation in the neutral Falicov-Kimball model, J. Stat. Phys. 91:829–843 (1998).Google Scholar
  17. 17.
    T. Kennedy and E. H. Lieb, An itinerant electron model with crystalline or magnetic long range order, Physica A 138:320–358 (1986).Google Scholar
  18. 18.
    R. Kotecký and D. Ueltschi, Effective interactions due to quantum fluctuations, Comm. Math. Phys. 206:289–335 (1999).Google Scholar
  19. 19.
    J. L. Lebowitz and N. Macris, Long range order in the Falicov-Kimball model: Extension of Kennedy-Lieb theorem, Rev. Math. Phys. 6:927–946 (1994).Google Scholar
  20. 20.
    P. Li and S.-T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88:309–318 (1983).Google Scholar
  21. 21.
    E. H. Lieb, Two theorems on the Hubbard model, Phys. Rev. Lett. 62:1201–1204. Errata 62:1927 (1989).Google Scholar
  22. 22.
    E. H. Lieb, The Hubbard model: Some rigorous results and open problems, in XIth International Congress of Mathematical Physics ( Paris, 1994 )(Internat. Press, 1995), pp. 392–412.Google Scholar
  23. 23.
    E. H. Lieb and M. Loss, Analysis, 2nd Ed. (Amer. Math. Soc., 2001).Google Scholar
  24. 24.
    A. D. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proc. Amer. Math. Soc. 131:631–636 (2002).Google Scholar
  25. 25.
    A. Messager and S. Miracle-Solé, Low temperature states in the Falicov-Kimball model, Rev. Math. Phys. 8:271–299 (1996).Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • Daniel Ueltschi
    • 1
  1. 1.Department of MathematicsUniversity of ArizonaTucson

Personalised recommendations