Abstract
The Hubbard model describes a lattice system of quantum particles with local (on-site) interactions. Its free energy is analytic when βt is small, or βt 2/U is small; here, β is the inverse temperature, U the on-site repulsion, and t the hopping coefficient. For more general models with Hamiltonian H=V+T where V involves local terms only, the free energy is analytic when β ‖T‖ is small, irrespective of V. There exists a unique Gibbs state showing exponential decay of spatial correlations. These properties are rigorously established in this paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
C. Borgs, R. Kotecký, and D. Ueltschi, Incompressible phase in lattice systems of interacting bosons, unpublished, available at http://dpwww.epfl.ch/instituts/ipt/publications.html (1997).
K. A. Chao, J. Spalek, and A. L. Oleś, Canonical perturbation expansion of the Hubbard model, Phys. Rev. B 18:3453–3464 (1978).
N. Datta, R. Fernaádez, and J. Fröhlich, Effective Hamiltonians and phase diagrams for tight-binding models, preprint, math-ph/9809007 (1998).
N. Datta, R. Fernández, J. Fröhlich, and L. Rey-Bellet, Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy, Helv. Phys. Acta 69:752–820 (1996).
R. L. Dobrushin, Estimates of semiinvariants for the Ising model at low temperatures, preprint ESI 125, available at http://esi.ac.at (1994).
F.J. Dyson, E. H. Lieb, and B. Simon, Phase transitions in quantum spin systems with isotropic and nonisotropic interactions, J. Stat. Phys. 18:335–383 (1978).
M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Boson localization and the superfluid-insulator transition, Phys. Rev. B 40:546–570 (1989).
Ch. Gruber and N. Macris, The Falicov-Kimball model: A review of exact results and extensions, Helv. Phys. Acta 69:850–907 (1996).
K. Huang, Statistical Mechanics, 2nd ed. (John Wiley & Sons, 1987).
J. Hubbard, Electron correlations in narrow energy bands, Proc. Roy. Soc. London A 276:238–257 (1963); II. The degenerate case 277:237–259 (1964); III. An improved solution 281:401–419 (1964).
T. Kennedy and E. H. Lieb, An itinerant electron model with crystalline or magnetic long range order, Physica A 138:320–358 (1986).
D. J. Klein and W. A. Seitz, Perturbation expansion of the linear Hubbard model, Phys. Rev. B 8:2236–2247 (1973).
R. Kotecky_ and D. Preiss, Cluster expansion for abstract polymer models, Commun. Math. Phys. 103:491–498 (1986).
R. Kotecký and D. Ueltschi, Effective interactions due to quantum fluctuations, preprint, available at http://dpwww.epfi.ch;/instituts/ipt/publications.html, to appear in Commun. Math. Phys. (1999).
J. L. Lebowitz and N. Macris, Lang range order in the Falicov-Kimball model: Extension of Kennedy-Lieb theorem, Rev. Math. Phys. 6:927–946 (1994).
E. H. Lieb, The Hubbard model: Some rigorous results and open problems, in Advances in Dynamical Systems and Quantum Physics (World Scientific, 1993).
A. H. MacDonald, S. M. Girvin, and D. Yoshioka, t/U expansion for the Hubbard model, Phys. Rev. B 37:9753–9756 (1988).
A. Messager and S. Miracle-Solè, Low temperature states in the Falicov-Kimball model, Rev. Math. Phys. 8:271–299 (1996).
Y. M. Park and H. J. Yoo, Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems, J. Stat. Phys. 80:223–271 (1995).
O. Penrose and L. Onsager, Bose-Einstein condensation and liquid Helium, Phys. Rev. 104:576–584 (1956).
C. N. Yang, Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors, Rev. Mod. Phys. 34:694–704 (1962).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ueltschi, D. Analyticity in Hubbard Models. Journal of Statistical Physics 95, 693–717 (1999). https://doi.org/10.1023/A:1004599410952
Issue Date:
DOI: https://doi.org/10.1023/A:1004599410952