Exact mean-field Hamiltonian for fermionic lattice models in high dimensions

P. G. J. van Dongen; D. Vollhardt

doi:10.1103/PhysRevLett.65.1663

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Exact mean-field Hamiltonian for fermionic lattice models in high dimensions

P. G. J. van Dongen and D. Vollhardt

Institut für Theoretische Physik C, Technische Hochschule Aachen, D-5100 Aachen, Federal Republic of Germany

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Phys. Rev. Lett. 65, 1663 – Published 24 September, 1990

DOI: https://doi.org/10.1103/PhysRevLett.65.1663

Abstract

We show that even for fermionic lattice models with on-site interaction a mean-field Hamiltonian can be constructed, which–in analogy with spin-lattice models–becomes exact in the limit of high spatial dimensions d. Here the mean fields are fermion operators, rather than numbers. We use this method to obtain the exact solution of a simplified Hubbard model on a Bethe lattice for all temperatures. The order parameter is found to have the conventional mean-field form, but exhibits unusually rich behavior as a function of the interaction.

References (18)

R. Brout, Phys. Rev. 118, 1009 (1960); C. J. Thompson, Commun. Math. Phys. 36, 255 (1974).
P. A. Pearce and C. J. Thompson, Commun. Math. Phys. 58, 131 (1978); T. Kennedy, E. H. Lieb and B. S. Shastry, Phys. Rev. Lett. 61, 2582 (1988); J. Stat. Phys. 53, 1019 (1988).
M. C. Gutzwiller, Phys. Rev. Lett. 10, 59 (1963); J. Hubbard, Proc. Roy. Soc. London A 276, 238 (1963); J. Kanamori, Prog. Theor. Phys. 30, 275 (1963).
J. Hubbard, Proc. Roy. Soc. London A 281, 401 (1964); M. C. Gutzwiller, Phys. Rev. 137, A1726 (1965).
L. M. Falicov and J. C. Kimball, Phys. Rev. Lett. 22, 997 (1969); in the rare-earth community the model proposed therein is referred to as the ``Falicov-Kimball model.''
T. Kennedy and E. H. Lieb, Physica (Amsterdam) 138A, 320 (1986); E. H. Lieb, 140A, 240 (1986).
U. Brandt and R. Schmidt, Z. Phys. B 63, 45 (1986); ibid. 67, 43 (1987).
W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989).
E. Müller-Hartmann, Z. Phys. B 74, 507 (1989); ibid. 76, 211 (1989).
H. Schweitzer and G. Czycholl, Solid State Commun. 69, 171 (1989); ibid. 74, 735 (1990).
U. Brandt and C. Mielsch, Z. Phys. B 75, 365 (1989); ibid. 79, 295 (1990).
For preliminary reviews, see D. Vollhardt, Int. J. Mod. Phys. B 3, 2189 (1989); E. Müller-Hartmann, 3, 2169 (1989).
For a general proof, see P. Lizana, R. Strack, and D. Vollhardt (to be published).
P. G. J. van Dongen (to be published).
The procedure followed here is also of interest for a fundamental reason. Strictly speaking it has never been rigorously proved (Ref. 6) that the low-temperature phase has AB structure for all T < $T_{c}$ . Assuming that this is the case, we find that the resulting expression for $T_{c}$ is identical to the value predicted in Ref. 11 when applied to the Bethe lattice. This effectively proves that our assumption is correct.
E. N. Economou, Green's Functions in Quantum Physics (Springer-Verlag, Berlin, 1979).
P. G. J. van Dongen, J. A. Vergés, and D. Vollhardt (to be published).
Note that this finding is not an artifact of the Bethe lattice structure, since the results qualitatively agree with those for a hypercubic lattice in d = inf (Ref. 11).

[c1] R. Brout, Phys. Rev. 118, 1009 (1960); C. J. Thompson, Commun. Math. Phys. 36, 255 (1974).

[c2] P. A. Pearce and C. J. Thompson, Commun. Math. Phys. 58, 131 (1978); T. Kennedy, E. H. Lieb and B. S. Shastry, Phys. Rev. Lett. 61, 2582 (1988); J. Stat. Phys. 53, 1019 (1988).

[c3] M. C. Gutzwiller, Phys. Rev. Lett. 10, 59 (1963); J. Hubbard, Proc. Roy. Soc. London A 276, 238 (1963); J. Kanamori, Prog. Theor. Phys. 30, 275 (1963).

[c4] J. Hubbard, Proc. Roy. Soc. London A 281, 401 (1964); M. C. Gutzwiller, Phys. Rev. 137, A1726 (1965).

[c5] L. M. Falicov and J. C. Kimball, Phys. Rev. Lett. 22, 997 (1969); in the rare-earth community the model proposed therein is referred to as the ``Falicov-Kimball model.''

[c6] T. Kennedy and E. H. Lieb, Physica (Amsterdam) 138A, 320 (1986); E. H. Lieb, 140A, 240 (1986).

[c7] U. Brandt and R. Schmidt, Z. Phys. B 63, 45 (1986); ibid. 67, 43 (1987).

[c8] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989).

[c9] E. Müller-Hartmann, Z. Phys. B 74, 507 (1989); ibid. 76, 211 (1989).

[c10] H. Schweitzer and G. Czycholl, Solid State Commun. 69, 171 (1989); ibid. 74, 735 (1990).

[c11] U. Brandt and C. Mielsch, Z. Phys. B 75, 365 (1989); ibid. 79, 295 (1990).

[c12] For preliminary reviews, see D. Vollhardt, Int. J. Mod. Phys. B 3, 2189 (1989); E. Müller-Hartmann, 3, 2169 (1989).

[c13] For a general proof, see P. Lizana, R. Strack, and D. Vollhardt (to be published).

[c14] P. G. J. van Dongen (to be published).

[c15] The procedure followed here is also of interest for a fundamental reason. Strictly speaking it has never been rigorously proved (Ref. 6) that the low-temperature phase has AB structure for all T < $T_{c}$ . Assuming that this is the case, we find that the resulting expression for $T_{c}$ is identical to the value predicted in Ref. 11 when applied to the Bethe lattice. This effectively proves that our assumption is correct.

[c16] E. N. Economou, Green's Functions in Quantum Physics (Springer-Verlag, Berlin, 1979).

[c17] P. G. J. van Dongen, J. A. Vergés, and D. Vollhardt (to be published).

[c18] Note that this finding is not an artifact of the Bethe lattice structure, since the results qualitatively agree with those for a hypercubic lattice in d = inf (Ref. 11).

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Exact mean-field Hamiltonian for fermionic lattice models in high dimensions

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