Abstract
A diagrammatic approach to the evaluation of correlated variational wave functions for strongly interacting fermions is presented. Diagrammatic rules for the calculation of the one-particle density matrix and the Hubbard interaction are derived which are valid for arbitraryd-dimensional lattices. An exact evaluation of expectation values is performed in the limitd=∞. The wellknown Gutzwiller approximation is seen to become the exact result for the expectation value of the Hubbard Hamiltonian in terms of the Gutzwiller wave function ind=∞. An efficient procedure to correct the Gutzwiller approximation in finite dimensions is developed. A detailed discussion of expectation values ind=∞ in terms of explicit antiferromagnetic wave functions is given. Thereby an approximate result for the ground state energy of the Hubbard model, obtained recently within a slave-boson approach, is recovered.
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Gutzwiller, M.C.: Phys. Rev. Lett.10, 159 (1963)
For a review, see Vollhardt, D.: In: Interacting electrons in reduced dimensions. Baeriswyl, D., Campbell, D. (eds.) New York: Plenum Press 1989
Anderson, P.W.: Science235, 1196 (1987)
Gutzwiller, M.C.: Phys. Rev.137A, 1726 (1965)
Ogawa, T., Kanda, K., Matsubara, T.: Prog. Theor. Phys.53, 614 (1975)
Vollhardt, D.: Rev. Mod. Phys.56, 99 (1984)
Takano, F., Uchinami, M.: Prog. Theor. Phys.53, 1267 (1975)
Florencio, J., Chao, K.A.: Phys. Rev. B14, 3121 (1976)
For a review, see H. Shiba in Proceedings of the Workshop on Two-Dimensional Strongly Correlated Electronic Systems, Beijing 1988 New York: Gordon and Breach 1989
Baeriswyl, D., Maki, K.: Phys. Rev. B31, 6633 (1985); Baeriswyl, D., Carmelo, J., Maki, K.: Synthetic Metals21, 271 (1987)
Metzner, W., Vollhardt, D.: Phys. Rev. Lett59, 121 (1987); Phys. Rev. B37, 7382 (1988)
Gebhard, F., Vollhardt, D.: Phys. Rev. Lett.59, 1472 (1987); Phys. Rev. B38, 6911 (1988)
Metzner, W., Vollhardt, D.: Phys. Rev. Lett.62, 324 (1989)
Kotliar, G., Ruckenstein, A.E.: Phys. Rev. Lett.57, 1362 (1986)
Hubbard, J.: Proc. R. Soc. London Ser. A276, 238 (1963); Kanamori, J.: Prog. Theor. Phys.30, 275 (1963)
The graphs used here differ slightly from the graphs described in Ref. 11,. Lines are now directed by an arrow and the spin is indicated by a free spin variable (not by drawing two types of lines) which has to be summed up. In this way one no longer needs to bother about the rather complicated “weight factors” introduced in Ref. 11. Metzner, W., Vollhardt, D.: Phys. Rev. Lett59, 121 (1987); Phys. Rev. B37, 7382 (1988)
See, for example, Negele, J.W., Orland, H.: Quantum manyparticle systems. New York: Addison-Wesley 1988
Note thatS (m)σ (k) is related to the quantityf mσ(k) introduced in Ref. 11,. byS (m)σ (k)=(g 2−1)m f mσ(k). Equation (25) is thus equivalent to (48) in Ref. 11. Metzner, W., Vollhardt, D.: Phys. Rev. Lett59, 121 (1987); Phys. Rev. B37, 7382 (1988)
We assume that |Φ0〉 does not break the point-group symmetry. A generalization is straightforward, however
Dongen, van P.G.J., Gebhard, F., Vollhardt, D.: Z. Phys. B—Condensed Matter76, 199 (1989)
Müller-Hartmann, E.: Z. Phys. B—Condensed Matter74, 507 (1989)
Schweitzer, H., Czycholl, G.: Solid State Commun69, 171 (1989)
Yokoyama, H., Shiba, H.: J. Phys. Soc. Jpn.56, 1490 (1987)
Penn, D.R.: Phys. Rev.142, 350 (1966)
Yokoyama, H., Shiba, H.: J. Phys. Soc. Jpn.56, 3570 (1987); J. Phys. Soc. Jpn.56, 3582 (1987)
Zhang, F.C., Gros, C., Rice, T.M., Shiba, H.: Supercond. Sci. Technol.1, 36 (1988)
I am grateful to F. Gebhard for pointing this out to me
For small densities or for ferromagnetic states the absence of a cutoff in the Gaussian DOS becomes important. However, these cases are not investigated here
For largeU the half-filled Hubbard model becomes equivalent to the Heisenberg model. The ground state of the latter is the Neél state ind=∞; see T. Kennedy, Lieb, E.H., Shastry, B.S.: Phys. Rev. Lett61, 2582 (1988)
Metzner, W.: (unpublished)
Note that we did not allow for ferromagnetism or phase separation. A more complete phase diagram including these possibilities is presently being elaborated by Menge, B., Müller-Hartmann, E.: Private communication
Gebhard, F., Vollhardt, D.: In: Interacting electrons in reduced dimensions. Baeriswyl, D., Campbell, D. (eds.) New York: Plenum Press 1989
Gros, C., Joynt, R., Rice, T.M.: Z. Phys. B—Condensed Matter68, 425 (1987); Gros, C.: Phys. Rev. B38, 931 (1988); Yokoyama, H., Shiba, H.: J. Phys. Soc. Jpn.57, 2482 (1988)
Stollhoff, G., Fulde, P.: Z. Phys. B—Condensed Matter and Quanta26, 257 (1977); Kaplan, T.A., Horsch, P., Fulde, P.: Phys. Rev. Lett.49, 889 (1982)
Fazekas, P.: Preprint (submitted to Phys. Scr.)
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Metzner, W. Variational theory for correlated lattice fermions in high dimensions. Z. Physik B - Condensed Matter 77, 253–266 (1989). https://doi.org/10.1007/BF01313669
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DOI: https://doi.org/10.1007/BF01313669