WEAK-COUPLING TREATMENT OF THE HUBBARD-MODEL IN ONE-DIMENSION, 2-DIMENSION AND 3-DIMENSION

We study the Hubbard model within the second order U-perturbation treatment around the non-magnetic Hartee-Fock solution. When starting from the standard expression, the explicit calculations are very cumbersome, as a 2d-fold momentum integration is required to obtain the selfenergy contribution in second order in U for a d-dimensional system. We show that these computational efforts are strongly reduced when starting from the limit of infinite dimension, d = infinity, and taking into account the effects of the finite dimension d by means of a 1/d-expansion. This 1/d-expansion converges rapidly for d = 3 and d = 2, and it converges (but slowly) even for d = 1. The frequency (E) and momentum (k) dependence of the selfenergy has been calculated. For d = 1 we obtain a very strong k-dependence, and - for zero temperature - the selfenergy imaginary part vanishes for most k-values within an interval around the Fermi energy, i.e. Im SIGMA(k) (E + i0) approximately \E\0, but it vanishes linearly at the Fermi "surface", i.e. Im SIGMA(kF) (E + i0) approximately \E\, in accordance with the Luttinger theorem for a one-dimensional system. For d = 2 the k-dependence is less strong, and for most k-values the selfenergy imaginary part vanishes quadratic with the energy, i.e. Im SIGMA(k) (E + i0) approximately E2 around the Fermi energy, i.e. the Luttinger theorem is fulfilled and one has a normal Fermi liquid behaviour within this approach. But for the half-filled Hubbard model on a quadratic lattice one gets - similar as in one dimension - a linear vanishing, if k is on the Fermi surface, i.e. Im SIGMA(kF) (E + i0) approximately \E\; therefore, in this case the 2-dimensional Hubbard model has the properties of a "marginal Fermi liquid" within the second order U-perturbation treatment. For d = 3 the k-dependence is very weak and nearly negligible, i.e. d = 3 is already very near to d = infinity, and one always gets a normal Fermi liquid behaviour. The spectral functions are also calculated; for sufficiently large U one obtains satellite peaks separated by U and, in addition, around the chemical potential a quasiparticle peak with d-dimensional van-Hove singularities.