It is proved explicitly that the strong on-site attractive interaction acts to form \ensuremath{\eta} pairings which make full use of the phase coherence of Bloch electrons. The U(1) gauge symmetry breaks spontaneously in the formation, which is accompanied by the creation of \ensuremath{\eta}-paired boson, ``\ensuremath{\eta} pairon.'' The condensation of \ensuremath{\eta} pairons takes place at a finite temperature to make up the ground-state condensate which has off-diagonal long-range order. The resulting ground state is a singlet and the energy spectrum has a gap. This type of superconductivity based on \ensuremath{\eta} pairing exhibits the same peculiar properties as observed in high-Tc oxide superconductors.
Using the Dyson equation of motion for both the diagonal one-particle Green function, G(k,ω) and off--diagonal Green function, F(k,ω), at the level of the Hubbard--I decoupling scheme, we have found that they have four poles symmetric in pairs, justifying a more elaborated calculation done by the Zürich group by means of the T-Matrix approach (Pedersen et al, Z. Physik B {\bf 103}, 21 (1997)) and the moment approach of Nolting (Z. Physik {\bf 255}, 25 (1972)). We find that the energy spectra and the weights of G(k,ω) and F(k,ω) have to be calculated self-consistently. G(k,ω) satisfies the first two moments while F(k,ω) the first sum rule. Our {\it order parameter} α(T) is given by 1/Ns∑kε(k)Δ(k). Due to the fact that we have a purely local attractive interaction Δ(k) can be of {\it any} s--type wave. However, for a {\it pure s--wave}, for which α(T)=0, we go back to the mean--field BCS results, with a renormalized chemical potential. In this case, the off--diagonal Green function, F(k,ω), satisfies the first two off--diagonal sum rules. We explicitly state the range of validity of our approximation.
The ground-state properties of the Hubbard model with attractive local pairing interactions are investigated in the framework of lattice density-functional theory. A remarkable correlation is revealed between the interaction-energy functional $W[\mathbit{\ensuremath{\eta}}]$ corresponding to the Bloch-state occupation-number distribution ${\ensuremath{\eta}}_{\ensuremath{\kappa}\ensuremath{\sigma}}$ and the entropy $S[\mathbit{\ensuremath{\eta}}]$ of a system of noninteracting fermions having the same ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$. The relation between $W[\mathbit{\ensuremath{\eta}}]$ and $S[\mathbit{\ensuremath{\eta}}]$ is shown to be approximately linear for a wide range of ground-state representable occupation-number distributions ${\ensuremath{\eta}}_{\mathbit{k}\ensuremath{\sigma}}$. Taking advantage of this statistical analogy, a simple explicit ansatz for $W[\mathbit{\ensuremath{\eta}}]$ of the attractive Hubbard model is proposed, which can be applied to arbitrary periodic systems. The accuracy of this approximation is demonstrated by calculating the main ground-state properties of the model on several 1D and 2D bipartite and nonbipartite lattices and by comparing the results with exact diagonalizations.