Crossover from BCS to Bose-Einstein in Hubbard Model

We generalize the standard attractive Hubbard, having an on-site pair creation operator Q$\text{}_{i}$ = a$\text{}_{i↑}$a$\text{}_{i↓}$, to one with n components Q$\text{}_{iα}$ = a$\text{}_{iα↑}$a$\text{}_{iα↓}$, α=1,2,...,n. In the limit n → ∞ we obtain the Ginzburg-Landau functional. On this basis we explore the crossover from weak (BCS) to strong coupling (Bose-Einstein condensation) superconductivity. The associated self-consistent equations for the Ginzburg-Landau parameters are similar to those of the T-matrix approach. The evolution of the band structure with increasing interaction strength is studied and correlated with the behavior of the pair propagator and the transition temperature. We find that the pairing interaction creates a new band which moves downwards in energy as the interaction strength increases and separates into a lower Hubbard band when the interaction strength becomes comparable to the band width. In the strong coupling regime, a third band with small spectral weight is also found in between the lower and upper Hubbard bands