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The two-dimensional Hubbard model is studied for small values of theinteraction strength (U of the order of the hopping amplitude t), using avariational ansatz well suited for this regime. The wave function, a refinedGutzwiller ansatz, has a BCS mean-field state with d-wave symmetry as itsreference state. Superconducting order is found for densities n <1 (but not forn=1). This resolves a discrepancy between results obtained with the functionalrenormalization group, which do predict superconducting order for small valuesof U, and numerical simulations, which did not find superconductivity for U<4t.Both the gap parameter and the order parameter have a dome-like shape as afunction of n with a maximum for n about 0.8. Expectation values for theenergy, the particle number and the superconducting order parameter arecalculated using a linked-cluster expansion up to second order in U. In thisway large systems (millions of sites) can be readily treated and well convergedresults are obtained. A big size is indeed required to see that the gap becomesvery small at half filling and probably tends to zero in the thermodynamiclimit, whereas away from half filling a finite asymptotic limit is reached. Fora lattice of a given size the order parameter becomes finite only above aminimal coupling strength U_c. This threshold value decreases steadily withincreasing system size, which indicates that superconductivity exists forarbitrarily small U for an infinite system. For moderately large systems thesize dependence is quite irregular, due to variations in level spacings at theFermi energy. The fluctuations die out if the gap parameter spans several levelspacings.

Superconductivity from repulsion: Variational results for the 2D Hubbard model in the limit of weak interaction