Pairing mechanism in the doped Hubbard antiferromagnet: the 4 x 4 model asa test case

We introduce a local formalism, in terms of eigenstates of number operators , having well defined point symmetry, to solve the Hubbard model at weak co upling on a N x N square lattice (for even N). The key concept is that of W = 0 states, that are the many-body eigenstates of the kinetic energy with vanishing Hubbard repulsion. At half filling, the wave function demonstrate s an antiferromagnetic order, a lattice step translation being equivalent t o a spin flip. Further, we state a general theorem which allows to find all the MI = 0 pairs (two-body W = 0 singlet states). We show that, in special cases, this assigns the ground state symmetries at least in the weak coupl ing regime. The AT = 4 case is discussed in detail. To study the doped half filled system, we enhance the group theory analysis of the 4 x 4 Hubbard m odel introducing an Optimal Group which explains all the degeneracies in th e one-body and many-body spectra. We use the Optimal Group to predict the p ossible ground state symmetries of the 4 x 4 doped antiferromagnet by means of our general theorem and the results are in agreement with exact diagona lization data. Then we create W = 0 electron pairs over the antiferromagnet ic state. We show analitycally that the effective interaction between the e lectrons of the pairs is attractive and forms bound states. Computing the c orresponding binding energy we are able to definitely predict the exact gro und state symmetry.