Antiferromagnetism of the two-dimensional Hubbard model at half-filling: the analytic ground state for weak coupling

We introduce a local formalism to deal with the Hubbard model on an N x N s quare lattice (for even N) in terms of eigenstates of number operators, hav ing well defined point symmetry. For U --> 0, the low-lying shells of the k inetic energy are filled in the ground state. At half-filling, using the 2N - 2 one-body states of the partially occupied shell S-hf, we build a set o f ((2N-2)(N-1)) degenerate unperturbed ground states with S-z = 0 which are then resolved by the Hubbard interaction W = U Sigma (r) n(r up arrow)n(r down arrow). We study the many-body eigenstates in S-hf Of the kinetic ener gy with vanishing eigenvalue of the Hubbard repulsion (W = 0 states). In th e S-z = 0 sector, this is an N-times-degenerate multiplet. From the singlet component one obtains the ground state of the Hubbard model for U = 0(+), which is unique, in agreement with a theorem of Lieb. The wave function dem onstrates an antiferromagnetic order, a lattice step translation being equi valent to a spin flip. We show that the total momentum vanishes, while the point symmetry is s or d for even or odd N/2, respectively.