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

Abstract

We introduce a local formalism to deal with the Hubbard modelon an N×N square lattice (for even N) in terms ofeigenstates of number operators, having well defined point symmetry.For U→0, the low-lying shells of the kinetic energyare filled in the ground state. At half-filling, using the 2N-2one-body states of the partially occupied shell \U0001d4aehf, webuild a set of degenerate unperturbed groundstates with Sz = 0 which are then resolved by the Hubbardinteraction Ŵ = U∑rn̂r↑n̂r↓. We study the many-body eigenstates in\U0001d4aehf of the kinetic energy with vanishing eigenvalue ofthe Hubbard repulsion (W = 0 states). In the Sz = 0 sector, thisis an N-times-degenerate multiplet. From the singlet componentone obtains the ground state of the Hubbard model for U = 0+,which is unique, in agreement with a theorem of Lieb. The wavefunction demonstrates an antiferromagnetic order, a lattice steptranslation being equivalent to a spin flip. We show that the totalmomentum vanishes, while the point symmetry is s or d for even orodd N/2, respectively.