Test of the Bardeen-Cooper-Schrieffer and the resonant-valence-bond wave functions as solutions of the Hubbard model : a small-cluster calculation

With the purpose of assessing the validity of commonly used approximations, best variational solutions are calculated for a four-site (tetrahedral-cluster) Hubbard model with periodic boundary conditions and compared with exact results. In particular, the optimum number-conserving Bardeen-Cooper-Schrieffer-type variational solution is obtained for two, four, and six electrons, and the optimal resonanting-valence-bond state for four electrons. For an attractive interaction [ital U], the exact ground state of the system is obtained in all four cases. If the interaction [ital U] is repulsive, the exact ground state is obtained only when the cluster contains two and six electrons. For four electrons the BCS approximation is only satisfactory for relatively small [ital U]. For any positive [ital U] the resonating-valence-bond state on the other hand, gives very good (albeit not exact) energies for the ground state.