Lm. Falicov et Cr. Proetto, TEST OF THE BARDEEN-COOPER-SCHRIEFFER AND THE RESONANT-VALENCE-BOND WAVE-FUNCTIONS AS SOLUTIONS OF THE HUBBARD-MODEL - A SMALL-CLUSTER CALCULATION, Physical review. B, Condensed matter, 47(21), 1993, pp. 14407-14412
With the purpose of assessing the validity of commonly used approximat
ions, best variational solutions are calculated for a four-site (tetra
hedral-cluster) Hubbard model with periodic boundary conditions and co
mpared with exact results. In particular, the optimum number-conservin
g Bardeen-Cooper-Schrieffer-type variational solution is obtained for
two, four, and six electrons, and the optimal resonating-valence-bond
state for four electrons. For an attractive interaction U, the exact g
round state of the system is obtained in all four cases. If the intera
ction U is repulsive, the exact ground state is obtained only when the
cluster contains two and six electrons. For four electrons the BCS ap
proximation is only satisfactory for relatively small U. For any posit
ive U the resonating-valence-bond state on the other hand, gives very
good (albeit not exact) energies for the ground state.