  It is rigorously shown that the two-dimensional Hubbard model with narrow
bands (including next nearest-neighbor hopping, etc.) does not exhibit $d_{x^2
-y^2}$-wave pairing long-range order at any nonzero temperature. This kind of
pairing long-range order will also be excluded at zero temperature if an
excited energy gap opens in the charge excitation spectrum of the system. These
results hold true for both repulsive and attractive Coulomb interactions and
for any electron fillings, and are consistent with quantum Monto Carlo
calculations.

Strongly Correlated Electrons;Superconductivity